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+Detecting Pump&Dump Stock Market Manipulation from Online
+Forums
+D. Nam
+D.B. Skillicorn
+School of Computing
+Queen’s University
+Kingston. Canada
+[email protected]
+Abstract
+The intersection of social media, low-cost trading platforms, and naive investors has created an
+ideal situation for information-based market manipulations, especially pump&dumps. Manipulators
+accumulate small-cap stocks, disseminate false information on social media to inflate their price,
+and sell at the peak.
+We collect a dataset of stocks whose price and volume profiles have the
+characteristic shape of a pump&dump, and social media posts for those same stocks that match the
+timing of the initial price rises. From these we build predictive models for pump&dump events based
+on the language used in the social media posts.
+There are multiple difficulties: not every post will cause the intended market reaction, some
+pump&dump events may be triggered by posts in other forums, and there may be accidental con-
+fluences of post timing and market movements. Nevertheless, our best model achieves a prediction
+accuracy of 85% and an F1-score of 62%. Such a tool can provide early warning to investors and
+regulators that a pump&dump may be underway.
+1
+Introduction
+New financial products and technologies have allowed naive investors to easily enter financial mar-
+kets.
+This has increased the risk of manipulation, and detecting and investigating fraudulent
+activities has become much more difficult. Many go undetected [8].
+Social media has created new methods for manipulating markets. A scheme known as Pump
+and Dump (P&D) is one popular mechanism. Fraudsters buy quantities of a stock, disseminate
+false information about it to artificially raise its price, and then sell their purchased shares at the
+higher price. Social media provides a channel for rapid dissemination and a pool of investors with
+little knowledge or experience who may not detect that the information is false.
+Conventional approaches to detecting manipulation look for known patterns, and for anomalous
+activity such as exceeded thresholds for prices and trading volumes. Suspicious activities can be
+detected using sets of rules and triggers that cause notifications of potential manipulation. However,
+those methods struggle in the presence of behaviours that deviate from historical patterns [16].
+Previous work has also focused on detecting manipulations so that regulators can penalise those
+who carry them out. This does little to help investors, either to prevent their being deceived or
+recovering their investments.
+1
+arXiv:2301.11403v1  [cs.SI]  26 Jan 2023
+
+Data-analytic techniques have the potential to detect false information as it being disseminated
+[11, 25]. Natural language analytics can detect the posts in social media that are intended to pump
+particular stocks, providing a real-time warning to potential investors. We investigate how well
+P&D schemes can be detected in posts on social media, by matching the language patterns in the
+posts to the pattern of stock price corresponding to a P&D manipulation.
+A penny stock is a stock that is traded by a small public company for less than $5 per share
+[24].
+Many of these companies are known for their volatility due to their limited coverage by
+analysts and interest from institutional buyers. Because of their low price, retail investors can buy
+large quantities of these stocks without having to invest much money. This, however, makes their
+prices volatile and so creates the potential for large returns on investments; but also leaves them
+vulnerable to manipulation by malicious actors. One study found that 50% of manipulated stocks
+are those with a small market capitalization [1].
+It might be supposed that the connection between a social media post and a P&D event is too
+tenuous to be detected – after all, not every post will have the desired effect, and a P&D might be
+triggered by some less visible social media activity. We show that, at least for penny stocks, the
+connection is reasonably detectable, and we achieve prediction accuracies (that a post is intended
+to cause a P&D event) of 85%, with an F1 score of 67% (± 12 percentage points) from posts alone,
+and 62% (± 3 percentage points) from posts and comments.
+2
+Tools
+Stance detection is a technique to determine the attitude or viewpoint of a text towards a target.
+It aims to detect whether the author of the text is in support of or against a given entity [21].
+Some applications of stance detection have been in political debates, fake news, and social media
+[15, 26, 30].
+Empath is a tool that was developed by Fast et al. [13] for researchers to generate and validate
+new lexical categories on demand. It uses deep learning to establish connections between words
+and phrases used in modern fiction. Given a small set of seed words that represents a category,
+Empath can provide new related terms using its neural embeddings. It also employs the use of
+crowd-sourcing to validate the terms that it considers are related. Along with the ability to create
+new categories, Empath comes with 200 built-in, pre-validated categories for common topics (e.g.,
+neglect, government, social media).
+SHAP (SHapley Additive exPlanation) is a tool that was developed by Lundberg and Lee [22]
+to determine the impact of each attribute on the output of a predictive model. It is based on
+Shapley values, a concept from game theory that determines a fair way to distribute the payoff for
+players that have worked in coalition towards an outcome [33].
+Extreme Gradient Boosting is a decision-tree based ensemble algorithm that has become known
+for its speed and performance [5]. Decision trees are built sequentially so that each one reduces
+the errors of the previous one [35]. Random Forests is a decision-tree based ensemble algorithm
+with each tree built from a subset of the rows and columns of the dataset [34]. This allows for
+variation among the trees and results in lower correlation among their predictions [37]. Support
+Vector Machines are a supervised learning algorithm that finds a hyperplane that best separates
+the data points from two classes [14].
+Artificial Neural Networks are computational networks that are inspired by the biological ner-
+vous system [10]. ANNs excel at prediction for data where the amount of information in each
+2
+
+Figure 1: Stages of Pump and Dump
+attribute is small and there are non-linear interactions among them. Deep learning models are a
+class of extensions to ANNs that have solved long standing prediction problems in image recogni-
+tion and natural language [20]. Convolutional Neural Networks (CNNs) are a class of deep learning
+networks that were designed initially to work with images but work surprisingly well with sequence
+data such as texts as well. Long Short-Term Memory (LSTM) deep learning networks are a type of
+recurrent neural network designed to handle the long-term dependencies present in sequence pre-
+diction problems [4]. Understanding text often requires looking ahead (think of verbs in German)
+and so processing text in both directions, using a biLSTM, provides better results for language [6].
+3
+Experiments
+Within a typical online forum, there are two different categories of texts. The first is a post, which
+initiates a discussion.
+The second is a set of comments responding to the post.
+For example,
+an individual may post saying that, in their opinion, a stock’s price is about to rise, with others
+respond by sharing their opinions in the same thread. Responders may agree with the original post,
+or disagree.
+P&D is an information-based manipulation, artificially raising the price of a stock through the
+dissemination of false information. As shown in Figure 1, this manipulation strategy involves three
+different stages [19]. The operators of the scheme first purchase the stock that they are planning
+to manipulate (Accumulation). Once they have acquired enough shares, they will disseminate false
+information to make it appear more desirable, driving up the price (Pump). Once the price has
+risen to the desired level of profit, the operators sell off their shares before anyone uncovers that
+the information has no basis or the hype dies down (Dump).
+To identify P&Ds within the market, patterns associated with the scheme must be established.
+3
+
+Price
+Accumulation
+Pump
+Dump
+TimeWhile the method of conducting a P&D may vary, two indicators that can identify them are sharp
+changes in price and volume [19]. A P&D will cause a significant price increase within a short
+amount of time, larger than the fluctuations that the stock typically experiences; followed by a
+decrease once the dump phase has begun. The volume also increases as the stock gains interest
+among investors during and after the dissemination phase. However, the volume will typically not
+immediately experience as sharp a decline as the price when the operators begin to dump their
+shares because of the reluctance of investors to believe that the price is illusory.
+If the profile of a P&D manipulation can be detected in the market, then the post that putatively
+caused it can be straightforwardly labelled and its language patterns investigated. (Of course, it is
+possible that some of the apparent connections are spurious, but it is relatively unlikely that a post
+touting a particular stock will be disseminated exactly when the stock’s price and volume begin a
+sharp rise).
+Labelling comments is more complex, since the comments may agree with the original post, or
+disagree. Only the language of those that agree can contribute to predicting a P&D event.
+3.1
+Data Sources
+Two different data sources were utilized. The first is the popular online website Reddit, where users
+discuss the stock market. The second is Yahoo Finance, a financial market website that provides
+historical data about companies.
+Reddit contains forums referred to as subreddits, each dedicated to the discussion of a specific
+topic. Popular forums for the discussions of stocks are r/pennystocks, r/wallstreetbets, r/stocks,
+r/RobinHoodPennyStocks, r/TheWallStreet. We use r/pennystocks and r/RobinHoodPennyStocks,
+Yahoo Finance is a website provided by Yahoo for investors to access financial news, market
+data, and basic financial tools. Given a stock symbol or company name, it provides the relevant
+market data.
+Classification techniques such as Extreme Gradient Boosting (XGBoost), Random Forests, Sup-
+port Vector Machine (SVM), and Artificial Neural Networks (ANNs) were used to learn predictive
+models, and then to identify which attributes (i.e. words) are most predictive. Figure 2 shows the
+experimental workflow.
+Figure 2: Experiment workflow
+Data from Reddit and Yahoo Finance were collected daily for the period October 1, 2019, to
+June 28, 2020. A breakdown of the data is shown in Table 1. The majority of the data is retrieved
+4
+
+Redldit
+Yahoo!
+Finance
+Anomaly
+Detection
+Text
+Labelling
+Model
+Model
+Data
+Preprocessing
+Training
+Testing
+Historical Data
+Agreement
+Model
+Data Retriever
+Data Preparation
+Dataset
+Modelling
+Model
+Comparison/
+EvaluationSubreddit
+Number of Posts
+Number of Comments
+Total
+r/pennystocks
+12,049
+234,149
+246,198
+r/RobinHoodPennyStocks
+6,506
+78,429
+84,935
+Total
+18,555
+312,578
+331,133
+Table 1: Breakdown of records collected from subreddits
+Figure 3: Data Collection Volumes
+from r/pennystocks, with about a third from r/RobinHoodPennyStocks. The number of comments
+is much larger than the number of posts, with posts making up only about 5% of the texts.
+As shown in Figure 3, there was a sharp increase in the number of submissions over the period
+of data collection:
+• r/pennystocks - 139,000 Members ⇒ 257,000 Members
+• r/RobinHoodPennyStocks - 52,000 Members ⇒ 133,0000 Members
+This seems to reflect an increase in amateur stock market investing because of the covid-19 pan-
+demic, and a corresponding increase in manipulation. i.e, as manipulators look to take advantage of
+new, naive investors during the pandemic. Alerts and press releases by the SEC and the Canadian
+Securities Administrators warned new investors to be vigilant about the increasing number of P&D
+schemes that have occurred around that time [9, 28, 29].
+The median number of words per post or comment was 22, and the total number of distinct
+words was 4,862.
+Replacing stock symbols by the market sector to which each business belongs allows us to see
+which sectors are discussed the most, and which are the targets of P&D. Figure 4 shows that
+healthcare stocks are the most mentioned, followed by technology stocks. The pandemic clearly
+had an effect on both attention to markets and manipulations. Temporal trends in the healthcare
+5
+
+10000
+8000
+Number of Records
+6000
+40.00
+2000
+DatesFigure 4: Histogram of market sectors discussed within subreddits
+sector, Figure 5 , show an increase in online activity at the beginning of the pandemic, and then a
+further increase in the middle of 2020. Figure 6 shows that P&D manipulations also increased in
+2020.
+Table 2 shows the information collected for each post and comment.
+Data from Yahoo Finance was scraped using the yfinance tool [2]. Stock symbols were extracted
+from Reddit posts. This step is non-trivial and required regular expression extraction, and look ups
+against the publicly traded exchanges. Posts which mentioned more than one stock were discarded,
+partly because of the complexity of deciding which stock may be being touted, and partly because
+P&D posts typically focus on one particular stock they are pumping. If a stock symbol was found,
+yfinance was used to collect the financial information described in Table 3.
+As shown in Figure 7, the daily Open, High, Low, Close, and Volume (OHLCV) data was
+collected over nine business days surrounding an event. Data was collected over five days before each
+post event to establish a baseline for price and volume. Penny stocks almost always shows minor
+variation in price and volume so this baseline is typically quite flat. The remaining four days contain
+the pump event (sharp increase) followed by a decrease in price and a slower decrease in volume.
+6
+
+00008
+70000
+60000
+of Records
+50000
+40000
+30000
+20000
+10000
+SectorConglomerates
+ SectorServices
+SectorUtilities
+SectorConsumerDefensive
+SectorBasicMaterials
+ SectorRealEstate
+ SectorFinancialServices
+SectorEnergy
+ Sectorlndustrials
+SectorCommunicationServices
+SectorUnknown
+SectorTechnology
+ SectorHealthcare
+SectorsFigure 5: Trend of posts and comments that discussed healthcare stocks
+Feature
+Description
+Post Title
+Title of the post.
+Post ID
+Unique identification code for post.
+Post Author
+Author of the post.
+Post Created
+Unix Timestamp of when post was submitted.
+Post Body
+Text of the post.
+Comment ID
+Unique identification code for comment.
+Comment Author
+Author of the comment.
+Comment Created
+Unix Timestamp of when comment was submit-
+ted.
+Comment Body
+Text of the comment.
+Table 2: Features of collected Reddit data
+Sabherwal et al. [27] studied the effects of online message boards on market manipulation and
+found that dumps typically occur within four days and this is plausible because the manipulators
+want to sell as soon as the price reaches a peak.
+Texts from subreddits were preprocessed using the following steps: remove URLs, expand con-
+tractions, remove HTML Tags, remove punctuation, remove extra whitespaces, remove numbers,
+lemmatization, and remove stopwords.
+Stock symbols within the text were replaced by dummy stock names representing the market
+sector associated with each business. This is required because the name of the particular stock
+being pumped and dumped in one case has nothing to do with the name of the stock being used
+in another case – but there might be correspondences within sectors. Here is an example:
+7
+
+4000
+3500 
+3000
+ of Records
+25:00
+Yumber
+2000
+15:00
+1000 
+500 
+DatesFigure 6: Trend of posts that have been labelled as P&D
+Feature
+Description
+Open
+Opening price of the stock for the given period.
+High
+Highest price for the stock within the given pe-
+riod.
+Low
+Lowest price for the stock within the given pe-
+riod.
+Close
+Closing price of the stock for the given period.
+Volume
+Total number of shares traded within the given
+period.
+Market Sector
+Associated industry that the company is in.
+Market Capitalization
+Total market value of the company’s outstand-
+ing shares.
+Table 3: Features of Yahoo! Finance data
+• “AYTU perfect time to buy” ⇒ “SectorHealthcare perfect time to buy”
+3.2
+Data Labelling
+To label each post, stock data surrounding the day in which the post was submitted to Reddit
+were analyzed. If the market data exhibited that pattern associated with P&D (a notable rise from
+the time of the post, followed by a sharp drop) then the post was labelled accordingly. A rise was
+detected by calculating the average price and volume in the five-day window before the post. The
+8
+
+120
+100
+Posts
+Number of i
+60
+40
+20
+DatesFigure 7: Time window used to collect market data.
+daily average price (DAP) of the values was first calculated for each of the five days.
+DAP(Xt) = 1
+4(Xtopen + Xthigh + Xtlow + Xtclose)
+(1)
+and then the baseline average price (BAP) was calculated by
+BAP(Xest) = 1
+5 ·
+T1
+�
+t=T0
+DAP(Xt)
+(2)
+The baseline average volume (BAV) was calculated by taking the average of the volume values
+over the estimation window.
+BAV (Xest) = 1
+5 ·
+T1
+�
+t=T0
+Xtvolume
+(3)
+A threshold was set at two standard deviations above the average price within the five-day
+estimation window. Price increases above this threshold were considered to be pump events. A
+similar threshold was used to define a volume anomaly. Events were considered to be the result of
+P&D if they exceeded the threshold for both price and volume. Figure 8 shows a comparison of
+the stock behaviours labelled using this approach.
+A sudden price rise or volume increase might coincide with a post, but is not necessarily caused
+by it. The rising region of each stock trend of a potential P&D event was min-max normalised,
+9
+
+Reddit Post Date
+Price
+Event
+To
+T2
+Volume
+27
+29
+May
+5
+11
+Estimation
+Event
+Window (5 Days)
+Window (4 Days)Figure 8: Comparison of stock behaviours that have been labelled using anomaly detection
+and its slope calculated. Steep price increases are more likely to arise from genuine information
+and less likely to have resulted from a single manipulation post, so the median slope across the
+entire dataset was calculated, and only slopes below the median were considered as potential P&D
+events. Figure 9 shows the distribution of stock price trend slopes from the entire the dataset. The
+median value is 0.18.
+3.3
+Agreement Model
+The comments associated with the P&D post cannot all be labelled as examples of P&D language,
+since not all of them will be supportive of the post they are responding to. Manipulators, of course,
+will post comments in support of the post, either from the same identity or from others.
+We developed an agreement model, using ideas from stance detection. This was done using
+Empath to generate a lexicon of agreement, seeding it with the words: bought, agree, positive,
+increasing, good, and now. Empath returned the words listed in Table 4. Posts touting stocks
+also use a specialised vocabulary, shown in these examples.
+• “probably go to shoot up tomorrow”
+10
+
+TRNX2020-04-16
+CCO2020-03-31
+USWS2020-06-07
+GNUS2020-06-09
+Stockbehaviours
+Stockbehaviours
+labelledasP&D
+labelledasnotP&DFigure 9: Distribution of stock price trend slopes
+only
+done
+better
+true
+knew
+besides
+like
+maybe
+wanted
+liked
+also
+important
+buying
+understand
+good
+understood
+needed
+work
+because
+successful
+knowing
+grateful
+plus
+much
+reasonable
+should
+give
+happy
+course
+glad
+well
+considering
+anyway
+agree
+meaning
+great
+probably
+sure
+thought
+guaranteed
+more
+honestly
+positive
+thankful
+actually
+agreed
+special
+doubt
+guess
+though
+bet
+buy
+surpass
+worth
+suppose
+although
+especially
+definitely
+certain
+figured
+given
+means
+Table 4: List of generated agreement words from Empath
+• “this bad boy just rocket”
+• “i will see you on the moon”
+An extended lexicon was determined manually by inspecting posts associated with manipulation.
+Table 5 contains the list of words that were chosen using this approach.
+Comments were labelled as associated with pumping if they contained two or more of the
+11
+
+1200
+1000
+800
+Number of Posts
+600
+400
+200moon
+fast
+massive
+rich
+surprise
+rocket
+profit
+top
+easy
+move
+pump
+rally
+peak
+early
+load
+soar
+climb
+worth
+shoot
+quick
+jump
+rise
+sale
+money
+burst
+pop
+high
+gain
+breakout
+drive
+hype
+spike
+run
+cash
+nice
+fly
+go
+up
+hit
+bank
+awesome
+confident
+surpass
+more
+zoom
+big
+great
+potential
+advantage
+Table 5: List of custom words used in the Agreement Model
+agreement words, or if they were (visibly) authored by the original poster. The following are some
+examples of comments that were labelled as not P&D related based on the agreement model:
+• “it be the american dream to fall for snake oil salesman and then lose everything it be a story
+as old as humanity”
+• “clearly a pump and dump scheme”
+• “do not touch it if the chart look like a hockey stick”
+This labelling of comments is limited by the completeness of the agreement lexicon, and also does
+not account for negations.
+P&D posts and comments are relatively rare and so the dataset is naturally imbalanced. Tech-
+niques such as SMOTE [3] and ADASYN [17] were tried but proved ineffective. Instead, where
+predictors allowed it, class weight parameters were set to penalise mistakes in the minority class.
+3.4
+Modelling
+The following predictors were used:
+• Extreme Gradient Boosting (XGBoost)
+• Random Forest (RF)
+• Support Vector Machine (SVM)
+• Artificial Neural Networks
+– Multilayer Perceptron (MLP)
+– Convolutional Neural Network (CNN)
+– Bidirectional Long Short Term Memory (BiLSTM)
+In each case the standard performance measures (accuracy, precision, recall, F1-Score, confusion
+matrix) were calculated, as well as the Shapley values which rank words by their importance to the
+predictions.
+12
+
+Record Type
+P&D
+Not P&D
+Total
+Posts
+3,006
+15,549
+18,555
+Comment
+26,727
+285,851
+312,578
+Total
+29,733
+312,142
+331,133
+Table 6: Dataset class distribution
+Model
+TP
+FP
+TN
+FN
+Accuracy
+Precision
+Recall
+F1-Score
+XGBoost Posts
+1728
+6615
+8934
+1278
+57.46 (±3.73)
+20.71 (±0.48)
+57.49 (±0.68)
+30.45 (±2.25)
+XGBoost Posts and Comments
+2007
+7646
+7903
+999
+53.41 (±1.42)
+20.79 (±0.85)
+66.77 (±1.58)
+31.71 (±0.96)
+RF Posts
+271
+646
+14903
+2735
+81.78 (±0.51)
+29.55 (±1.40)
+9.01 (±0.52)
+13.81 (±0.78)
+RF Posts and Comments
+414
+211
+15338
+2592
+84.89 (±0.69)
+66.24 (±1.69)
+13.77 (±0.47)
+22.80 (±0.75)
+SVM Posts
+1752
+5263
+10286
+1254
+64.88 (±1.14)
+24.98 (±0.76)
+58.28 (±1.05)
+34.97 (±1.16)
+SVM Posts and Comments
+2125
+4559
+10990
+881
+70.6 (±0.49)
+31.79 (±0.43)
+70.69 (±0.56)
+43.86 (±0.57)
+MLP Posts
+2382
+1718
+13831
+624
+87.38 (±6.66)
+58.10 (±11.65)
+79.24 (±12.76)
+67.04 (±12.12)
+MLP Posts and Comments
+2103
+2602
+12947
+903
+81.11 (±3.71)
+44.70 (±4.28)
+69.96 (±3.80)
+54.55 (±4.36)
+CNN Posts
+2373
+1709
+13840
+633
+87.38 (±7.04)
+58.13 (±12.02)
+78.94 (±12.76)
+66.96 (±12.37)
+CNN Posts and Comments
+2304
+2068
+13481
+702
+85.07 (±1.25)
+52.70 (±2.33)
+76.65 (±3.45)
+62.46 (±2.64)
+biLSTM Posts
+2297
+2495
+13054
+709
+82.73 (±8.11)
+47.93 (±9.92)
+76.41 (±10.94)
+58.91 (±10.82)
+biLSTM Posts and Comments
+2288
+2370
+13179
+718
+83.36 (±2.27)
+49.12 (±3.25)
+76.11 (±3.86)
+59.71 (±3.54)
+Table 7: Summary of model performance
+4
+Results
+Table 6 shows the class distribution for the dataset. Less than 9% of the records are labelled as
+being P&D. This is typical of datasets where fraud is present; indeed it is striking that the rate of
+fraud is this high.
+The results of each of the predictive model are reported in Table 7 using 5-fold cross validation
+and upweighting the fraud class when the model permits it.
+The neural network models perform well as expected.
+Models such as XGBoost, Random
+Forests, and SVM had disappointing performance, and a heterogeneous stacked classifier combining
+their predictions did not improve on the performance of the individual predictors, suggesting that
+they make their errors on the same records.
+At first glance, the ANN models using posts perform better than those using posts and com-
+ments. However, the standard deviations of the performance numbers show that the inclusion of
+comments provides stability for correctly identifying P&D posts. The best performing model over-
+all is CNN, especially with comments included. Its precision is relatively low; of all the records that
+the model predicts to be P&D, only 52.7% are actually correct. If we look at the rate at which each
+class is predicted to be positive, a better outlook of the model is provided. Given a positive P&D
+text, the model has a 76.65% chance of classifying it correctly, whereas, if it is given a negative
+text, it has a 13.3% chance of classifying it incorrectly as positive. It is perhaps a little surprising
+that biLSTM did not perform best since they are typically strong predictors for natural language
+problems.
+The SHAP Explainers produce diagrams that rank the attributes by their impact on outcomes.
+Figure 10 shows the diagram for the CNN predictor for posts and comments and the 30 most
+impactful words. Although the influence of any single word is inevitably weak, there are visible
+red dots to the right for many of these words, indicating that higher frequencies of these words are
+associated with P&D events. The names of the popular sectors are indicator of P&Ds, as are words
+13
+
+Predicted Label
+Actual Label
+Misclassified Post
+P&D
+Not P&D
+sectorunknown about to soar
+P&D
+Not P&D
+sectorunknown fitness equipment maker owner
+of bow flex completely sell out of most retail
+store how be this look just buy in share
+P&D
+Not P&D
+quick all in sectorcommunicationservices pump
+my first time actually do something right the
+lambos go to be green for gain
+P&D
+Not P&D
+blast off look like gold and oil will be big player
+this i also suggest look at sectortechnology
+P&D
+Not P&D
+sectorenergy drop time to buy it be drop below
+which be its day low be it a good time to buy
+Not P&D
+P&D
+sectortechnology release patent news on thermal
+tech could be a mark sympathy play bust out
+over
+Not P&D
+P&D
+sectorhealthcare do anyone understand why sec-
+torhealthcare shoot up soo much i be not able
+to find any real catalyst
+Not P&D
+P&D
+sectorhealthcare on the move this have potential
+reach today
+Not P&D
+P&D
+sectorhealthcare to the moon
+Not P&D
+P&D
+any thought on when to sell sectorenergy bought
+in late i be up after hour should i wait til tomor-
+row or sell as soon as possible in the am
+Table 8: Examples of misclassified posts from CNN model
+from the agreement model such as “buy” and “go”. Across the best performing models, the same
+set of words emerge as the most impactful features (not shown).
+Misclassifications by the model have different impacts depending on how and where it is used.
+For an ordinary investor, a false positive (a post predicted to be a P&D when it isn’t) means a
+missed opportunity for profit, but a false negative means a financial loss. For a regulatory body, a
+false positive is problematic, but a false negative less so. Table 8 shows some of the examples of
+misclassifications by the CNN model.
+Some false positives, predicted to be P&D from the text, but without a corresponding market
+movement may be instances where the post failed to attract enough attention to cause a measurable
+market movement, or was so blatant that it was not credible to typical investors.
+Some false
+negatives may be because the posts were too short to contain the required two words, because the
+pumping took place on another platform or because a market movement happened to match the
+timing of the post.
+14
+
+5
+Related work
+The application of data analytics for detecting market manipulation is a relatively new in the
+field of finance. Most research has focused on detecting trade-based manipulation because it is
+most common [32]. Huang and Chang found that of the manipulation cases prosecuted in Taiwan
+from 1991 to 2010, 96.61% were trade-based, and only 3.39% were information-based [18]. Some
+examples detecting trade-based manipulation are: Ogut et al. [38] in the emerging Istanbul Stock
+Exchange, Wang et al. [32] for prosecuted manipulation cases reported by the China Securities
+Regulatory Commission, Cao et al. [7] using real trading data from four popular NASDAQ stocks
+with synthetic cases of manipulation (spoofing and quote stuffing), Cao et al. [36] using seven
+popular NASDAQ and LSE stocks data injecting ten simulated stock price manipulations, Diaz et
+al. [12] using manipulation cases pursued by the U.S. Securities and Exchange Commission (SEC)
+in 2003, and Golomohammadi et al. [16] trying to detect three groups of manipulation schemes:
+marking the close, wash trades, and cornering the market.
+For information-based manipulation, Victor and Hagemann [31] looked at 149 confirmed P&D
+schemes coordinated through Telegram chats and pumped via Twitter. Using XGBoost, they built
+a model that achieved a sensitivity of 85% and specificity of 99%. They concluded that P&Ds were
+frequent among cryptocurrencies that had a market capitalization of $50 million or below and often
+involved trading volumes of several hundred thousand dollars within a short time-frame.
+Mirtaheri et al. [23] looked specifically at forecasting P&Ds by combining the information from
+Twitter and Telegram. They manually labelled known P&D operation messages on Telegram, and
+then used SVMs with a stochastic gradient descent optimizer to label the remaining messages as
+P&D or not. They used Random Forests to detect whether a manipulation event was going to take
+place within the market. Their results showed that they were able to detect, with reasonable accu-
+racy, whether there is an unfolding manipulation scheme occurring on Telegram. Their proposed
+model was able to achieve an accuracy of 87% and an F1-Score of 90%.
+Some partially automated tools have also been developed.
+These flag suspicious activities
+that can then by investigated by regulators.
+Delort et al.
+[11] used Naive Bayes classifiers to
+examine collected messages from HotCopper, an Australian stock message board. They successfully
+identified messages of concern, but the number of false positives was too high to use the model
+in an automated way. Owda et al. [25] compared messages to lexicon templates of known illegal
+financial activities (e.g. Pump and Dump, Insider Information). They found that, of the 3000
+comments that were collected on a daily basis, 0.2% were deemed suspicious.
+6
+Conclusion
+The intersection of social media with low-cost trading platforms and naive investors has made
+market manipulation an attractive strategy.
+Pump&dump is particularly simple to implement
+since it requires only the dissemination of fictional information about the future prospects for a
+stock. This is particular easy for penny stocks where validating information is difficult for ordinary
+investors, and where relatively small purchase volumes can cause large price movements.
+We investigate protecting investors, and assisting regulators, by building predictive models that
+label social media posts (and the responses they elicit) as potential drivers of P&D events. We
+do this by collecting posts and comments, developing a model for a P&D event based on patterns
+of price and volume changes, using the match between posts and P&D events to label posts, and
+15
+
+extending this labelling to comments using an agreement model. Natural language predictors then
+learn the language patterns associated with P&D manipulations, so that new manipulations can
+be detected before they affect the market.
+Data is imbalanced, since manipulations are rare, but our best predictive model achieves an
+F1-score of 62% and an accuracy of 85%. Improvements in performance are limited by potential
+coincidences between a post and a price and volume change that mimics a P&D, posts that fail to
+reach a sufficient audience to cause the desired buying behaviour, and natural language issues that
+arise from informal and short texts, and a specialised vocabulary used in stock discussion forums.
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+Algorithm:
+XGBoost
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+Machine
+Learning,
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+analyticsvidhya.com/blog/2018/09/an-end-to-end-guide-to-understand-the-math-
+behind-xgboost/, Accessed: 2021-09-15.
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+18
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+emerging market: The case of Turkey. Expert Systems with Applications, 36(9):11944–11949,
+November 2009.
+19
+
+Figure 10: CNN SHAP Summary Plot for posts and comments
+20
+
+High
+buy
+sectorhealthcare
+stock
+get
+one
+sectorindustrials
+go
+look
+poob
+share
+sell
+sectortechnology
+price
+sectorconsumercyclical
+Feature value
+day
+megathread
+would
+company
+like
+make
+sectorcommunicationservices
+news
+see
+hold
+today
+post
+week
+market
+sectorunknown
+think
+0.2
+0.1
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+LOW
+SHAP value (impact on model output)

-tE4T4oBgHgl3EQf4A1b/content/tmp_files/2301.05310v1.pdf.txt

@@ -0,0 +1,1583 @@
+Optimization of Hybrid Power Plants:
+When Is a Detailed Electrolyzer Model Necessary?
+Manuel Tobias Baumhof, Enrica Raheli, Andrea Gloppen Johnsen, and Jalal Kazempour
+Department of Wind and Energy Systems, Technical University of Denmark, Kgs. Lyngby, Denmark
+{mtba, enrah, anglopj, jalal}@dtu.dk
+Abstract—Hybrid power plants comprising renewable power
+sources and electrolyzers are envisioned to play a key role in
+accelerating the transition towards decarbonization. It is common
+in the current literature to use simplified operational models for
+electrolyzers. It is still an open question whether this is a good
+practice, and if not, when a more detailed operational model is
+necessary. This paper answers it by assessing the impact of adding
+different levels of electrolyzer details, i.e., physics and operational
+constraints, to the optimal dispatch problem of a hybrid power
+plant in the day-ahead time stage. Our focus lies on the number
+of operating states (on, off, standby) as well as the number
+of linearization segments used for approximating the non-linear
+hydrogen production curve. For that, we develop several mixed-
+integer linear models, each representing a different level of
+operational details. We conduct a thorough comparative ex-post
+performance analysis under different price conditions, wind farm
+capacities, and minimum hydrogen demand requirements, and
+discuss under which operational circumstances a detailed model
+is necessary. In particular, we provide a case under which a
+simplified model, compared to a detailed one, results in a decrease
+in profit of 1.8% and hydrogen production of 13.5% over a year.
+The key lesson learned is that a detailed model potentially earns
+a higher profit in circumstances under which the electrolyzer
+operates with partial loading. This could be the case for a certain
+range of electricity and hydrogen prices, or limited wind power
+availability. The detailed model also provides a better estimation
+of true hydrogen production, facilitating the logistics required.
+Index Terms—hybrid power plants, electrolyzer, hydrogen,
+mixed-integer linear programming
+I. INTRODUCTION
+A. Background
+In order to limit global warming to a maximum of 1.5 °C,
+greenhouse gas emissions must be reduced to net zero by 2050,
+as called for in the European Green Deal 2019 [1]. Renewable
+hydrogen produced through electrolysis could aid in two
+major challenges on the path towards the net zero goal. First,
+electrolyzers can act as flexible loads and therefore potential
+frequency restoration ancillary service providers, contributing
+to maintaining the power balance in power systems with
+increased penetration of renewable energy sources. Second,
+renewable hydrogen can be further synthesized into other
+green fuels, eventually enabling decarbonization in the hard-
+to-abate sectors, such as heavy transport and industry.
+Hybrid power plants comprising of renewable power sources
+(wind and/or solar) and electrolyzers are the key components
+to accelerate the current energy transition through hydrogen
+[2]. Nonetheless, uncertainties in terms of the cost-benefit of
+electrolyzers in the long run have challenged the widespread
+investment in said technologies and thereby large-scale pro-
+duction of renewable-based green hydrogen [3]. In Denmark,
+there is currently a special focus on green hydrogen at the
+governmental level and also, among the regulator, system
+operator, and many industry stakeholders, envisioning a large
+deployment of electrolyzers and other power-to-X facilities in
+the coming years. In 2021 the Danish government published
+a strategy for the national power-to-X development, aiming to
+build 4 to 6 GW of electrolysis capacity by 2030, doubling the
+current Danish peak demand [4]. This emerging trend is not
+limited to Denmark, and many other countries both in Europe
+and globally see hydrogen as a key solution for the realization
+of green societies of the future [2], [5].
+B. Aim and Literature Review
+It is a common practice in the current literature to use a
+simplified operational model for electrolyzers e.g., by using
+a constant power-to-hydrogen conversion ratio irrespective of
+whether the electrolyzer operates in full capacity or not [6]–
+[9]. In addition, some papers do not consider operational states
+of the electrolyzer [6], [9]. This paper challenges these simpli-
+fication practices. While a simplified model works satisfacto-
+rily under certain operational circumstances, there are several
+other circumstances under which a simplified one yields a
+sub-optimal operation of electrolyzers, underestimating their
+value. This paper answers when a detailed operational model
+should be applied, and to what extent the profit and hydrogen
+production can be increased by using a detailed model. We will
+also discuss to what extent a detailed model brings additional
+computational burden.
+In general, two main physical aspects of electrolyzers need
+to be modeled for operation in the day-ahead time stage:
+1) Electrolyzer efficiency: The power-to-hydrogen conver-
+sion efficiency is a function of the power consumption
+of the electrolyzer. To accurately model the hydrogen
+production of the electrolyzer, the varying efficiency
+should be captured, which introduces non-linearities to
+the model. The simple models usually use a constant
+efficiency, while more accurate modeling incorporates
+the non-linearities, which can be later linearized.
+2) Number of operating states: Proper operational modeling
+of electrolyzers may require introducing three states,
+namely on, off, and standby, to ensure no hydrogen pro-
+duction below a given minimum allowed partial loading,
+for which additional binary variables are needed. Many
+1
+arXiv:2301.05310v1  [math.OC]  12 Jan 2023
+
+papers in the literature do not even model states, thus
+assuming the electrolyzer is always on, or model two
+states only, i.e., on and off, similar to conventional power
+generators1.
+Various studies have incorporated different levels of opera-
+tional details of the electrolyzer into their optimization prob-
+lems. In [7] and [8], a constant efficiency is applied but two
+and three states are modeled, respectively, by adding binary
+variables. In [10], three states are modeled, while assuming a
+linear hydrogen production curve, despite showing that the
+production curve is not well approximated by a first-order
+interpolation. A hybrid power plant including an electrolyzer
+is modeled in [11], where the non-linear hydrogen production
+is linearized between two points, with a single binary variable
+representing the on/off state of the electrolyzer. In [12] a
+quadratic production curve is applied and the resulting non-
+linear program is eventually solved by a heuristic. In [13],
+three states are included, and differently from the other papers,
+the operating temperature is considered as a variable, provid-
+ing an extra degree of freedom in the electrolyzer operation.
+This model allows to take into account the temperature impact
+on the conversion efficiency and the quality of the generated
+heat. The non-linear hydrogen production is then linearized
+around a fixed reference operating point to formulate the
+problem as a mixed-integer linear program (MILP).
+C. Contributions and Paper Organization
+To the best of our knowledge, there is a lack of a com-
+prehensive analysis in the current literature, identifying the
+operational circumstances under which a simple model ends
+up in a sub-optimal operation of electrolyzers, resulting in a
+reduced profit and hydrogen production2. This paper bridges
+such a gap through the following contributions:
+• To embed constraints describing the physics of electrolyz-
+ers while keeping the final model as a MILP,
+• To thoroughly investigate ex-post the impact of the in-
+clusion of different operational details on the final profit
+of the hybrid power plant and the amount of hydrogen
+produced,
+• and finally, to provide a set of recommendations in
+terms of including operational details of electrolyzers,
+depending on the application, the range of electricity
+prices, and the hydrogen price.
+Without loss of generality, this paper focuses on alkaline
+electrolyzers, as they are currently the most mature tech-
+nology [14]. The proposed model can be extended to other
+low-temperature electrolyzers, such as polymer electrolyte
+membrane (PEM). More operational characteristics may be
+necessary for modeling solid-oxide electrolyzers (SOEC).
+1We will discuss later in Section IV that under some operational conditions,
+a two-state model including on and standby states works well too. In contrast,
+the two-state model on-off is not satisfactory neither in terms of dispatch
+decisions nor the computational performance.
+2Reference [13] provides a similar analysis, however, the Faraday efficiency
+is assumed to be one. The consequences of this assumption will be further
+discussed in Section II-B.
+The rest of the paper is organized as follows. Section II
+describes the electrolyzer physics, focusing on the operating
+states and the hydrogen production curve. Section III provides
+the proposed MILP, representing all three states of the elec-
+trolyzer. Section IV discusses the impact of the electrolyzer
+modeling choices by means of a test case and a thorough
+sensitivity analysis. Section V concludes the paper. Finally,
+Appendices A and B provide two MILPs (simpler than the
+one proposed in Section III), both representing two states of
+the electrolyzer only, where one is a model with on-off states,
+and the other one is a model with on-standby states.
+II. ELECTROLYZER PHYSICS
+The core of the renewable-hydrogen hybrid power plant is
+the electrolyzer, where water is decomposed into hydrogen
+and oxygen by means of electrical power. The physics and
+operating characteristics of alkaline electrolyzers are described
+in this section and will be formulated as a set of mixed-integer
+linear constraints in Section III.
+A. States
+To describe and model the real operation of an alkaline
+electrolyzer, it is necessary to distinguish three different states:
+1) On state: the electrolyzer operates within its feasible
+load range, consuming power and producing hydrogen with a
+conversion efficiency that depends on the partial load, which
+will be explained in Section II-B. The minimum operating
+power for alkaline electrolyzers is around 15-20% of the
+nominal power, below which the electrolyzer must go into
+standby or off.
+2) Standby state: the electrolyzer does not produce any
+hydrogen but consumes the power needed to maintain the
+system temperature and pressure so that it can rapidly resume
+production. The value of the standby power consumption is
+not usually disclosed by manufacturers, but values between
+1-5% of the electrolyzer full load capacity have been adopted
+in the literature [7], [8], [10]. The time needed to switch from
+standby to on, i.e., a warm start-up is of the order of 30 seconds
+[8].
+3) Off state: the electrolyzer is shut down completely and
+does not consume any power nor produce any hydrogen. How-
+ever, to switch back to on, a significant amount of electricity
+is needed, corresponding to a cold start-up cost. Moreover,
+at least 20 minutes are necessary before resuming hydrogen
+production [8]. Apart from the introduced cold start-up cost
+and start-up time, the frequent shut down of the electrolyzer
+may have a negative impact on the device degradation and
+lifetime [15].
+B. Efficiency and Production Curve
+The conversion efficiency of electricity into hydrogen is not
+constant but depends on the partial load, i.e., the ratio between
+power consumption at a specific time and the nominal power
+of the electrolyzer. The variation of the efficiency based on the
+operating set-point is mainly due to two phenomena: (i) the
+current-voltage relationship, also called the polarization curve,
+2
+
+10
+20
+30
+40
+50
+Power [MW]
+17.5
+18.0
+18.5
+19.0
+19.5
+Efficiency [kg/MWh]
+(a)
+10
+20
+30
+40
+50
+Power [MW]
+200
+400
+600
+800
+Hydrogen [kg/h]
+(b)
+Non-linear curve
+Approximated curve
+pe *
+h*
+hr
+} h
+Fig. 1. Plot (a): the efficiency curve, and plot (b): the hydrogen production
+curve of a 52.25-MW alkaline electrolyzer, as a function of the electric power
+consumption, working at 90 °C and 30 bar. The black curves represent the
+original non-linear curves. Approximated by two segments, the red curve in
+plot (b) is the piecewise linearized hydrogen production curve. The non-linear
+efficiency curve corresponding to this piecewise linearization is represented
+by the red curve in plot (a). In our formulation, we will only use the red
+piecewise linear production curve in plot (b). The inner plot of (b) shows
+the hydrogen production discrepancy ∆h between original and approximated
+curves, for a given power consumption level.
+and (ii) the Faraday efficiency. We explain both phenomena in
+the following.
+The current-voltage relationship describes the voltage in-
+crease (also called over-voltage or over-potential) with increas-
+ing current density, due to different losses, as explained in [16]
+and [13]. Ulleberg [17] introduced a widely adopted empirical
+formulation that describes the relationships between voltage,
+current density, and electrolyzer operating temperature. To fur-
+ther take into account the operating pressure, this formulation
+was modified by Sanchez et al. [18]. For a given temperature
+and pressure, this can be formulated as
+U cell(i) = U rev + K1i + K2log(K3i + 1),
+(1)
+where U cell(i) is the cell voltage as a function of the current
+density i. In addition, U rev is the open-circuit voltage (i.e.,
+voltage corresponding to current density equal to zero). The
+parameters K1, K2, K3 are constants obtained from experi-
+mental data and can be found in [18]. Voltage U rev can be
+calculated for a specific operating temperature according to
+an empirical equation that can be found in [18]. The power
+consumed by the electrolyzer pe(i) can be calculated as
+pe(i) = U cell(i)iA,
+(2)
+where A is the total area of the cells composing the elec-
+trolyzer. The Faraday law calculates the hydrogen production
+h(i) of the electrolyzer as
+h(i) = 3600 · ηF(i)M H2iA
+2F
+,
+(3)
+where h(i) is the hydrogen production rate in kg/h, M H2 is the
+molar mass of hydrogen in kg/mol, F is the Faraday constant,
+and ηF(i) is the Faraday efficiency as a function of current
+density. The latter is defined as the ratio between the actual and
+the theoretical maximum amount of hydrogen produced. The
+difference between actual and theoretical output is explained
+in [17], and it increases significantly when the electrolyzer
+is working at low-current densities. In [18], an empirical
+expression that captures the relationship between the Faraday
+efficiency and the current density at a given temperature is
+provided: ηF(i) is close to one for higher current densities, and
+it drops to zero when reducing the current. The electrolyzer
+efficiency is defined as
+η(i) = h(i)
+pe(i),
+(4)
+where generally η(i) is expressed in kg/MWh. For different
+values of i, the black curve in Figure 1(a) shows efficiency η(i)
+versus power consumption pe(i). In addition, the black curve
+in Figure 1(b) shows the hydorgen production h(i) versus
+power consumption pe(i). For notational clarity, we drop (i)
+in the rest of the paper. The black curves in Figure 1 show that
+the model is non-linear. The efficiency has a peak at around
+30% of the load. This characteristic peak in the efficiency
+curve is not captured when a constant conversion efficiency is
+used, as done in [6], [8], [10], or when the Faraday efficiency
+is assumed to be equal to one in the entire feasible operating
+range, as done in [13].
+To keep the final problem a MILP, but describe the hydrogen
+production with more details, we use a piecewise linearization
+of the hydrogen production curve as shown by the red curve in
+Figure 1(b), for two linearization segments. For each segment
+s ∈ S, the As (slope) and Bs (intercept) coefficients of the
+line can be calculated such that the approximated hydrogen
+production is Aspe + Bs. Later we will define a binary
+variable indicating which segment is active. The proposed
+approximation is exact only at the segment endpoints (i.e.,
+linearization points), otherwise, it is an underestimation of
+the original non-linear curve. For example, the optimal power
+set-point pe∗ in the inset of Figure 1(b) corresponds to the
+hydrogen production h∗ according to the proposed piecewise
+linear model with two segments3. However, the actual hydro-
+gen realization based on the electrolyzer physics is hr. The
+hydrogen production difference ∆h is reduced by increasing
+the number of segments, and the effect of the hydrogen surplus
+obtained when choosing only one segment, as done in [10], is
+discussed in Section IV.
+According to this piecewise linear formulation for the
+hydrogen production curve, the efficiency η for segment s
+can be calculated based on (4), resulting in η = As + Bs
+pe .
+This is depicted by the red dotted curve in Figure 1(a), given
+two linearization segments used. Note that it does not present
+a linear behavior. However, this non-linear efficiency curve
+does not appear in our optimization problem. The hydrogen
+production curve is used instead, which is linearized through
+segments, as illustrated by the red dotted curve in Figure 1(b).
+III. PROBLEM FORMULATION
+We consider a hybrid power plant, as depicted in Figure 2,
+consisting of a wind farm, an electrolyzer, a hydrogen com-
+pressor, and a hydrogen storage. The generated wind power
+can be either sold to the grid at the electricity market price,
+3Symbol ∗ refers to the optimal value.
+3
+
+Wind farm
+Grid
+Electrolyzer
+Compressor
+Hydrogen 
+storage
+Hydrogen 
+demand
+Electricity
+Hydrogen
+Fig. 2. Schematic representation of a hybrid power plant.
+or consumed by the electrolyzer to produce 100% renewable-
+based green hydrogen. The hydrogen produced can either be
+directly delivered to the demand or temporarily stored in an on-
+site hydrogen storage, with an associated cost for compressing
+the gas. The dashed blue line in Figure 2 represents the option
+to buy electricity from the grid only to supply the electrolyzer’s
+standby power when there is no wind power.
+The hydrogen price is assumed to be a single-value constant,
+and the hybrid power plant serves a minimum daily hydrogen
+demand. We assume the plant has perfect foresight of future
+wind power production and electricity price. Given the 1-hour
+time resolution in our model, we neglect the ramping limitation
+which are typically around ±20% of the nominal power per
+second [10], as well as the warm and cold start-up times of
+the electrolyzer.
+For the optimal operation of the hybrid power plant, we
+develop a complete MILP in Section III-A accounting for
+three states of the electrolyzer and then provide two simplified
+counterparts in Section III-B, each with two states of the
+electrolyzer.
+Notation: All parameters are upper-case or Greek letters,
+whereas all variables are lower-case letters. All binary vari-
+ables are noted by z.
+A. Three-state Model
+The most complete MILP includes the objective function
+(6) constrained by (7)-(29).
+1) Objective function: Over the set of hours t ∈ T , the
+objective function (6) maximizes the total profit of the hybrid
+power plant as
+max
+x
+�
+t∈T
+ptλDA
+t
++ dtλh − pin
+t λin
+t − zsu
+t λsu,
+(6)
+where the variable set x will be defined later. The first term
+corresponds to selling power pt to the grid at the day-ahead
+electricity market price λDA
+t
+. The second term pertains to
+delivered hydrogen dt at a fixed price λh. The third term
+represents the cost for purchasing standby power pin
+t to support
+the electrolyzer’s standby state in case the wind power is
+insufficient. The corresponding price is λin
+t
+= λDA
+t
++ λTSO,
+where λTSO is the grid tariff imposed by the Transmission
+System Operator (TSO). Finally, the fourth term corresponds
+to the cold start-up cost of the electrolyzer, where the binary
+variable zsu
+t
+indicates the start-up at hour t, associated with
+the cost per startup λsu.
+2) Power balance: In every hour t, the power pt sold in the
+day-ahead market is equal to the wind farm power production
+P w
+t plus power pin
+t bought from the grid to support the standby
+state of the electrolyzer, subtracted by the power consumption
+pe
+t of the electrolyzer and the power consumption pc
+t of the
+compressor, such that
+pt = P w
+t + pin
+t − pe
+t − pc
+t
+∀ t ∈ T .
+(7)
+3) Limit on pin
+t : The input power pin
+t
+is limited by the
+standby state consumption of the electrolyzer, implying that
+power cannot be bought from the grid to produce hydrogen:
+pin
+t ≤ P sbzsb
+t
+∀ t ∈ T ,
+(8)
+where the parameter P sb is the standby consumption, and the
+binary variable zsb
+t
+indicates whether the electrolyzer is in the
+standby mode in hour t.
+4) Electrolyzer operational states: Constraint (9) ensures
+that the electrolyzer can take only one out of three states at
+any hour t, namely online, standby, or off:
+zon
+t
++ zoff
+t
++ zsb
+t
+= 1
+∀ t ∈ T ,
+(9)
+where similar to zsb
+t , binary variables zon
+t
+and zoff
+t
+indicate
+whether in hour t the electrolyzer is on and off, respectively.
+The states are activated based on the electricity consumption of
+the electrolyzer. In the online state, the electricity consumption
+pe
+t of the electrolyzer can neither exceed the capacity Ce nor
+go below a minimum load limit P min. In the standby state, the
+electricity consumption must be equal to the standby power
+consumption P sb. These constraints are enforced by
+pe
+t ≤ Cezon
+t
++ P sbzsb
+t
+∀ t ∈ T
+(10)
+pe
+t ≥ P minzon
+t
++ P sbzsb
+t
+∀ t ∈ T .
+(11)
+To represent the cold start-up of the electrolyzer, the binary
+variable zsu
+t
+is defined, taking the value 1 in the case of
+a transition from off to on state in hour t, as enforces by
+constraints (12) and (13). Further, constraint (14) ensures
+that the transition from an off-state to a standby-state is not
+allowed, to avoid bypassing of the start-up cost.
+zsu
+t
+≥ zon
+t
+− zon
+t−1 − zsb
+t−1
+∀ t ∈ T \1,
+(12)
+zsu
+t=1 = 0,
+(13)
+zoff
+t−1 + zsb
+t
+≤ 1
+∀ t ∈ T \1.
+(14)
+5) Electrolyzer hydrogen production: The hydrogen pro-
+duction ht is a function of the electricity consumption of the
+electrolyzer. As explained in Section II-B, for each segment
+s ∈ S, a linear function of the segment power consumption
+ˆpe
+ts with slope As and intercept Bs is defined, such that
+ht =
+�
+s∈S
+(Asˆpe
+ts + Bszh
+ts)
+∀ t ∈ T ,
+(15)
+where the binary variable zh
+ts defines which segment s is active
+in hour t. Each segment is valid within a pre-defined interval
+of upper P s and lower P s power consumption levels, i.e.,
+P szh
+ts ≤ ˆpe
+ts ≤ P szh
+ts
+∀ t ∈ T , s ∈ S.
+(16)
+4
+
+VectorStock
+VectorStock.com/24756804shutterstock.com • 1658641081Constraint (17) ensures that hydrogen production happens
+in the online state only, while one segment only can be active
+at any hour t. In addition, (18) computes the total power
+consumption of the electrolyzer:
+zon
+t
+=
+�
+s∈S
+zh
+t,s
+∀ t ∈ T
+(17)
+pe
+t =
+�
+s∈S
+ˆpe
+ts + P sbzsb
+t
+∀ t ∈ T .
+(18)
+6) Hydrogen storage: Constraints (19)-(25) represent the
+storage operation:
+ht = hd
+t + sin
+t
+∀ t ∈ T ,
+(19)
+dt = hd
+t + sout
+t
+∀ t ∈ T ,
+(20)
+sout
+t
+≤ Sout
+∀ t ∈ T ,
+(21)
+pc
+t = Kcsin
+t
+∀ t ∈ T ,
+(22)
+st=1 = Sini + sin
+t=1 − sout
+t=1
+(23)
+st = st−1 + sin
+t − sout
+t
+∀ t ∈ T \1,
+(24)
+st ≤ Cs
+∀ t ∈ T .
+(25)
+The hydrogen produced ht can either go directly to the demand
+hd
+t or be injected into the hydrogen storage sin
+t , as enforced
+by (19). The total hydrogen dt delivered to the demand is
+equal to the sum of hydrogen directly from the electrolyzer
+and that from the storage sout
+t
+, as per (20). The storage
+output of every hour is limited by the output flow capacity
+Sout in (21). Further, the compressor consumes power pc to
+compress the hydrogen injected into the storage. Assuming
+adiabatic compression, the compression coefficient Kc can be
+calculated, as proposed by [13]. The power consumption for
+compression is then (22). The state of charge of the hydrogen
+storage in the initial and following hours is calculated by
+(23) and (24), where Sini is the hydrogen initially stored in
+the storage at the beginning of time horizon T . The storage
+hydrogen mass capacity Cs is enforced by (25). Note that we
+do not impose any constraint for the energy stored at the end
+of time horizon T . Therefore, pursuing profit maximization in
+this time horizon, the hybrid power plant will leave the storage
+empty in the last hour4.
+7) Hydrogen demand: Imagine within the underlying time
+horizon T , which could be, for example, a year, there are N
+number of time subsets, e.g., 365 days, indexed by n, such
+that there is a minimum hydrogen demand for each n:
+�
+t∈Hn
+dt ≥ Dmin
+n
+∀ n ∈ {1, ..., N},
+(26)
+where Hn is the set of hours within time subset n.
+8) Variable declaration: Constraint (27) declares the non-
+negativity conditions:
+dt, ht, hd
+t , pt, pc
+t, pin
+t , ˆpe
+ts, st, sin
+t , sout
+t
+∈ R+.
+(27)
+4One can enforce a constraint on the minimum stored hydrogen at the end
+of the time horizon, or add a value for this stored energy to the objective
+function.
+Constraint (28) lists binary variables:
+zsu
+t , zh
+ts, zon
+t , zoff
+t , zsb
+t
+∈ {0, 1}.
+(28)
+Therefore, the total number of binary variables is |T |(4 +
+|S|) binaries, where |T | and |S|, respectively, are the number
+of hours and the number of segments used to linearize the
+hydrogen production curve. Finally, the variable set x is
+defined as
+x = {dt, ht, hd
+t , pt, pc
+t, pin
+t , ˆpe
+ts,
+sin
+t , st, sout
+t
+, zsu
+t , zh
+ts, zon
+t , zoff
+t , zsb
+t }.
+(29)
+Accordingly, in addition to |T |(4+|S|) number of binary vari-
+ables, we have |T |(9 + |S|) number of continuous variables.
+B. Two-state Models
+The optimal operation problem (6)-(29) of the hybrid power
+plant accounting for three states of the electrolyzer can be
+simplified if two states only are considered, either on-off states
+or on-standby states. Both result in MILPs.
+In the latter, i.e., the MILP with on-off states, one binary
+variable (instead of three) per hour t is sufficient, such that it
+indicates whether the electrolyzer in the given hour is on or
+off. The resulting MILP is provided in Appendix A. The total
+number of binary variables in this MILP is |T |(2 + |S|).
+Similarly, a single binary variable per hour t is enough
+in the MILP with on-standby states, indicating whether the
+electrolyzer is online or in standby mode. Also, the start-up
+binary variable is not needed. The corresponding MILP is
+given in Appendix B, where among three MILPs, we need
+the lowest number of binary variables, i.e., |T |(1 + |S|).
+IV. NUMERICAL STUDY
+We apply the proposed MILPs of Section III to a case study
+and investigate how the optimal operation of the hybrid power
+and the resulting profit change by adding more operational
+details of the electrolyzer. All source codes and input data are
+publicly shared5. We consider several options for the number
+of linearization segments, i.e., |S|, used to approximate the
+hydrogen production curve of the electrolyzer, including 1, 2,
+4, 8, and 12 segments. Also, we consider three options for
+the number of electrolyzer states: three states on-off-standby
+(OOS), two states on-standby (OS), and two states on-off
+(OO). In the rest of this section, we will refer to various
+models as, for example, OOS-12, implying we consider three
+states (OOS) with 12 segments. Finally, we conduct a sensitiv-
+ity analysis to explore the impact of various input parameters,
+such as wind farm capacity, hydrogen demand, and hydrogen
+price, on the operation of the hybrid power plant.
+A. Case Study
+We consider a hybrid power plant whose structure equals
+the one in Figure 2, and its input data is provided in Table I.
+The capacity of the wind farm is 104.5 MW, corresponding to
+11 V164-9.5 MW™ Vestas turbines, located in Køge Bay,
+5GitHub: https://github.com/mtba-dtu/detailed-electrolyzer-model
+5
+
+TABLE I
+INPUT DATA FOR THE CASE STUDY
+Wind farm
+Capacity
+Cw
+104.5
+MW
+Electrolyzer
+Capacity
+Ce
+50%
+of Cw
+Standby load
+P sb
+1%
+of Ce
+Minimum load
+P min
+15%
+of Ce
+Pressure
+30
+bar
+Temperature
+90
+°C
+Max. current density
+5,000
+A/m2
+Start-up cost
+λsu
+2,612.50
+C [10]
+TSO tariff
+λTSO
+15.06
+C/MWh
+Storage
+Capacity
+Cs
+22,000
+kg
+Maximum output
+Sout
+912.13
+kg/h
+Compressor
+Inlet temperature
+40
+°C
+Inlet pressure
+30
+bar
+Outlet pressure
+200
+bar
+Mechanical efficiency
+75%
+Hydrogen
+Price
+λh
+2.10
+C/kg
+Minimum demand
+Dmin
+n
+3,667
+kg/day
+Denmark. The electrolyzer capacity is set to 50% of the
+wind farm capacity, amounting to 52.25 MW. The modeling
+horizon spans one year with an hourly temporal resolution.
+We apply hourly electricity price data for 2019, as price data
+for the following years might be distorted by macroeconomic
+impacts, such as COVID-19. Day-ahead electricity prices for
+the East Denmark area (DK2) are obtained from ENTSO-e
+Transparency platform [19] and hourly historical wind capac-
+ity factors at the given location for 2019 are retrieved from
+the Renewable.ninja web platform [20]. The average yearly
+capacity factor for the selected location is 43.7%. The hybrid
+power plant is only allowed to buy power from the grid to
+keep the electrolyzer in standby mode, in case the wind power
+is insufficient. In that case, the electricity is bought at the
+hourly day-ahead market price plus the grid tariff of the TSO.
+Since the wind farm is located in DK2, the consumption tariff
+imposed by the Danish TSO, Energinet, is applied [21]. The
+minimum daily demand can be met by the full-load operation
+of the electrolyzer for around four hours. The hydrogen storage
+is scaled to store all hydrogen produced if the electrolyzer
+operates at full capacity for 24 consecutive hours.
+B. Impacts of the Number of Segments
+Let us consider the OOS case with three states, for which
+we solve the proposed MILP (6)-(29). We start with OOS-1,
+where |S| = 1. This means the original non-linear hydrogen
+production curve, depicted in Figure 1(b), is approximated by
+a single linear curve. Here, the minimum power consumption
+P min and the capacity Ce of the electrolyzer are taken as
+two endpoints. By moving to OOS-2, where the number of
+segments |S| is 2, we consider an additional point P η,max,
+which refers to the power consumption level corresponding to
+the peak in the efficiency curve in Figure 1(a). By increasing
+|S| to 4, and then to 8, the mean load value between existing
+points is added, splitting one segment into two. The same
+procedure but only on the right side of P η,max is applied
+when we move from OOS-8 to OOS-12, as this side covers
+1
+6
+12
+18
+24
+0
+20
+40
+60
+Electrolyzer power [MW]
+(a)
+1
+6
+12
+18
+24
+Time [h]
+(b)
+1
+6
+12
+18
+24
+(c)
+Ce
+P , max
+Pmin
+Psb
+pe
+t
+DA
+t
+25
+32
+38
+45
+Day-ahead price [ /MWh]
+Fig. 3.
+The power consumption schedule of the electrolyzer (pe
+t) in an
+example high-wind day when its hydrogen production curve is linearized
+by (a) 1, (b) 4, and (c) 12 segments. These three plots, from left to right,
+correspond to cases OOS-1, OOS-4, and OOS-12, respectively.
+over around 70% of the feasible operating range. With the
+adoption of this procedure, all cases from OOS-2 to OOS-12
+include the point P η,max. In addition, points are not removed
+when refining the discretization. By adding more segments, the
+hydrogen production curve and thus the electrolyzer efficiency
+with partial loading is more accurately represented.
+The increase in the number of segments |S| enables the
+electrolyzer to consume power more flexibly, as depicted in
+Figure 3, where the optimal power consumption schedule of
+the electrolyzer for one example day of the year is shown
+for three different numbers of segments (1, 4, and 12). It is
+observed that when the optimal power consumption of the
+electrolyzer is not constrained by wind production shortage,
+as on the chosen day, the optimal consumption level is always
+one of the piecewise linearization points. There are instances,
+e.g., hour 5 in Figure 3, where OOS-1 goes into the standby
+state as the day-ahead price is too high for profitable hydrogen
+production. In contrast, OOS-4 and OOS-12 continue the
+operation in the on state, but at the power consumption level
+corresponding to the maximum efficiency, where hydrogen
+production is still profitable.
+The number of segments |S| plays an important role in the
+optimal dispatch decision when the day-ahead price lies within
+a specific price range. The upper bound of this price range
+corresponds to the highest price for which the production of
+hydrogen is still profitable. The lower bound is the price below
+which the optimal dispatch decision is always the maximum
+electrolyzer consumption. Figure 4 shows the distribution of
+the day-ahead price λDA
+t
+over 8,760 hours of year 2019 in
+DK2 with the bounds of the price range of interest are marked
+by the red and green dotted lines. The upper bound is found
+as the day-ahead price for which the hydrogen production is
+only feasible at the maximum efficiency, denoted by α in the
+inner plot of Figure 4. The lower bound corresponds to the
+efficiency at the full load, denoted by β. If the day-ahead price
+of a given hour lies outside of this range, the dispatch decision
+for any number of segments would be the same; produce at
+the maximum possible load or cease the production, and there
+would be no added value of a detailed production curve 6.
+This will be further investigated in Section IV-F.
+6These two price thresholds are calculated by multiplying the hydrogen
+price and the efficiency at points β and α, respectively.
+6
+
+25
+30
+35
+40
+45
+50
+55
+Day-ahead price [ /MWh]
+0
+50
+100
+150
+200
+250
+300
+350
+Frequency
+Price
+Price mean
+Power [%]
+Efficiency 
+[kg/MWh]
+Fig. 4. Histogram of the day-ahead electricity price over 8,760 hours of year
+2019 in DK2. Prices λα and λβ correspond to electricity prices for which
+the electrolyzer operates at points α and β, indicated in the inner plot.
+C. Impacts of the States
+We consider three cases OOS, OO, and OS, each for both
+1 and 12 segments. Recall that their corresponding MILPs
+are different7. Comparing the results of MILPs with the same
+number of segments, we observe OS and OOS perform almost
+equally, as observed in Figure 5. The reason for this is the low
+frequency of consecutive hours of too high day-ahead prices,
+where a complete shut-off would be preferred over the standby
+state. Over 8,760 hours, OOS-1 starts up only 2 times, with a
+total of 286 hours offline. The difference in results obtained for
+OS and OOS increases if a higher standby power consumption
+or lower cold start-up cost for the electrolyzer is assumed,
+which would lead to more frequent shut-offs. On the contrary,
+OO earns the lowest profit, mainly due to the high start-up
+cost, which decreases the operational flexibility as even a short
+pause in production incurs a high cost.
+D. Ex-post Performance Analysis
+Recall that three MILPs solve the problem based on the
+linearized hydrogen curve. Through the following ex-post
+performance analysis, it is seen that this leads to both sub-
+optimal dispatch decisions and an underestimation of the true
+amount of hydrogen produced. We have already observed in
+Figure 1(b) that the linearized red curve is below the original
+black non-linear hydrogen production curve, implying that the
+hydrogen production might be underestimated. This means that
+we can expect to produce more hydrogen than what MILPs
+calculate. Such a difference is expected to be reduced by
+using more segments |S| to approximate the original non-
+linear hydrogen production curve.
+Pursuing a fair comparison among models, we conduct an
+ex-post performance analysis. Once the MILPs are solved and
+the optimal power consumption pe∗
+t
+of the electrolyzer ob-
+tained, we re-calculate the true amount of hydrogen produced
+based on the original non-linear hydrogen production curve.
+Note that we do not re-optimize the problem8. We refer to
+the amount of extra hydrogen and its corresponding profit as
+7While we solve the proposed MILP (6)-(29) for OOS, the MILPs
+presented in Appendixes A and B are solved for OO and OS, respectively.
+8To avoid re-optimization, we assume the extra hydrogen is directly sold to
+the demand and is not stored in the hydrogen storage. Otherwise, one needs to
+re-optimize a posteriori to optimize the operation of storage and compressor.
+12
+8
+4
+2
+1
+Number of segments
+97
+98
+99
+100
+Profit [%]
+OOS
+OS
+OO
+States
+15.8
+15.9
+16.1
+16.2
+Profit [million ]
+OOS
+Realized surplus
+1 seg.
+12 seg.
+Fig. 5.
+Estimated and realized surplus profit. The first five bars from the
+left correspond to OOS-1 to OOS-12. The next six bars show the results for
+OO-1, OO-12, OS-1, OS-12, OOS-1, and OOS-12, respectively. The right
+vertical axis is the profit in million C, whereas the left vertical axis is the
+relative profit in % in comparison to the highest profit achieved by OOS-12.
+12
+8
+4
+2
+1
+Number of segments
+80
+85
+90
+95
+100
+H2 production [%]
+OOS
+OS
+OO
+States
+2.3
+2.5
+2.6
+2.8
+2.9
+H2 production [thousand tons]
+OOS
+Realized surplus
+1 seg.
+12 seg.
+Fig. 6. Estimated and realized surplus hydrogen produced. The first five bars
+from the left correspond to OOS-1 to OOS-12. The next six bars show the
+results for OO-1, OO-12, OS-1, OS-12, OOS-1, and OOS-12, respectively.
+“realized surplus”. We assume that all extra hydrogen is sold
+at the same constant price, i.e., C2.10/kg.
+Figure 5 provides the estimated and realized surplus profit
+among different cases. The estimated profit (gray area) is
+the optimal value obtained for the objective function of the
+corresponding MILP, while the realized profit (dark area),
+calculated ex-post, takes into account the profit of selling extra
+hydrogen. Similarly, Figure 6 shows the total estimated and
+realized surplus hydrogen produced. Note that the compressor
+would need to consume more power (around 1 MWh/ton) due
+to extra hydrogen. We draw two conclusions from Figures 5
+and 6:
+(1) Realized surplus: This surplus for profit and hydrogen
+production is reduced by increasing the number of segments,
+due to the improved approximation of the original non-linear
+curve. The realized surplus profit decreases from C71,199
+(0.44%) for OOS-1 to C602 (below 0.01%) for OOS-12.
+Similarly, the hydrogen production surplus is significantly
+decreased, yielding a realized surplus of ∼ 34 tons (1.27%) for
+OOS-1 and only 0.3 tons (0.01%) for OOS-12. By choosing
+a low number of segments, the hydrogen production is under-
+estimated which may lead to logistic issues and inefficiencies
+in the real-life operation of the hybrid power plant.
+(2) Ex-post profit and hydrogen production: Adding more
+electrolyzer details (segments or/and states) always leads to
+an increase in the ex-post profit. To compare various models,
+7
+
+TABLE II
+COMPUTATIONAL ASPECTS
+Case
+Computational time [s]
+No. of binary variables
+OS-1
+1.4
+2×8760
+OS-12
+12.7
+13×8760
+OOS-1
+137.8
+5×8760
+OOS-2
+135.8
+6×8760
+OOS-4
+236.3
+8×8760
+OOS-8
+350.3
+12×8760
+OOS-12
+473.7
+16×8760
+OO-1
+767.1
+3×8760
+OO-12
+1,763.1
+14×8760
+OOS-12 is taken as a benchmark, as it leads to the highest
+profit. First, the impact of the number of segments is examined,
+while keeping the number of states fixed and equal to 3. The
+ex-post profit reduction applying 1 instead of 12 segments is
+0.72%, corresponding to around 117.6 kC for the entire hybrid
+power plant. The ex-post hydrogen production is increased by
+8.32%, corresponding to around 241 tons. This percentage
+deviation is notably higher in part because the increase in
+hydrogen profit is dampened by the reduction in electric-
+ity profit (3.86% electricity profit increase for 1 segment
+compared to 12 segments). For OOS-1, the profit share of
+selling hydrogen is much lower than the profit share of selling
+electricity (around 34%). By introducing more segments, the
+contribution of hydrogen sales is increased to 38% at the
+expense of electricity sales. More profit and different business
+models are therefore unlocked by including more electrolyzer
+details in the MILP formulation. Figures 5 and 6 show that the
+errors are considerably reduced by implementing 4 segments
+instead of 1. Second, we assess the impact of the states on the
+ex-post profit and hydrogen production. While OS performs
+just as well as OOS as described in Section IV-C, OO with 12
+segments results in a 1.22% lower ex-post profit, and in a 4%
+lower hydrogen production. For OO-1, a profit reduction of
+around 1.8% and a reduced hydrogen production of 13.5% are
+observed, compared to the benchmark. Finally, we observe that
+neglecting the standby state in the model formulation leads to
+the worst outcome in terms of profit and hydrogen production
+potential.
+E. Computational Analysis
+All MILPs have been solved using the Gurobi solver in Julia
+on a MacBook Pro M1 2020 with 16 GB RAM. The optimality
+gap is fixed to 0.01% when we solve every MILP. The
+increase in the number of linearization segments |S| leads to an
+increase in computational time due to introducing more binary
+variables. For OOS, the computational time is increased from
+138 seconds for 1 segment to 474 seconds for 12 segments,
+as reported in Table II. Removing the off state significantly
+reduces the computational time, with OS-1 being by far the
+fastest MILP to be solved (1.4 seconds). The OO models
+require the highest computational time, although they embody
+fewer binary variables than their OS and OOS counterparts.
+We hypothesize the reason is that the start-up cost constraints
+with inter-temporal nature are more often active when the
+option of standby state is not present. Therefore, we do
+not recommend using OO as its corresponding profit is the
+lowest among all cases (Figure 5), and it is being solved
+comparatively slower. Further, if computational efficiency is
+crucial, it may be beneficial to neglect the off state and run
+the OS model for improved computational performances. In
+general, the computational time increases with the number
+of segments but is deemed reasonable for the OS and OOS
+models, considering that our optimization problem is run over
+8,760 hours. As operational problems are typically solved
+for a shorter time horizon, e.g., 24 hours for day-ahead
+scheduling, the computational cost of adding more details to
+the electrolyzer would be minimal.
+F. Sensitivity Analysis with Respect to Input Data
+In the previous sections, we have shown that adopting a
+simplified electrolyzer model can lead to an underestimation
+of the profit and hydrogen production for the hybrid power
+plant. We have also shown that the benefit of added details is
+case-specific, and depends on the input parameters. We now
+aim at assessing the impact of input parameters and system
+configuration on these results, through a sensitivity analysis.
+In particular, we will focus on wind over electrolyzer capacity
+ratio, hydrogen demand over electrolyzer capacity ratio, and
+the hydrogen price. The sensitivity analysis is performed on
+the OOS-1 and OOS-12 models.
+1) Wind size: Recall from Table I that the wind farm
+capacity is 2 times that of the electrolyzer. To assess the impact
+of the wind-to-electrolyzer capacity ratio, two additional cases
+are considered, under which such a ratio is 1, 2 (reference),
+and 8. When this ratio is reduced from 2 to 1, the number
+of hours where the power input to the electrolyzer is limited
+by the wind availability is increased from 5,326 to all hours.
+Conversely, when the ratio is increased from 2 to 8, the number
+of power-limited hours is reduced to 1,236. We observe that
+the realized surplus for hydrogen production increases with
+the number of hours with limited wind power. The reason for
+this is that the piecewise approximation is exact only on the
+linearization points, and the limited wind availability forces
+the electrolyzer to operate out of those points. Conversely,
+when the number of wind power-limited hours is reduced,
+the electrolyzer operates more often on the linearization
+points, where the approximation is exact. It follows that the
+underestimation of hydrogen production is greater the more
+the electrolyzer is limited from operating at the linearization
+points. With a wind-to-electrolyzer ratio of 1, the difference
+in ex-post hydrogen production between 1 and 12 segments
+is 13%, which is reduced to 3% when the ratio increases to
+8. Therefore, incorporating electrolyzer details is crucial for
+hybrid power plants where the wind-to-electrolyzer capacity
+ratio is small.
+2) Hydrogen demand size: To investigate the sensitivity of
+optimization outcomes with respect to the hydrogen demand,
+the minimum daily demand is doubled, corresponding to
+around 8 full-load hours of hydrogen production. We observe
+that the impact of adding more segments to the electrolyzer
+8
+
+production curve diminishes when the demand constraint is
+tighter, i.e., with a higher minimum daily demand. For the case
+with the reference demand, the difference between the ex-post
+profit for OOS-12 and OOS-1 is 8%. This difference, when the
+hydrogen demand is doubled, is reduced to 2%. The increase
+in demand forces the electrolyzer to operate more frequently
+at its maximum load, where both OOS-1 and OOS-12 share
+the same linearization point and efficiency.
+3) Hydrogen price: To explore the impact of the hydrogen
+price, we increase it from C2.10/kg to C5.00/kg. As already
+discussed in Section IV-B, adding more segments impact
+the optimal solution and profit as long as the electricity
+price in the given hour is in the range [λβ, λα], shown in
+Figure 4. Since λα and λβ are proportional to the hydrogen
+price, by increasing the hydrogen price, the range [λβ, λα] is
+widened and moved towards higher electricity prices, where
+the frequency of occurrence is reduced. When the MILP is
+solved with the hydrogen price of C5/kg, it is more frequently
+optimal to operate the electrolyzer at full load (39% of the
+time, compared to 11% for the case with the hydrogen price of
+C2.1/kg) and the linearization segments are utilized less. This
+also results in a significantly decreased computational time
+(below 20 seconds for OOS-12). The profit contribution from
+the hydrogen sale is increased significantly to 92%. The ex-
+post profit and hydrogen production difference between OOS-
+1 and OOS-12 are reduced to 0.01% and 0.03%, respectively
+(they are 0.72% and 8.32% for the C2.1/kg case).
+The modeling of segments is relevant if higher hydrogen
+prices are coupled with also higher electricity prices. In
+this way, the electricity price range [λβ, λα] would still be
+overlapping with the majority of day-ahead price occurrences.
+For example, we test an artificial case where the day-ahead
+electricity price time series was multiplied by a constant factor
+to increase the mean price to around C90/MWh (similar to
+the mean value for 2021 in DK2). In this case, with the
+hydrogen price of C5/kg, similar results to the 2019 test case
+with the hydrogen price of C2.1/kg were obtained in terms of
+the impact of the number of segments. For a given hydrogen
+price and efficiency curve, checking if the price range [λβ, λα]
+overlaps with the expected electricity price is therefore crucial
+to assess a priori the impact of choosing a simplified model
+for the production curve (e.g., 1 linearization segment only)
+and support the modeling choices.
+V. DISCUSSION AND CONCLUSION
+Several studies have focused on the optimal dispatch of
+hybrid renewable-hydrogen power plants assuming simplified
+models for the electrolyzer component. This paper investigates
+the impact of choosing different levels of operational details
+for the electrolyzer model on the dispatch decisions, profit, the
+amount of hydrogen produced, and computational time. The
+impact of two modeling choices is considered: the operating
+states (on, off, standby), and the number of segments used
+to linearize the hydrogen production curve. The problems are
+formulated as MILPs, where the number of binary variables
+depends on the number of states and segments.
+For fixed states, adding more linearization segments for
+approximating the hydrogen production curve results in a
+higher profit, and a reduced surplus in the ex-post profit
+calculation, meaning that the model is able to estimate the
+actual cost and revenue streams more accurately. Moreover,
+a better estimation of the produced hydrogen is achieved.
+In fact, the linearization results in an underestimation of the
+produced hydrogen, but the underestimation is reduced by
+increasing the number of segments. Apart from introducing
+errors in the actual realized profit, thus potentially impacting
+the investment decisions in these types of technologies, the
+systematic underestimation of the hydrogen produced by the
+electrolyzer might introduce logistical inefficiencies, e.g., truck
+scheduling, and storage discharging/filling.
+The impact of adding more piecewise segments to the
+hydrogen production curve depends on the distribution of day-
+ahead electricity prices in the given time horizon. The model
+formulations with 1 and 12 segments take significantly dif-
+ferent dispatch decisions when the day-ahead electricity price
+is within a certain range, which depends on the electrolyzer
+efficiency (minimum and maximum) and the hydrogen price.
+Out of this day-ahead electricity price range, the model with 1
+and 12 segments takes the same dispatch decisions. Therefore,
+the value of adding more details to the hydrogen production
+curve could differ by varying input data and case studies. It
+is observed that this value decreases when the electrolyzer
+operates less at partial loading, e.g. when the input power is
+less limited by available wind power or with high-demand
+constraints. In this paper, revenues from other than the day-
+ahead market are not considered but this may also impact the
+dispatch strategy and therefore benefit from more segments.
+Choosing to represent only on and off states leads to the
+highest profit underestimation and worst ex-post performance
+while modeling only on and standby states lead to similar
+profit and dispatch decisions to the three-state model. This
+result is, however, significantly affected by the assumption
+made on the standby power consumption of the electrolyzer
+and its start-up cost. These parameters are highly uncertain
+due to the lack of data on large-scale electrolyzers.
+In conclusion, adopting more simplified models for the
+electrolyzer always leads to a reduced profit and sub-optimal
+scheduling. However, the impact of adding more details may
+vary depending on the case study considered and especially
+the range of day-ahead electricity prices, hydrogen price, wind
+power production compared to the electrolyzer installed capac-
+ity, standby power consumption, and start-up cost. Among all
+considered models, the most complete one (three states with
+12 segments) was solved for a 1-year horizon in less than
+10 minutes. The increase in computational time by adding
+more details would be marginal if a day-ahead scheduling
+problem is considered instead. Moreover, reducing the three-
+state model to two states only is not always faster, as it was
+observed that the two-state on-off model with 12 segments
+was the longest to solve among all the cases considered. A
+more detailed representation of the electrolyzers should be
+preferred for operational problems. For investment problems,
+9
+
+we hypothesize that it may be adequate to adopt a more
+simplified model of the electrolyzer, but this should be further
+assessed and it was out of the scope of the current paper.
+Further research should be conducted to assess the impact of
+modeling choices when additional revenue streams are consid-
+ered, such as flexibility provisions in ancillary service markets,
+which may impact the dispatch decisions of the hybrid power
+plant. Additionally, as there is a high uncertainty related to the
+start-up and standby costs, the sensitivity of these parameters
+on the impact of added details should be assessed further.
+Moreover, the level of detail needed for investment problems
+should be further investigated. The modeling of electrolyzer
+cell degradation over time should be investigated and included
+in the model with additional constraints. Finally, uncertainties
+in wind power supply and electricity prices should be included.
+ACKNOWLEDGEMENT
+This research was supported by the Energy Cluster Den-
+mark through the “Sustainable P2X Business Model” project,
+and by the Danish Energy Development Programme (EUDP)
+through the HOMEY project (64021-7010). We would like to
+thank Jens Jakob Sørensen (Ørsted), Alexander Holm Kiilerich
+(Ørsted), Roar Hestbek Nicolaisen (Hybrid Greentech), Yan-
+nick Werner (DTU), and Matˇej Novotn´y for collaborations,
+thoughtful discussions, and constructive feedback.
+APPENDIX
+A. The simplified MILP with On-Off States
+This appendix provides the MILP (A.30a), where the on
+and off states of the electrolyzer are only modeled. This is a
+simplified model compared to the one proposed in Section III
+with three states of the electrolyzer.
+max
+Ω
+�
+t∈T
+ptλDA
+t
++ dtλh − zsu
+t λsu
+(A.30a)
+s.t.
+pt = P w
+t − pe
+t − pc
+t
+∀ t ∈ T ,
+(A.30b)
+pe
+t ≤ Cezoo
+t
+∀ t ∈ T ,
+(A.30c)
+pe
+t ≥ P minzoo
+t
+∀ t ∈ T ,
+(A.30d)
+zsu
+t
+≥ zoo
+t
+− zoo
+t−1
+∀ t ∈ T \1,
+(A.30e)
+zoo
+t
+=
+�
+s∈S
+zh
+t,s
+∀ t ∈ T ,
+(A.30f)
+pe
+t =
+�
+s∈S
+ˆpe
+ts
+∀ t ∈ T ,
+(A.30g)
+(13), (15) − (16), (19) − (26),
+(A.30h)
+dt, ht, hd
+t , pt, pc
+t, ˆpe
+ts, st, sin
+t , sout
+t
+∈ R+,
+(A.30i)
+zsu
+t , zh
+ts, zoo
+t
+∈ {0, 1},
+(A.30j)
+Ω = {dt, ht, hd
+t , pt, pc
+t, ˆpe
+ts, sin
+t , sout
+t
+, zsu
+t , zh
+ts, zoo
+t }.
+(A.30k)
+B. The simplified MILP with On-Standby States
+This appendix presents the simplified MILP (A.31), taking
+into account on and standby states of the electrolyzer.
+max
+Γ
+�
+t∈T
+ptλDA
+t
++ dtλh − pin
+t λin
+t
+(A.31a)
+s.t.
+pin
+t ≤ P sb(1 − zos
+t )
+∀ t ∈ T ,
+(A.31b)
+pe
+t ≤ Cezos
+t + P sb(1 − zos
+t )
+∀ t ∈ T ,
+(A.31c)
+pe
+t ≥ P minzos
+t + P sb(1 − zos
+t )
+∀ t ∈ T ,
+(A.31d)
+zos
+t =
+�
+s∈S
+zh
+t,s
+∀ t ∈ T ,
+(A.31e)
+pe
+t =
+�
+s∈S
+ˆpe
+ts + P sb(1 − zos
+t )
+∀ t ∈ T ,
+(A.31f)
+(7), (15) − (16), (19) − (26),
+(A.31g)
+dt, ht, hd
+t , pt, pc
+t, pin
+t , ˆpe
+ts, st, sin
+t , sout
+t
+∈ R+,
+(A.31h)
+zh
+ts, zos
+t ∈ {0, 1},
+(A.31i)
+Γ = {dt, ht, hd
+t , pt, pc
+t, pin
+t , ˆpe
+ts, sin
+t , st, sout
+t
+, zsu
+t , zos
+t }.
+(A.31j)
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+10
+

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+arXiv:2301.04848v1  [math.FA]  12 Jan 2023
+τ-QUANTIZATION AND τ-COHEN CLASSES DISTRIBUTIONS
+OF FEICHTINGER OPERATORS
+FEDERICO BASTIANONI AND FRANZ LUEF
+Abstract. We investigate the τ-quantizations and Cohen’s class distributions
+of a suitable class of trace-class operators, called Feichtinger’s operators, and
+show that it is a convenient substitute for the class of Schwartz operators. Many
+well-known concepts and results for functions in time-frequency analysis have an
+operator-analog in our setting, e.g. that Cohen’s classes are convolutions of Wigner
+functions with distributions or characterization of the class of Schwartz operators
+as an intersection of weighted variants of the class of Feichtinger operators.
+Contents
+1.
+Introduction
+1
+2.
+Preliminaries
+3
+2.1.
+A family of time-frequency representations
+3
+2.2.
+Basics of QHA and novel tools
+5
+2.3.
+τ-quantization of functions
+6
+3.
+Feichtinger operators
+7
+3.1.
+τ-quantization of operators
+9
+3.2.
+A convenient environment for QHA
+13
+3.3.
+τ-Cohen’s class of operators
+23
+4.
+A characterization of Schwartz operators
+28
+Acknowledgments
+30
+References
+30
+1. Introduction
+There is a vast literature on the boundedness of pseudodifferential operators
+for certain classes of symbols in various quantization schemes along the lines of
+H¨ormander classes or alternatively using Sj¨ostrand’s class or Shubin’s classes, e.g.
+2010 Mathematics Subject Classification. 42B35;46E35;47G30;47B10.
+Key
+words
+and
+phrases. Cohen’s
+class,
+τ-quantization,
+Feichtinger’s
+algebra,
+Wigner
+distribution.
+1
+
+2
+FEDERICO BASTIANONI AND FRANZ LUEF
+[1, 3, 4, 11, 22]. In the present work, we put our focus on Shubin’s τ-quantization
+and the associated time-frequency representations, the τ-Cohen classes.
+Our approach to this circle of ideas is based on the framework of quantum har-
+monic analysis with the goal to lift the well-known results concerning functions to
+an appropriate class of functions, which we call Feichtinger operators, S0, and which
+is the operator analog of the well-known Feichtinger algebra S0.
+We also discuss the relation between Feichtinger operators S0 and the class of
+Schwartz operators introduced by Keyl, Kiukas and Werner in [14]. There the idea
+is put forward that one should look for analogs of function spaces in the setting of
+classes of operators, which has been realized in the case of Sobolev spaces in [15]
+and for modulation spaces in [6].
+For τ ∈ [0, 1] the τ-quantization of a symbol a ∈ S′(R2d), the space of tempered
+distributions, is given by
+(1)
+Opτ(a)f(t) :=
+�
+R2d e2πi(t−y)ξa((1 − τ)t + τy, ξ)f(y) dydξ f ∈ S(Rd),
+where the operator Opτ(a) is understood to be defined in the weak sense. A well-
+known fact is that one can relate ⟨Opτ(a)f, g⟩ to a time-frequency representation,
+Wτ(f, g), the cross-τ-Wigner distribution of f and g:
+⟨Opτ(a)f, g⟩ = ⟨a, Wτ(g, f)⟩,
+for all
+f, g ∈ S(Rd).
+Given an operator S, we denote by aS
+τ its τ-symbol, i.e. the tempered distribution
+such that Opτ
+�
+aS
+τ
+�
+= S and Opτ is called the τ-Shubin quantization. For f, g ∈
+L2(Rd) we denote the rank-one operator by f ⊗ g and note that af⊗g
+τ
+= Wτ(g, f),
+i.e. there is an intrinsic relation between quantization schemes and time-frequency
+representations.
+We show that for well-behaved operators, e.g. trace class operators or Feichtinger
+operators, this relation might be extended to operators.
+Recall that Wigner in
+his ground-breaking work on quasi-probability distributions introduced the cross-
+Wigner distribution for certain classes of operators [24], which was later extended
+to more general classes of operators by Moyal in [19].
+Let S be a continuous operator between the Feichtinger algebra S0 and its contin-
+uous dual space S′
+0. We denote by KS the kernel of S, which exists by Feichtinger’s
+kernel theorem and is a mild distribution on R2d.
+We define Feichtinger operators, S0, to be the following class of continuous and
+linear operators S0 := S : S′
+0(Rd) → S0(Rd) that map norm bounded w-∗ convergent
+sequences in S′
+0 into norm convergent sequences in S0. In [10] it was shown that
+these are precisely the linear continuous operators from S′
+0 to S0 that have a kernel
+in Feichtinger’s algebra, the so-called inner kernel theorem.
+One of our main tools is that Feichtinger operators have a nice spectral decomposi-
+tion. If S is in S0, then there exist two (non-unique) sequences {fn}n, {gn}n ⊆ S0(Rd)
+
+τ-QUANTIZATION AND τ-COHEN CLASSES OF FEICHTINGER OPERATORS
+3
+such that
+S =
+∞
+�
+n=1
+fn ⊗ gn,
+∞
+�
+n=1
+∥fn∥S0 ∥gn∥S0 < ∞,
+KS =
+∞
+�
+n=1
+Kfn⊗gn.
+Hence, Feichtinger operators are trace class operators and we can compute their trace
+as follows tr(S) =
+�
+Rd KS(x, x) dx. In [7] operators having such a decomposition
+have been studied and called Feichtinger states in case tr(S) = 1, but there the link
+between these operators and the work [10] was not established, which is one of our
+main observations.
+Then the τ-Wigner distribution of S is defined in the following way
+(2)
+WτS(x, ω) :=
+�
+Rd e−2πitωKS(x + τt, x − (1 − τ)t) dt.
+Our key observation is the following identity:
+⟨a,WτS⟩ = tr(Opτ(a)S∗) =: ⟨Opτ(a),S⟩,
+for S in S0 or J 1, and WτS is the τ-Wigner distribution of S. Consequently, we
+interpret WτS as the τ-quantization of an operator in S0 or J 1.
+Note, that if S is the rank-one operator f ⊗ g this becomes the aforementioned
+relation between the τ-Wigner distribution and the Shubin τ-transform.
+Based on this framework we deduce operator analogs of well-known results on
+τ-Wigner distributions and τ-Shubin quantization, which indicates that this is a
+very convenient setting for this type of investigation. In addition, we extend the
+Cohen class of an operator, introduced in [17], to the τ-setting and show that it
+can be written as the convolution of the Wigner distribution of an operator with a
+distribution as in the function setting.
+We close our discussion with the introduction of weighted versions of S0 and prove
+that the intersection of all these is the class of Schwartz operators in [14]. As in the
+case of functions, we hope that this global description of the Schwartz operators will
+also turn out to be useful in subsequent studies and it also hints at operator analogs
+of Gelfand-Shilov classes or other classes of test functions and the corresponding
+class of ultradistributions.
+2. Preliminaries
+In this paper, the parameter τ always belongs to [0, 1], even when not specified.
+2.1. A family of time-frequency representations. For x, ω ∈ Rd we define the
+translation and modulation operator by
+Txf(t) := f(t − x),
+Mωf(t) := e2πiωtf(t),
+∀t ∈ Rd,
+respectively. Their composition is denoted by π(x, ω) := MωTx.
+
+4
+FEDERICO BASTIANONI AND FRANZ LUEF
+Given τ ∈ [0, 1], the τ-time-frequency shift (τ-TFS) at (x, ω) ∈ R2d is defined to
+be
+(3)
+πτ(x, ω) := e−2πiτxωMωTx = M(1−τ)ωTxMτω.
+For τ = 0 we recover the usual time-frequency shifts π0 = π. The following relations
+are consequences of elementary computations, which are left to the reader:
+πτ(x, ω)πτ(x′, ω′) = e−2πi[(1−τ)xω′−τx′ω]πτ(x + x′, ω + ω′),
+πτ(x, ω)πτ(x′, ω′) = e−2πi[xω′−x′ω]πτ(x′, ω′)πτ(x, ω),
+πτ(x, ω)∗ = π1−τ(−x, −ω) = e−2πi(1−τ)xωπ(−x, −ω).
+In the present paper the symbol ⟨·,·⟩ either denotes the inner product in L2(Rd)
+or a duality pairing between a Banach space X and its dual space X′, which is
+compatible with the latter, i.e. ⟨·,·⟩ is assumed to be linear in the first argument
+and conjugate-linear in the second one. In particular, the dual pairs considered in
+this work are (L2, L2), (S′
+0, S0), (S′
+0, S0), respectively.
+Above, S0 is the Feichtinger algebra (24), for the definitions of S0 and S′
+0 see the
+equations (25),(26) and (28). We introduce for f, g ∈ L2(Rd), or for any suitable
+dual pair, the τ-short-time Fourier transform (τ-STFT) of f w.r.t g:
+(4)
+V τ
+g f(x, ω) := ⟨f, πτ(x, ω)g⟩,
+∀x, ω ∈ Rd.
+As can be easily verified, the mapping
+πτ : R2d → U(L2(Rd)),
+where U(L2(Rd)) denotes the unitary operators on L2(Rd), is a projective represen-
+tation of R2d for any τ. Consequently, V τ is the wavelet transform associated to πτ,
+thus V τ
+g f is a continuous function.
+Remark 2.1. For τ = 0 we obtain the usual STFT V 0
+g f = Vgf and we have
+(5)
+V τ
+g f(x, ω) = e2πiτxωVgf(x, ω).
+By the preceding identity, we have that V
+1
+2
+g f is the cross-ambiguity function of f and
+g:
+(6)
+V
+1
+2
+g f(x, ω) = A(f, g)(x, ω).
+We recall another frequently used time-frequency representation, the so-called
+cross-τ-Wigner distribution of f and g in L2(Rd) defined by
+(7)
+Wτ(f, g)(x, ω) :=
+�
+Rd e−2πitωf(x + τt)g(x − (1 − τ)t) dt.
+We aim to extend the definition of Wτ from functions to operators, see (15).
+
+τ-QUANTIZATION AND τ-COHEN CLASSES OF FEICHTINGER OPERATORS
+5
+2.2. Basics of QHA and novel tools. In this subsection we introduce the ba-
+sic definitions of quantum harmonic analysis (QHA) following the seminal work of
+Werner [23].
+For z ∈ R2d and A ∈ B(L2(Rd)) the translation of the operator A by z is
+(8)
+αz(A) := π(z)Aπ(z)∗,
+which satisfies αzαz′ = αz+z′. By the parity operator, we mean
+(9)
+Pf(t) := ˇf(t) := f(−t),
+for any f ∈ L2(Rd), which induces an involution of A ∈ B(L2(Rd)):
+(10)
+ˇA := PAP.
+We denote by J 1 the space of all trace class operators on L2(Rd). Given a ∈ L1(R2d)
+and S ∈ J 1. The convolution between a and S is the operator
+(11)
+a ⋆ S := S ⋆ a :=
+�
+R2d a(z)αz(S) dz,
+were the integral may be interpreted in the weak sense. For operators S, T ∈ J 1,
+their convolution is the function defined for every z ∈ R2d as
+(12)
+S ⋆ T(z) := tr
+�
+Sαz( ˇT)
+�
+.
+In this paper, we reserve the symbol ⊗ for rank-one operators.
+Namely, given
+f, g ∈ L2(Rd):
+(13)
+(f ⊗ g)ψ := ⟨ψ, g⟩f,
+∀ψ ∈ L2(Rd).
+The kernel of an operator S will always be denoted by KS. Evidently, the kernel of
+the operator f ⊗ g is the tensor product of functions f(x)g(y):
+(f ⊗ g)ψ(t) = ⟨ψ, g⟩f(t) =
+�
+Rd f(t)g(x)ψ(x) dx.
+In the sequel we denote the tensor product of two functions by f(x)g(y), we shall
+adopt the notation
+(14)
+Kf⊗g(x, y) = f(x)g(y).
+We now interpret (7) as the cross-τ-Wigner distribution of the rank-one operator
+f ⊗ g.
+Hence, it is natural to define the τ-Wigner distribution of an operator S with kernel
+KS in the following way:
+(15)
+WτS(x, ω) :=
+�
+Rd e−2πitωKS(x + τt, x − (1 − τ)t) dt.
+For S ∈ J 1 and τ ∈ [0, 1], we define the Fourier-τ-Wigner transform of S to be:
+(16)
+FWτS(z) := tr (πτ(z)∗S) ,
+∀z ∈ R2d.
+
+6
+FEDERICO BASTIANONI AND FRANZ LUEF
+For τ = 1/2 we recover the usual Fourier-Wigner transform [23].
+The τ-spreading representation of S ∈ B(L2) is the decomposition
+(17)
+S =
+�
+R2d h(z)πτ(z) dz,
+where the integral is understood in the weak sense. The function h is called the
+τ-spreading function of S.
+In the following, we shall consider the τ-spreading representation as a quantization
+scheme that assigns to a function an operator. Namely, h ∈ L1(R2d) gets associated
+to the operator
+(18)
+SRτ(h) :=
+�
+R2d h(z)πτ(z) dz.
+Let Fσ denote the symplectic Fourier transform. In the following lemma we collect
+a number of important relations between these notions. The proofs are elementary
+computations and based on the spectral decomposition of the trace class operators
+S and T ([21]), which we leave to the interested reader.
+Lemma 2.2. Let f, g, ∈ L2(Rd), S, T ∈ J 1, a ∈ L1(R2d) and τ ∈ [0, 1]. Then:
+(i) Fσ(Wτ(f ⊗ g)) = V τ
+g f;
+(ii) FWτ(f ⊗ g) = V τ
+g f;
+(iii) WτS = FσFWτS;
+(iv) FWτS(x, ω) = e−2πi(1/2−τ)xωFW1/2S(x, ω);
+(v) Fσ(S ⋆ T) = FWτS · FW1−τT = FW1−τS · FWτT;
+(vi) FWτ(a ⋆ S) = Fσa · FWτS;
+(vii) FWτS is the τ-spreading function of S, i.e. S =
+�
+R2d FWτS(z)πτ(z) dz.
+We notice that if we consider the rank-one operator S = f ⊗g, then the assertions
+(iii) and (ii) of the previous lemma imply
+(19)
+Wτ(f, g) = Wτ(f ⊗ g) = FσV τ
+g f.
+2.3. τ-quantization of functions. The τ-quantization of a symbol a ∈ S′(R2d),
+the space of tempered distributions, is formally given by
+(20)
+Opτ(a)f(t) :=
+�
+R2d e2πi(t−y)ξa((1 − τ)t + τy, ξ)f(y) dydξ,
+where f ∈ S(Rd). Opτ(a) may be described rigorously in the weak sense:
+⟨Opτ(a)f, g⟩ = ⟨a, Wτ(g, f)⟩,
+∀f, g, ∈ S(Rd).
+Given an operator S, we denote by aS
+τ its τ-symbol, i.e. the tempered distribution
+such that
+Opτ
+�
+aS
+τ
+�
+= S.
+
+τ-QUANTIZATION AND τ-COHEN CLASSES OF FEICHTINGER OPERATORS
+7
+Remark 2.3. Under suitable assumptions, for example a ∈ L1(R2d), straightforward
+calculations give
+Opτ(a) =
+�
+R2d Fσa(z)πτ(z) dz,
+and since also FWτ Opτ(a) is the τ-spreading function of Opτ(a), we have
+(21)
+a = FσFWτ Opτ(a).
+Hence, for S ∈ J 1
+(22)
+aS
+τ = FσFWτS = WτS.
+Given a ∈ S′
+0(R2d) and f, g ∈ S0(Rd), we recall the definition of cross-τ-Cohen’s
+class representation of f and g, with kernel a:
+(23)
+Qτ
+a(f, g) := a ∗ Wτ(f, g).
+3. Feichtinger operators
+In this section we summarize some important results concerning a class of op-
+erators studied in [10]. For such operators, introduced below, we adopt the name
+“Feichtinger operators” for reasons which will become evident later.
+We recall that the Feichtinger algebra over Rd [9] is the Banach space
+(24)
+S0(Rd) := {f ∈ L2(Rd) | Vgf ∈ L1(R2d)},
+for some g ∈ L2(Rd) ∖ {0}, endowed with the norm
+∥f∥S0 := ∥Vgf∥L1 =
+�
+R2d |Vgf(x, ω)| dxdω.
+We refer the reader to [13] for a detailed survey on S0(Rd). In this work, S′
+0(Rd)
+denotes the conjugate-dual of S0(Rd).
+Definition 3.1. The set of Feichtinger operators is defined to be
+S0 :={S : S′
+0(Rd) → S0(Rd) | S is linear, continuous and
+maps norm bounded w-∗ convergent sequences in S′
+0
+(25)
+into norm convergent sequences in S0}.
+We adopt the following notation:
+(26)
+S′
+0 := B(S0(Rd), S′
+0(Rd))
+and state the so called Outer Kernel Theorem [10, Theorem 1.1]:
+Theorem 3.2. The Banach space S′
+0 is isomorphic to S′
+0(R2d) via the map T �→ KT,
+where the relation between T and its kernel KT is given by
+⟨Tf,g⟩ = ⟨KT,Kg⊗f⟩,
+∀ f, g, ∈ S0(Rd).
+
+8
+FEDERICO BASTIANONI AND FRANZ LUEF
+The following statement goes under the name of Inner Kernel Theorem.
+We
+present it in our setting. To this end, we introduce the following notation: given
+σ, ν ∈ S′
+0(Rd), we denote by ν �⊗σ the unique element of S′
+0(R2d) such that
+⟨ν �⊗σ,Kψ⊗ϕ⟩ = ⟨ν,ψ⟩⟨σ,ϕ⟩,
+∀ ψ, ϕ ∈ S0(Rd).
+We refer the reader to [10, Theorem 1.3], Lemma 3.1 and Corollary 3.10, too.
+Theorem 3.3. The space of Feichtinger operators S0 is a Banach space if endowed
+with the norm of B(S′
+0, S0) and it is naturally isomorphic as Banach space to S0(R2d)
+through the map T �→ KT, where the relation between T and its kernel KT is given
+by
+⟨ν,Tσ⟩ = ⟨ν �⊗σ,KT⟩,
+∀ σ, ν, ∈ S′
+0(Rd).
+Moreover, S0 is Banach algebra under composition. If S, T ∈ S0, then
+(27)
+KS◦T(y, u) =
+�
+Rd KT(y, t)KS(t, u) dt.
+By the above theorems 3.2 and 3.3, S′
+0 is the (conjugate) topological dual of S0
+and the duality is given by
+(28)
+⟨T,S⟩ = ⟨KT,KS⟩.
+Lemma 3.4. Suppose S ∈ S0. Then there exist two non-unique sequences {fn}n, {gn}n ⊆
+S0(Rd) such that
+S =
+∞
+�
+n=1
+fn ⊗ gn,
+∞
+�
+n=1
+∥fn∥S0 ∥gn∥S0 < +∞,
+KS =
+∞
+�
+n=1
+Kfn⊗gn.
+Moreover,
+S0 ֒→ J 1
+with
+tr(S) =
+�
+Rd KS(x, x) dx.
+Proof. We just have to prove the continuous inclusion of Feichtinger operators into
+J 1, all the remaining statements can be found in [10], see in particular Corollary
+3.15 and Remark 9. The claim follows from an elementary computation:
+∥S∥J 1 = |tr(A)| ≤
+�
+Rd
+∞
+�
+n=1
+|fn(x)gn(x)| dx =
+∞
+�
+n=1
+�
+Rd |fn(x)gn(x)| dx
+≤
+∞
+�
+n=1
+∥fn∥L2 ∥gn∥L2 ≲
+∞
+�
+n=1
+∥fn∥S0 ∥gn∥S0 < ∞.
+Since S0(R2d) = S0(Rd)ˆ⊗S0(Rd), see e.g. [10, Lemma 2.1], we get
+∥S∥J 1 ≲ ∥KS∥S0 ≍ ∥S∥S0 ,
+
+τ-QUANTIZATION AND τ-COHEN CLASSES OF FEICHTINGER OPERATORS
+9
+which gives the desired assertion.
+□
+The preceding result and the observations in [10, p. 4] yield
+(29)
+S0 ֒→ J 1 ֒→ J 2 ֒→ B(L2(Rd)) ֒→ S′
+0.
+The fact that all Feichtinger operators are trace class implies the validity of Lemma
+2.2.
+3.1. τ-quantization of operators. The following remark is the key insight for the
+subsequent results concerning Opτ and Wτ.
+Remark 3.5. Let us consider f, g ∈ L2(Rd) such that f ̸= 0, a ∈ L2(R2d) and {fj}j
+o.n.b. for L2 with f1 = f. Then we compute as follows:
+⟨Opτ(a)f, g⟩ = ⟨Opτ(a)f,
+∞
+�
+j=1
+⟨g, fj⟩fj⟩ =
+∞
+�
+j=1
+⟨Opτ(a) (⟨fj, g⟩f) , fj⟩
+=
+∞
+�
+j=1
+⟨Opτ(a)(f ⊗ g)fj, fj⟩ = tr (Opτ(a)(f ⊗ g)) .
+Taking into account the weak definition of Opτ(a) and (15) we can write
+(30)
+⟨Opτ(a)f, g⟩ = ⟨a, Wτ((f ⊗ g)∗)⟩ = tr (Opτ(a)(f ⊗ g)) = ⟨Opτ(a), (f ⊗ g)∗⟩(J 1,J ∞).
+By computations similar to the ones above for S ∈ J 1 with the spectral decomposition
+�∞
+k=1 λkfk ⊗ gk after extending {fk}k to an orthonormal basis of L2(Rd) implies
+(31)
+⟨a, WτS⟩ = tr (Opτ(a)S∗) = ⟨Opτ(a), S⟩(J 1,J ∞).
+Theorem 3.6. For every τ ∈ [0, 1] the following mappings are linear and continu-
+ous:
+Opτ : L2(R2d) → J ∞,
+Wτ : J 1 → L2(R2d).
+Moreover, Opτ is the Banach space adjoint of Wτ: Opτ = W ∗
+τ .
+Proof. The boundedness of Opτ is evident; the proof of the continuity of Wτ follows
+by a similar reasoning as the proof of the subsequent Theorem 3.7. The last claim
+is just (31).
+□
+Theorem 3.7. For every τ ∈ [0, 1] the following mappings are linear and continu-
+ous:
+Opτ : S′
+0(R2d) → S′
+0,
+Wτ : S0 → S0(R2d).
+Moreover, Opτ is the Banach space adjoint of Wτ: Opτ = W ∗
+τ , i.e.
+for every
+a ∈ S′
+0(R2d) and S ∈ S0
+(32)
+⟨a,WτS⟩ = ⟨Opτ(a),S⟩.
+
+10
+FEDERICO BASTIANONI AND FRANZ LUEF
+Proof. The boundedness and linearity of Opτ follow from the definitions. By using
+the formal representation of Opτ(a) we can derive an expression for its kernel:
+(33)
+KOpτ (a)(t, x) =
+�
+Rd e2πi(t−x)ωa((1 − τ)t + τx, ω) dω.
+Let us consider first f, g ∈ S0. Then a standard argument, see e.g. [5, Proposition
+1.3.25], gives that
+Wτ(f ⊗ g) = Wτ(f, g) ∈ S0(R2d)
+with
+∥Wτ(f ⊗ g)∥S0 ≲ ∥f∥S0 ∥g∥S0 .
+Since Lemma 2.2 holds for S0, we write Wτ = FσFWτ and use the spectral decom-
+position for S of the form �∞
+n=1 fn ⊗ gn as shown in Lemma 3.4. Now, we compute:
+FWτS(z) = tr(πτ(z)∗S) = tr(
+∞
+�
+n=1
+πτ(z)∗(fn ⊗ gn))
+=
+∞
+�
+n=1
+⟨πτ(z)∗fn, gn⟩ =
+∞
+�
+n=1
+V τ
+gnfn(z).
+(34)
+Taking a suitable window for the norm on S0(R2d) [13, Theorem 5.3] we have
+∥FWτS∥S0 ≤
+∞
+�
+n=1
+��V τ
+gnfn
+��
+S0 =
+∞
+�
+n=1
+∥fn∥S0 ∥gn∥S0 < +∞.
+Consequently,
+∥FWτS∥S0 ≤ inf{
+∞
+�
+n=1
+∥fn∥S0 ∥gn∥S0 , S =
+∞
+�
+n=1
+fn ⊗ gn}
+≤ inf{
+∞
+�
+n=1
+∥fn∥S0 ∥gn∥S0 , KS =
+∞
+�
+n=1
+Kfn⊗gn}
+= ∥KS∥S0 ≍ ∥S∥S0 .
+We proved the boundedness of FWτ : S0 → S0(R2d), the continuity of the symplectic
+Fourier transform Fσ : S0(R2d) → S0(R2d) is well-known, and thus the continuity of
+
+τ-QUANTIZATION AND τ-COHEN CLASSES OF FEICHTINGER OPERATORS
+11
+Wτ : S0 → S0(R2d) follows. Concerning the last claim, we proceed as follows:
+⟨Opτ(a),S⟩ = ⟨KOpτ (a),KS⟩ = ⟨KOpτ (a),
+∞
+�
+n=1
+Kf⊗gn⟩
+=
+∞
+�
+n=1
+⟨KOpτ (a),Kf⊗gn⟩ =
+∞
+�
+n=1
+⟨Opτ(a)gn,fn⟩
+=
+∞
+�
+n=1
+⟨a,Wτ(fn ⊗ gn)⟩ = ⟨a,
+∞
+�
+n=1
+Wτ(fn ⊗ gn)⟩
+= ⟨a,WτS⟩,
+which concludes the proof.
+□
+On account of Theorem 3.6 and 3.7, it seems reasonable to interpret WτS as the
+τ-quantization of an operator in S0 or J 1.
+Corollary 3.8.
+(i) For every τ ∈ [0, 1] the mapping Wτ : S0 → S0(R2d) is a
+topological isomorphism with inverse given by Opτ : S0(R2d) → S0;
+(ii) A linear and continuous operator S : S0(Rd) → S′
+0(Rd) belongs to S0 if and
+only if WτS ∈ S0(R2d) for some (and hence any) τ ∈ [0, 1].
+Proof. (i) We observed in (22) that WτS is just the τ-symbol aS
+τ of a trace class
+operator S, in particular this holds for S ∈ S0. Therefore,
+Opτ ◦WτS = Opτ(aS
+τ ) = S.
+We now show that if we start with a ∈ S0(R2d), then Opτ(a) belongs to S0. From
+(33), we have that the kernel of Opτ(a) can be written as
+KOpτ (a)(t, x) =
+�
+Rd e2πi(t−x)ωa((1 − τ)t + τx, ω) dω = ΨτF −1
+2 a(t, x),
+where F −1
+2
+is the inverse of the partial Fourier transform with respect to the second
+variable; Ψτ is the change of variables induced by the matrix
+(35)
+�
+1 − τ
+τ
+1
+−1
+�
+,
+ΨτF(t, x) := F((1 − τ)t + τx, t − x).
+From the assumption a in the Feichtinger algebra S0(R2d) we have F −1
+2 a ∈ S0(R2d),
+thus ΨτF −1
+2 a is in S0(R2d), i.e. Opτ(a) is an element of S0. The fact that Opτ is
+continuous from S0(R2d) into S0 is evident from the applications of F −1
+2
+and Ψτ.
+Hence we have shown that
+Wτ ◦ Opτ(a) = aOpτ (a)
+τ
+= a.
+(ii) The claim is a straightforward consequence of (i).
+□
+
+12
+FEDERICO BASTIANONI AND FRANZ LUEF
+Corollary 3.9.
+(i) For every τ ∈ [0, 1] FWτ : S0 → S0(R2d) is a topological
+isomorphisms with inverse given by the τ-spreading representation
+(36)
+SRτ : S0(R2d) → S0 , a �→
+�
+R2d a(z)πτ(z) dz;
+(ii) Let us define
+(37)
+SRτ : S′
+0(R2d) → S′
+0 a �→
+�
+R2d a(z)πτ(z) dz,
+where the integral has to be understood weakly as follows:
+⟨SRτ(a)f,g⟩ := ⟨a,V τ
+f g⟩,
+a ∈ S′
+0(R2d), f, g ∈ S0(Rd).
+Then SRτ as in (37) is well-defined, linear, continuous, extends (36) and it
+is the Banach space adjoint of FWτ in (i):
+(38)
+SRτ = F ∗
+Wτ,
+in the sense that for every a ∈ S′
+0(R2d) and S ∈ S0
+⟨a,FWτS⟩ = ⟨SRτ(a),S⟩ = ⟨KSRτ (a),KS⟩;
+(iii) Every function F ∈ S0(R2d) admits an expansion of the following type:
+F =
+∞
+�
+n=1
+V τ
+gnfn,
+for some sequences {fn}n, {gn}n ⊆ S0(Rd) such that �∞
+n=1 ∥fn∥S0 ∥gn∥S0 <
+∞.
+Proof. (i) First we notice that if we start with a ∈ S0(R2d), then SRτ(a) is the
+Feichtinger operator with kernel
+KSRτ (a)(y, u) =
+�
+Rd a(y − u, ω)e2πiyω dω = F −1
+2 [a(y − u, ·)](y).
+Clearly SRτ is continuous from S0(R2d) into S0.
+Since we have Wτ = FσFWτ and Fσ is an automorphism of S0(R2d), we can write
+FWτ = FσWτ and which is an isomorphism due to Corollary 3.8. To prove that SRτ
+
+τ-QUANTIZATION AND τ-COHEN CLASSES OF FEICHTINGER OPERATORS
+13
+is the inverse of FWτ we use (34), take S = �∞
+n=1 fn ⊗ gn ∈ S0 and ψ, ϕ ∈ S0(Rd):
+⟨(SRτ ◦ FWτS)ψ,ϕ⟩ =
+�
+R2d FWτS(z)⟨πτ(z)ψ,ϕ⟩ dz
+=
+∞
+�
+n=1
+�
+R2d V τ
+gnfn(z)V τ
+ψ ϕ(z) dz
+=
+∞
+�
+n=1
+⟨fn,ϕ⟩⟨gn,ψ⟩
+= ⟨
+∞
+�
+n=1
+⟨ψ,gn⟩fn,ϕ⟩
+= ⟨
+∞
+�
+n=1
+(fn ⊗ gn)ψ,ϕ⟩
+= ⟨Sψ,ϕ⟩,
+in the third equality we used Moyal’s identity. For the composition FWτ ◦SRτ, notice
+that this is the identity on S0(R2d) due lo Lemma 2.2 (vii).
+(ii) Well-posedness, linearity and continuity of SRτ from S′
+0(R2d) into S′
+0 are stan-
+dard. Trivially (37) extends (36). To see that SRτ is the Banach space adjoint of
+FWτ from S0 into S0(R2d), take a ∈ S′
+0(R2d) and S ∈ S0. In the following calcula-
+tions we use: the prior stated (34), the representation for Feichtinger operators and
+their kernel given in Lemma 3.4, the Outer and Inner Kernel Theorems:
+⟨a,FWτS⟩ =
+∞
+�
+n=1
+⟨a,V τ
+gnfn⟩ =
+∞
+�
+n=1
+⟨SRτ(a)gn,fn⟩
+=
+∞
+�
+n=1
+⟨KSRτ (a),Kfn⊗gn⟩ = ⟨KSRτ (a),KS⟩
+= ⟨SRτ(a),S⟩.
+(iii) The last claim is a direct consequence of the computations in (34) and the
+surjectivity of FWτ.
+□
+3.2. A convenient environment for QHA. In Section 2 we introduced convo-
+lutions between a function and an operator and two operators. Keyl, Kiukas and
+Werner [14] showed that such convolutions make sense for wider classes of (gen-
+eralized) functions and operators. We summarize here the main results; in what
+follows S denotes the set of pseudo-differential operators with Weyl symbol in the
+Schwartz class S(R2d) and S′ those pseudo-differential operators with Weyl symbol
+in S′(R2d). On account of the Schwartz Kernel Theorem we can identify S′ with
+the continuous and linear operators from S(Rd) into S′(Rd).
+
+14
+FEDERICO BASTIANONI AND FRANZ LUEF
+Proposition 3.10.
+(i) Suppose S, T ∈ S, A ∈ S′, b ∈ S(R2d) and a ∈ S′(R2d).
+Then the following convolutions are well-defined and they extend the ones
+defined in Subsection 2.2:
+S ⋆ T ∈ S(R2d),
+S ⋆ A ��� S′(R2d),
+b ⋆ S ∈ S,
+a ⋆ S, b ⋆ A ∈ S′;
+(ii) The Fourier-Wigner transform can be extended to a topological isomorphism
+FW1/2 : S′ → S′(R2d);
+(iii) We have Fσ(S ⋆ T) = FW1/2S · FW1/2T and FW1/2(b ⋆ S) = Fσb · FW1/2S
+whenever S, T and b are such that the convolutions are defined as in part (i);
+(iv) The Weyl symbol of A ∈ S′ is given by FσFW1/2A.
+The authors of [14] proved that the class of so-called Schwartz operators S has
+the structure of a Fr´echet space. We propose that the Banach space of Feichtinger
+operators S0 is an alternative to S that is a much bigger class of “nice” operators.
+We start with some preliminaries on S0 and S0.
+Lemma 3.11. Given f ∈ S′
+0(Rd), there exists a sequence {fn}n ⊆ S0(Rd) which
+w-∗ converges to f and it is bounded by ∥f∥S′
+0, i.e.
+lim
+n→+∞⟨fn, g⟩ = ⟨f,g⟩
+∀ g ∈ S0(Rd),
+sup
+n ∥fn∥S0 ≤ ∥f∥S′
+0 .
+Proof. Let us fix f ∈ S′
+0(Rd) ∖ {0} and set R := ∥f∥S′
+0. By [13, Proposition 6.15],
+there exists a net {fα}α∈A ⊆ S0(Rd) which converges w-∗ to f in S′
+0 and such that
+∥fα∥S′
+0 ≤ R for every α ∈ A. Set
+BR :=
+�
+f ∈ S′
+0(Rd) | ∥f∥S′
+0 ≤ R
+�
+and
+ER := S0(Rd) ∩ BR,
+where S0 is identified with its natural embedding in S′
+0, i.e.
+ER ⊆ BR ⊆ ER
+w−∗.
+ER
+w−∗ is bounded in S′
+0(Rd).
+In fact, if f0 ∈ ER
+w−∗, then there exists a net {fα}α∈A ⊆ ER that it converges w-∗
+to f0. Hence, we obtain
+∥f0∥S′
+0 ≤ lim inf
+α∈A
+∥fα∥S′
+0 = lim
+α∈A inf{∥fβ∥S′
+0 | α ⪯ β} ≤ lim
+α∈A R = R.
+In particular, this shows that ER
+w−∗ ⊆ BR and we get
+ER
+w−∗ = BR.
+Since S0 separable, and the relative w-∗ topology on BR is induced by a metric by
+[18, Therem 2.6.23]. Hence the topological w-∗ closure of ER equals its sequential
+w-∗ closure. Consequently, there exists a sequence {fn}n ⊆ ER which converges w-∗
+to f in S′
+0(Rd).
+□
+
+τ-QUANTIZATION AND τ-COHEN CLASSES OF FEICHTINGER OPERATORS
+15
+Remark 3.12. The above lemma holds also for any LCA second countable group G
+replacing Rd, see [8, Theorem 2] for the separability of S0(G).
+Lemma 3.13. For any S ∈ S′
+0, there exists a sequence {Sn}n ⊆ S0 such that
+(i) ∥Sn∥S′
+0 ≲ ∥S∥S′
+0;
+(ii) limn→+∞ |⟨(S − Sn)f,g⟩| = 0 for all f, g ∈ S0(Rd).
+Proof. This is a straightforward application of the Kernel Theorems 3.2 and 3.2 and
+of Lemma 3.11.
+□
+Convergence as in item (ii) of the above lemma will be also denoted by
+Sn
+w−∗
+−→
+n
+S
+in
+S′
+0
+or
+S = w- ∗ -limn Sn
+in
+S′
+0.
+Lemma 3.14. Let S : S0 → S′
+0 be in S0. Then the Banach space adjoint S∗: S′
+0 →
+S0 is in S0 with kernel
+(39)
+KS∗(y, u) = KS(u, y).
+Proof. We take f, g ∈ S0(Rd), then
+⟨Sf,g⟩ =
+�
+R2d KT(y, u)g(y)f(u) dydu
+=
+�
+Rd f(u)
+�
+Rd KS(y, u)g(y) dy du
+= ⟨f,S∗g⟩.
+Hence, S∗g(y) =
+�
+Rd KS(u, y)g(u) du, i.e. KS∗(y, u) = KS(u, y) which is an element
+of S0(R2d).
+□
+Corollary 3.15. S0 is a Banach ∗-algebra.
+We notice that (S∗)ˇ= ( ˇS)∗, so that from now on we shall simply write ˇS∗ when
+necessary.
+Lemma 3.16.
+(i) The following applications are surjective isometries:
+(i − a) αz : S0 → S0, for any z = (x, ω) ∈ R2d, and
+(40)
+KαzS(y, u) = e2πi(y−u)ωKS(y − x, u − x);
+(i − b) ˇ·: S0 → S0 and
+(41)
+K ˇS(y, u) = KS(−y − u);
+(i − c) αz : S′
+0 → S′
+0, for any z ∈ R2d;
+(i − d) ˇ·: S′
+0 → S′
+0;
+(ii) Let S, T ∈ S0 and b ∈ S0(R2d). Then
+S ⋆ T ∈ S0(R2d),
+b ⋆ S ∈ S0;
+
+16
+FEDERICO BASTIANONI AND FRANZ LUEF
+(iii) The kernel of the mixed-state localization operator b ⋆ S is given by
+(42)
+Kb⋆S(y, u) =
+�
+Rd b(x, ω)e2πi(y−u)ωKS(y − x, u − x) dxdω;
+for very z = (x, ω) ∈ R2d the kernel of Sαz ˇT is
+(43)
+KSαz ˇT(y, u) =
+�
+Rd e2πi(y−t)ωKT(x − y, x − t)KS(t, u) dt.
+Proof. (i) We leave the elementary computations to the interest reader, and note that
+in order to prove αzS, ˇS ∈ S0 the result [10, Corollary 3.3] is useful. A continuous
+and linear operator S : S0 → S′
+0 is a Feichtinger operator if and only if
+�
+R2d
+�
+R2d |⟨Sπ(z)g1,π(w)g2⟩| dzdw
+is finite for any g1, g2 ∈ S0(Rd).
+(ii) We first address the convolution between two Feichtinger operators. By item
+(i) and the fact that S0 is a Banach algebra under composition, we have that Sαz ˇT
+is in S0 for any z = (x, ω) ∈ R2d. We have by [10, Corollary 3.15]:
+S ⋆ T(z) = tr(Sαz ˇT) =
+�
+Rd KSαz ˇT(y, y) dy =
+�
+R2d Kαz ˇT(y, t)KS(t, y) dtdy
+=
+�
+R2d e2πi(y−t)ωKT(x − y, x − t)KS(t, y) dtdy
+=
+�
+Rd
+��
+Rd KT(x − y, x − t)KS(t, y)e−2πitω dt
+�
+e2πiyω dy
+= F −1
+2 F1
+�
+ΦT(x,x)KT · KS
+�
+(ω, ω),
+where ΦF(t, y) := F(−y, −t), F1 and F2 are the partial Fourier transforms with re-
+spect to the first and second variable, respectively. Consider now f, g, h, l ∈ S0(Rd),
+it is useful to compute the following where P is the parity operator:
+F −1
+2 F1
+�
+ΦT(x,x)Kh⊗l · Kf⊗g
+�
+(ω, ω) =
+�
+Rd
+��
+Rd h(x − y)l(x − t)f(t)g(y)e−2πitω dt
+�
+e2πiyω dy
+=
+�
+Rd f(t)e−2πitωl(x − t) dt ·
+�
+Rd g(y)e2πiyωh(x − y) dy
+= VP lf(−x, ω) · VP hg(−x, ω).
+Hence F −1
+2 F1
+�
+ΦT(x,x)Kh⊗l · Kf⊗g
+�
+(ω, ω) is in S0(R2d) as a function of (x, ω). We
+consider now two representations S = �∞
+n=1 fn ⊗ gn and T = �∞
+n=1 hn ⊗ ln, see
+
+τ-QUANTIZATION AND τ-COHEN CLASSES OF FEICHTINGER OPERATORS
+17
+Lemma 3.4, so that
+KS =
+∞
+�
+n=1
+Kfn⊗gn,
+KT =
+∞
+�
+n=1
+Khn⊗ln.
+It follows that we can write
+S ⋆ T(z) = F −1
+2 F1
+�
+ΦT(x,x)
+∞
+�
+M
+Khm⊗lm ·
+∞
+�
+n=1
+Kfn⊗gn
+�
+(ω, ω)
+=
+∞
+�
+m=1
+∞
+�
+n=1
+F −1
+2 F1
+�
+ΦT(x,x)Khm⊗lm · Kfn⊗gn
+�
+(ω, ω)
+=
+∞
+�
+m=1
+∞
+�
+n=1
+VP lmfn(−x, ω) · VP hmgn(−x, ω) ∈ S0(R2d),
+the convergence is guaranteed by Lemma 3.4.
+Concerning b⋆S, the following estimate for any f, g ∈ S0(Rd) proves that b⋆S ∈ S′
+0:
+|⟨(b ⋆ S)f,g⟩| ≤
+�
+R2d |b(z)| |⟨Sπ(z)∗f,π(z)∗g⟩| dz ≲ ∥b∥L1 ∥S∥S′
+0 ∥f∥S0 ∥g∥S0 .
+We exploit [10, Theorem 3.2 (ii)] to show that b ⋆ S is in S0. For g1, g2 ∈ S0(Rd) we
+have
+�
+R2d
+�
+R2d |⟨(b ⋆ S)π(w)g1,π(u)g2⟩| dwdu ≤
+�
+R2d
+�
+R2d
+�
+R2d |b(z)|
+× |⟨Sπ(w − z)g1,π(u − z)g2⟩| dzdwdu
+=
+�
+R2d
+�
+R2d |⟨Sπ(w′)g1,π(u′)g2⟩| dw′du′ ·
+�
+R2d |b(z)| dz < +∞.
+(iii) We compute explicitly the kernel of the operator given by the convolution b⋆S:
+⟨(b ⋆ S)f,g⟩ =
+�
+R2d b(x, ω)
+�
+R2d KS(y, u)π(−z)g(y)π(−z)f(u) dydu dxdω
+=
+�
+R2d
+�
+R2d b(x, ω)e2πi(y−u)ωKS(y, u)g(y + x)f(u + x) dxdω dydu,
+for z = (x, ω) ∈ R2d. The change of variables y′ = y + u, u′ = u + x gives the desired
+result.
+The last claim is just a direct application of (40), (41) and the Banach
+algebra property for S0 [10, Lemma 3.10].
+□
+Corollary 3.17. Let S, T ∈ S0 with spectral decompositions S = �∞
+n=1 fn ⊗ gn and
+T = �∞
+n=1 hn⊗ln, where {fn}n, {gn}n, {hn}n, {ln}n ⊆ S0(Rd) with �∞
+n=1 ∥fn∥S0 ∥gn∥S0 <
+
+18
+FEDERICO BASTIANONI AND FRANZ LUEF
++∞, �∞
+n=1 ∥hn∥S0 ∥ln∥S0 < +∞. Then, with the notations introduced in the proof
+of Lemma 3.16, for every z = (x, ω) ∈ R2d:
+S ⋆ T(z) = F −1
+2 F1
+�
+ΦT(x,x)KT · KS
+�
+(ω, ω)
+=
+∞
+�
+m=1
+∞
+�
+n=1
+VP lmfn(−x, ω) · VP hmgn(−x, ω).
+(44)
+Definition 3.18. Let A ∈ S′
+0, a ∈ S′
+0(R2d), S ∈ S0 and b ∈ S0(R2d). Consider any
+sequences {An}n ⊆ S0 and {an}n ⊆ S0(R2d) such that
+An
+w−∗
+−→
+n
+A
+in
+S′
+0
+and
+an
+w−∗
+−→
+n
+a
+in
+S′
+0(R2d).
+Then we define:
+S ⋆ A := w- ∗ -limn S ⋆ An
+in
+S′
+0(R2d);
+(45)
+a ⋆ S := S ⋆ a := w- ∗ -limn an ⋆ S
+in
+S′
+0;
+(46)
+b ⋆ A := A ⋆ b := w- ∗ -limn b ⋆ An
+in
+S′
+0.
+(47)
+Remark 3.19. The reader may find it useful to keep in mind the following simple
+identities, which will be used in the proof of the subsequent proposition. Consider
+S ∈ S0, ψ, ϕ, f, g ∈ S0(Rd) and z ∈ R2d:
+αz(ψ ⊗ ϕ) = π(z)ψ ⊗ π(z)ϕ;
+(ψ ⊗ ϕ)(Kf⊗g) = ⟨f, ϕ⟩(ψ ⊗ g);
+(ψ ⊗ ϕ) ⋆ ˇS(z) = ⟨π(z)Sπ(z)∗ψ,ϕ⟩.
+Proposition 3.20. The convolutions introduced in Definition 3.18:
+(i) They do not depend on the sequences chosen; moreover, taking A, a, S, b as
+in Definition 3.18:
+⟨S ⋆ A,b⟩ = ⟨KA,Kb⋆ ˇS∗⟩;
+(48)
+⟨(a ⋆ S)f,g⟩ = ⟨a,(g ⊗ f) ⋆ ˇS∗⟩;
+(49)
+⟨(b ⋆ A)f,g⟩ = ⟨KA,Kb∗⋆(g⊗f)⟩,
+(50)
+where b∗(z) := b(−z);
+(ii) These extend the definitions given in Subsection 2.2;
+(iii) They are commutative;
+
+τ-QUANTIZATION AND τ-COHEN CLASSES OF FEICHTINGER OPERATORS
+19
+(iv) Moreover, they are associative. In particular, if z ∈ R2d, T, Q ∈ S0, σ ∈
+S0(R2d) and A, a, S, b as in Definition 3.18 then:
+(S ⋆ (T ⋆ b))(z) = ((S ⋆ T) ∗ b)(z);
+(51)
+S ⋆ (T ⋆ Q) = (S ⋆ T) ⋆ Q;
+(52)
+(S ⋆ b) ⋆ σ = S ⋆ (b ∗ σ);
+(53)
+S ⋆ (T ⋆ a) = (S ⋆ T) ∗ a;
+(54)
+A ⋆ (T ⋆ b) = (A ⋆ T) ⋆ b;
+(55)
+S ⋆ (T ⋆ A) = (S ⋆ T) ⋆ A;
+(56)
+in the above identities ∗ denotes the usual convolution between two functions
+or a function and a distribution.
+Proof. (i) It suffices to show (48), (49) and (50), since the other assertions in (i) are
+evident.
+We start with(48).
+Let b ∈ S0(R2d) and z = (x, ω) ∈ R2d, in the subsequent
+computations we use Lemma 3.14 and 3.16:
+⟨S ⋆ A,b⟩ = lim
+n→+∞⟨S ⋆ An,b⟩ = lim
+n→+∞
+�
+R2d tr(Sαz ˇAn)b(z) dz
+= lim
+n→+∞
+�
+R2d
+�
+Rd KSαz ˇ
+An(y, y) dyb(z)dz
+= lim
+n→+∞
+�
+R2d
+�
+Rd
+�
+Rd e2πi(y−t)ωKAn(x − y, x − t)KS(t, y) dtdy b(z) dz
+= lim
+n→+∞
+�
+R2d
+�
+Rd
+�
+Rd e2πi(t′−y′)ωKAn(y′, t′)KS(x − t′, x − y′) dt′dy′ b(z) dz
+= lim
+n→+∞
+�
+Rd
+�
+Rd KAn(y′, t���)
+��
+R2d KS(x − t′, x − y′)e2πi(y′−t′)ωb(z) dz
+�
+dy′dt′
+= lim
+n→+∞
+�
+Rd
+�
+Rd KAn(y′, t′)
+��
+R2d K ˇS(t′ − x, y′ − x)e2πi(y′−t′)ωb(z) dz
+�
+dy′dt′
+= lim
+n→+∞
+�
+Rd
+�
+Rd KAn(y′, t′)
+��
+R2d K ˇS∗(y′ − x, t′ − x)e2πi(y′−t′)ωb(z) dz
+�
+dy′dt′
+= lim
+n→+∞
+�
+Rd
+�
+Rd KAn(y′, t′)Kb⋆ ˇS∗(y′, t′) dy′dt′.
+
+20
+FEDERICO BASTIANONI AND FRANZ LUEF
+About (49), we take f, g ∈ S0(Rd) and compute directly keeping in mind Remark
+3.19:
+⟨(a ⋆ S)f,g⟩ =
+lim
+n→+∞
+�
+R2d an(z)⟨π(z)Sπ(z)∗f,g⟩ dz
+=
+lim
+n→+∞
+�
+R2d an(z)⟨π(z)S∗π(z)∗g,f⟩ dz
+=
+lim
+n→+∞
+�
+R2d an(z)(g ⊗ f) ⋆ ˇS∗(z) dz.
+Let us address (50):
+⟨(b ⋆ A)f,g⟩ =
+lim
+n→+∞⟨Kb⋆An,Kg⊗f⟩
+=
+lim
+n→+∞
+�
+R2d
+� �
+R2d b(x, ω)e2πi(y−u)ωKAn(y − x, u − x) dxdω
+�
+× g(y)f(u) dydu
+=
+lim
+n→+∞
+�
+R2d KAn(y′, u′)
+� �
+R2d b(x, ω)e−2πi(y′−u′)ω
+× g(y′ + x)f(u′ + x) dxdω
+�
+dy′du′
+=
+lim
+n→+∞
+�
+R2d KAn(y′, u′)
+� �
+R2d b∗(x′, ω′)e2πi(y′−u′)ω′
+× g(y′ − x′)f(u′ + x′) dx′dω′
+�
+dy′du′
+=
+lim
+n→+∞
+�
+R2d KAn(y′, u′)Kb∗⋆(g⊗f)(y′, u′) dy′du′,
+where for sake of brevity we set b∗(z) := b(−z).
+(ii) and (iii) are trivial.
+(iv) We prove just (51), (52) and (53). The remaining identities can be derived in
+a similar manner.
+
+τ-QUANTIZATION AND τ-COHEN CLASSES OF FEICHTINGER OPERATORS
+21
+In order to show (51) we compute for z ∈ R2d:
+(S ⋆ (T ⋆ b))(z) = tr
+�
+S ◦ αz
+���
+R2d b(z)αwT dw
+�
+ˇ
+��
+= tr
+�
+S ◦
+��
+R2d b(w)αz ((αwT)ˇ) dw
+��
+= tr
+�
+S ◦
+�
+R2d b(w)αzα−w ˇT dw
+�
+= tr
+�
+S ◦
+�
+R2d b(−w′)αw′αz ˇT dw′
+�
+=
+�
+R2d b(−w′) tr
+�
+Sαw′+z ˇT
+�
+dw′,
+where the last equality is due, e.g., to [20, Proposition 2.9]. we can the rephrase the
+last right-side term as
+�
+R2d b(z − w′′) tr
+�
+Sαw′′ ˇT
+�
+dw′′ =
+�
+R2d b(z − w′′)(S ⋆ T)(w′′) dw′′
+= ((S ⋆ T) ∗ b)(z).
+For the proof of (52), the following property of the trace is useful:
+�
+R2d tr(SαwT) dw = tr(S) tr(T),
+where S, T ∈ J 1. Take now f, g ∈ S0(Rd):
+⟨(S ⋆ (T ⋆ Q))f,g⟩ =
+�
+R2d tr(Tαz ˇQ)⟨αzSf,g⟩ dz
+=
+�
+R2d tr(Qαz ˇT) tr((αzS)(f ⊗ g)) dz
+=
+�
+R2d
+�
+R2d tr(Q(αz ˇT)αw((αzS)(f ⊗ g))) dwdz
+=
+�
+R2d
+�
+R2d tr((f ⊗ g)(αwQ)αz((αw ˇT)S)) dzdw
+=
+�
+R2d tr(Sαw ˇT) tr((αwQ)(f ⊗ g)) dw
+= ⟨((S ⋆ T) ⋆ Q)f,g⟩.
+
+22
+FEDERICO BASTIANONI AND FRANZ LUEF
+Also the last identity (53) may be deduced by a direct computation. For f, g ∈
+S0(Rd) we have
+⟨((S ⋆ b) ⋆ σ)f,g⟩ =
+�
+R2d σ(z)⟨αz(S ⋆ b)f,g⟩ dz
+=
+�
+R2d σ(z)
+�
+R2d b(w)⟨(αwS)π(z)∗f,π(z)∗g⟩ dwdz
+=
+�
+R2d
+�
+R2d σ(z)b(w)⟨(αw+zS)f,g⟩ dwdz
+=
+�
+R2d
+�
+R2d σ(z)b(w) tr((αw+zS)(f ⊗ g)) dwdz
+=
+�
+R2d b(w)
+�
+R2d σ(z′ − w) tr((αz′S)(f ⊗ g)) dz′dw
+=
+�
+R2d(
+�
+R2d b(w)σ(z′ − w) dz′) tr((αz′S)(f ⊗ g)) dw
+=
+�
+R2d b ∗ σ(z′)⟨(αz′S)f,g⟩ dz′
+= ⟨(S ⋆ (b ∗ σ))f,g⟩.
+This concludes the proof.
+□
+Corollary 3.21. The mappings FWτ and Wτ defined on S0 can be extended to
+topological isomorphisms
+FWτ : S′
+0 → S′
+0(R2d)
+and
+Wτ : S′
+0 → S′
+0(R2d)
+by duality:
+(57)
+⟨FWτS,a⟩ := ⟨S,SRτa⟩,
+⟨WτS,a⟩ := ⟨S, Opτ a⟩,
+where S ∈ S′
+0 and a ∈ S0(R2d). The inverses are given by
+SRτ : S′
+0(R2d) → S′
+0
+and
+Opτ : S′
+0(R2d) → S′
+0,
+respectively.
+Proof. The definitions in (57) rely on the fact that Opτ = W ∗
+τ and SRτ = F ∗
+Wτ, see
+Theorem 3.7 and Corollary 3.9. It is straightforward to see that if S ∈ S′
+0, then
+FWτS and WτS defined as in (57) are in S′
+0(R2d). Also linearity and boundedness
+of FWτ : S′
+0 → S′
+0(R2d) and Wτ : S′
+0 → S′
+0(R2d) are easy to verify as well as the fact
+that they extend FWτ : S0 → S0(R2d) and Wτ : S0 → S0(R2d).
+We show that Wτ is an isomorphisms with inverse Opτ, then FWτ is treated in
+the same way.
+Wτ is injective because Opτ : S0(R2d) → S0 is an isomorphism.
+Fix now a ∈ S′
+0(R2d), there exists a sequence {an}n ⊆ S0(R2d) such that an
+w−∗
+−→
+n
+
+τ-QUANTIZATION AND τ-COHEN CLASSES OF FEICHTINGER OPERATORS
+23
+a
+in
+S′
+0(R2d). Since Wτ is an isomorphism between S0 and S0(R2d), there exists
+{An}n ⊆ S0 such that an = WτAn. We see that there is A ∈ S′
+0 such that An
+w−∗
+−→
+n
+A
+in
+S′
+0, in fact taking b ∈ S0(R2d)
+⟨a,b⟩ =
+lim
+n→+∞⟨WτAn,b⟩ = lim
+n→+∞⟨An, Opτ b⟩.
+Hence a = WτA, which proves that Wτ is onto. We show now that Wτ ◦ Opτ is the
+identity on S′
+0(R2d), take a ∈ S′
+0(R2d) and b ∈ S0(R2d):
+⟨Wτ ◦ Opτ a,b⟩ = ⟨Opτ a, Opτ b⟩ = ⟨a,Wτ ◦ Opτ b⟩ = ⟨a,b⟩.
+The first identity is just (57), the second one is (32) and the last one is (i) of
+Corollary 3.8. For the other direction, take S ∈ S′
+0 and T ∈ S0:
+⟨Opτ ◦WτS,T⟩ = ⟨WτS,WτT⟩ = ⟨S, Opτ ◦WτT⟩ = ⟨S,T⟩.
+The first identity is (32), the second one is (57) and the last one is (i) of Corollary
+3.8.
+□
+3.3. τ-Cohen’s class of operators. In the present subsection we define Qτ
+a(S)
+and recall the definition of Qτ
+S(f) from [17]. We shall see that Qτ
+a(S) relates to
+well-known objects and observe that it coincides with the τ-symbol of the mixed-
+state localization operator a ⋆ S. We continue with some statements concerning the
+interplay between the Gabor matrix of an operator Gϕ
+T, the τ-Cohen’s class, the
+trace and the τ-Wigner distribution.
+Definition 3.22. For a ∈ S′
+0(R2d) we define the τ-Cohen’s class distribution, with
+kernel a, of an operator S ∈ S0 as
+(58)
+Qτ
+a(S) := a ∗ WτS.
+Of course, the rank-one case f ⊗ g reduces to the definition given in (23). We
+recall also the definition given in [17] of Cohen’s class distribution of a function
+f ∈ S0(Rd) w.r.t. the operator S ∈ S′
+0 by
+(59)
+QSf := (f ⊗ f) ⋆ ˇS.
+It can be easily seen that for every z ∈ R2d
+QSf(z) = (f ⊗ f) ⋆ ˇS(z) = ⟨(αzS)f, f, ⟩.
+Remark 3.23. If a ∈ S′
+0(R2d) and S ∈ S0, then we see that the τ-Cohen’s class rep-
+resentation of S w.r.t. a is just the τ-symbol of the mixed-state localization operator
+a ⋆ S:
+aa⋆S
+τ
+= Wτ(a ⋆ S) = a ∗ WτS = Qτ
+a(S).
+
+24
+FEDERICO BASTIANONI AND FRANZ LUEF
+Lemma 3.24. Let S ∈ S0 have the spectral decomposition �∞
+n=1 fn⊗gn, for f, ϕ, ψ ∈
+S0(Rd) and {hn}n ⊆ S0(Rd) with
+∞
+�
+n=1
+∥hn∥2
+S0 < +∞.
+. Then for every z ∈ R2d:
+Qτ
+W1−τ ( ˇψ, ˇϕ)(S)(z) =
+∞
+�
+n=1
+Vϕfn(z)Vψgn(z);
+(60)
+Qτ
+W1−τ ( ˇϕ, ˇϕ)(
+∞
+�
+n=1
+hn ⊗ hn)(z) =
+∞
+�
+n=1
+|Vϕhn(z)|2 .
+(61)
+Proof. Clearly, it suffices to prove the first identity. We show first that for f, g ∈
+S0(Rd)
+(62)
+Qτ
+a(f, g) = (f ⊗ g) ⋆ Op1-τ(a).
+In fact, applying Fσ to the right-hand side first we get
+Fσ((f ⊗ g) ⋆ Op1-τ(a)) = FWτ(f ⊗ g) · FW1−τ Op1-τ(a) = V τ
+g f · Fσa.
+We apply Fσ a second time:
+(f ⊗ g) ⋆ Op1-τ(a) = FσV τ
+g f ∗ FσFσa = Wτ(f, g) ∗ a.
+We can now proceed as follows:
+Qτ
+W1−τ ( ˇψ, ˇϕ)(S) = W1−τ( ˇψ, ˇϕ) ∗ Wτ(
+∞
+�
+n=1
+fn ⊗ gn) =
+∞
+�
+n=1
+W1−τ( ˇψ, ˇϕ) ∗ Wτ(fn, gn)
+=
+∞
+�
+n=1
+(fn ⊗ gn) ⋆ Op1-τ(W1−τ( ˇψ, ˇϕ)) =
+∞
+�
+n=1
+(fn ⊗ gn) ⋆ ( ˇψ ⊗ ˇϕ)
+=
+∞
+�
+n=1
+Vϕfn(z)Vψgn(z),
+where the last equality is due to [17].
+□
+We call a bounded operator T on L2(Rd) positive, denoted by T ≥ 0, if
+⟨Tf, f⟩ ≥ 0,
+∀ f ∈ L2(Rd).
+An operator T ∈ J 1 and T ≥ 0 is also called a state in quantum mechanics.
+Let us take T ∈ S′
+0 and ϕ ∈ S, then the Gabor matrix of T (w.r.t. ϕ) is defined as
+(63)
+Gϕ
+T(z, w) := ⟨Tπ(w)ϕ, π(z)ϕ⟩,
+z = (x, ω), w = (u, v) ∈ R2d.
+
+τ-QUANTIZATION AND τ-COHEN CLASSES OF FEICHTINGER OPERATORS
+25
+We notice that the Gabor matrix of an operator does not depend on τ, in the sense
+that
+Gϕ
+T(z, w) = ⟨Tπ(w)ϕ, π(z)ϕ⟩ = ⟨Tπτ(w)ϕ, πτ(z)ϕ⟩,
+∀ τ ∈ [0, 1].
+Remark 3.25. We point out that the diagonal of the Gabor matrix of T, w.r.t. ϕ,
+is the Cohen’s class representation of ϕ w.r.t. T up to a reflection:
+(64)
+Gϕ
+T(−z, −z) = QTϕ(z).
+In fact
+Gϕ
+T(−z, −z) = ⟨Tπ(−z)ϕ, π(−z)ϕ⟩ = ⟨Tπ(z)∗ϕ, π(z)∗ϕ⟩
+= ⟨(αzT)ϕ, ϕ, ⟩ = QT ϕ(z).
+Let F and H be functions of (z, w) ∈ R4d and let Θ be a real 4d × 4d matrix.
+Then the twisted convolution induced by Θ is defined as
+(65)
+F ♮Θ H(z, w) :=
+�
+R2d
+�
+R2d F(z′, w′)H(z − z′, w − w′)e2πi(z,w)Θ(z′,w′) dz′dw′.
+Lemma 3.26. Let T, S ∈ J 1, T, S ≥ 0. Then for every τ ∈ [0, 1] we have
+(66)
+tr(TS) =
+�
+R2d WτT(z)WτS(z) dz.
+Proof. Since T and S are trace-class and positive, they can be described as
+T =
+∞
+�
+n=1
+λnfn ⊗ fn,
+S =
+∞
+�
+n=1
+µngn ⊗ gn
+for some orthonormal sets {fn}n and {gn}n in L2 and λn, µn ≥ 0. Let {en}n be an
+o.n.b. for L2(Rd):
+tr(TS) =
+∞
+�
+n=1
+⟨TSen, en⟩ =
+∞
+�
+i,j
+λjµi |⟨fj, gi⟩|2 .
+On the other hand,
+�
+R2d WτT(z)WτS(z) dz =
+∞
+�
+i,j
+λjµi
+�
+R2d Wτfj(z)Wτgi(z) dz =
+∞
+�
+i,j
+λjµi |⟨fj, gi⟩|2 ,
+where the last equality is due to Moyal’s identity. This concludes the proof.
+□
+Remark 3.27. Since we assume S ≥ 0, S is self-adjoint and for τ = 1/2 we have
+that W1/2S is real-valued. In fact, using the representation given in the proof of
+Lemma 3.26:
+W1/2S =
+∞
+�
+n=1
+µnW1/2gn
+
+26
+FEDERICO BASTIANONI AND FRANZ LUEF
+with every W1/2gn real-valued and µn ≥ 0. Hence, for τ = 1/2 we recover [12,
+Lemma 2.7].
+Lemma 3.28. Let T ∈ J 1 and consider ϕ ∈ S(Rd) such that ∥ϕ∥L2 = 1. Then
+(67)
+tr T =
+�
+R2d⟨(αzT)ϕ, ϕ⟩ dz =
+�
+R2d QT ϕ(z) dz =
+�
+R2d Gϕ
+T(z, z) dz.
+Proof. The proof follows from a direct computation using the representations pre-
+sented in the proof of Lemma 3.26 and Moyal’s identity involving the function ϕ:
+⟨fj, gi⟩ = ⟨Vϕfj, Vϕgi⟩,
+we leave details to the interested reader.
+□
+Lemma 3.29. Let T ∈ J 1, T ≥ 0 and let ϕ ∈ S(Rd) such that ∥ϕ∥L2 = 1. Then
+for every z ∈ R2d:
+(68)
+QTϕ(z) =
+�
+R2d WτT(w)Wτϕ(z + w) dw = WτT ∗ (Wτϕ)∗(z),
+where (Wτϕ)∗(w) = Wτϕ(−w).
+Proof. We compute directly
+QT ϕ(z) = ⟨π(z)Tπ(z)∗ϕ, ϕ⟩ = tr(T(π(z)∗ϕ ⊗ π(z)∗ϕ))
+=
+�
+R2d WτT(w)Wτ(π(z)∗ϕ ⊗ π(z)∗ϕ)(w) dw,
+the last equation holds because of Lemma 3.26. An elementary calculation gives
+Wτ(π(z)∗ϕ ⊗ π(z)∗ϕ)(w) = Wτϕ(z + w),
+which is also known as covariance property and this concludes the proof.
+□
+Lemma 3.30. Let T ∈ J 1, T ≥ 0 and consider ϕ ∈ S(Rd) such that ∥ϕ∥L2 = 1.
+Then for every z, w ∈ R2d:
+|Gϕ
+T(z, w)|2 ≤ QTϕ(−z)QT ϕ(−w).
+Proof. The claim follows from the Cauchy-Schwarz inequality for the inner product
+induced by the positive operator T and Remark 3.25.
+□
+Lemma 3.31. Let 0d and Id denote the zero and identity d×d matrices, respectively.
+Let us define
+Θ :=
+
+
+0d
+0d
+0d
+0d
+Id
+0d
+0d
+0d
+0d
+0d
+0d
+0d
+0d
+0d
+−Id
+0d
+
+ .
+
+τ-QUANTIZATION AND τ-COHEN CLASSES OF FEICHTINGER OPERATORS
+27
+Let T ∈ J 1 and consider ϕ ∈ S(Rd) such that ∥ϕ∥L2 = 1. For z = (x, ω), w =
+(u, v) ∈ R2d we have
+Gϕ
+T(z, w) = Gϕ
+T ♮Θ(Gϕ
+ϕ⊗ϕ)∗(z, w)
+(69)
+=
+�
+R2d
+�
+R2d Gϕ
+T(z′, w′)(Gϕ
+ϕ⊗ϕ)∗(z − z′, w − w′)e2πi(ωx′−u′v) dz′dw′,
+where z′ = (x′, ω′), w′ = (u′, v′) ∈ R2d.
+Proof. We apply twice Moyal’s identity:
+Gϕ
+T(z, w) =
+�
+R2d Vϕ[Tπ(w)ϕ](z′)Vϕ[π(z)ϕ](z′) dz′
+=
+�
+R2d
+�
+R2d Vϕ[π(w)ϕ](w′)Vϕ[T ∗π(z′)ϕ](w′)⟨π(z′)ϕ, π(z)ϕ⟩ dz′dw′
+=
+�
+R2d
+�
+R2d Gϕ
+T(z′, w′)⟨π(w)ϕ, π(w′)ϕ⟩⟨π(z′)ϕ, π(z)ϕ⟩ dz′dw′.
+It is then a direct, although tedious, calculation to show that
+⟨π(z)ϕ, π(z′)ϕ⟩⟨π(w′)ϕ, π(w)ϕ⟩ = (Gϕ
+ϕ⊗ϕ)∗(z − z′, w − w′)e2πi(ωx′−u′v).
+This concludes the proof.
+□
+Lemma 3.32. Let T ∈ J 1, T ≥ 0 and consider ϕ ∈ S(Rd) such that ∥ϕ∥L2 = 1.
+Then for any τ ∈ [0, 1]:
+(70)
+WτT(z) =
+�
+R2d
+�
+R2d e−2πi[(ωx′−ω′x)+( 1
+2− 3
+4 τ)x′ω′+x′v]Gϕ
+T
+�z′
+2 − w, −z′
+2 − w
+�
+dwdz′,
+where z = (x, ω), z′ = (x′, ω′), w = (u, v) ∈ R2d.
+Proof. We start rephrasing the τ-Wigner distribution of T:
+WτT(z) = FσFWτT(z) =
+�
+R2d e−2πi(ωx′−ω′x) tr(πτ(z′)∗T) dz′.
+Recalling the properties for πτ, see Section 2, we see that
+πτ(z′/2 + z′/2) = e2πi[(1−τ) x′ω′
+4
+−τ x′ω′
+4
+]πτ(z′/2)πτ(z′/2)
+= e
+π
+2 i(1−2τ)x′ω′πτ(z′/2)πτ(z′/2).
+
+28
+FEDERICO BASTIANONI AND FRANZ LUEF
+Taking the adjoint we get πτ(z′)∗ = e− π
+2 i(1−2τ)x′ω′πτ(z′/2)∗πτ(z′/2)∗ and we write
+using Lemma 3.28:
+tr(πτ(z′)∗T) = e− π
+2 i(1−2τ)x′ω′ tr(πτ(z′/2)∗Tπτ(z′/2)∗)
+= e− π
+2 i(1−2τ)x′ω′ �
+R2d⟨Tπτ(z′/2)∗πτ(w)∗ϕ, πτ(z′/2)πτ(w)∗ϕ⟩ dw
+= e− π
+2 i(1−2τ)x′ω′e− π
+2 i(1−τ)x′ω′
+×
+�
+R2d⟨Tπτ(−z′/2)πτ(−w)ϕ, πτ(z′/2)πτ(−w)ϕ⟩ dw
+= e− π
+2 i(2−3τ)x′ω′ �
+R2d⟨Tπ(−z′/2)π(−w)ϕ, π(z′/2)π(−w)ϕ⟩ dw
+= e− π
+2 i(2−3τ)x′ω′ �
+R2d e−2πix′v⟨Tπ(−z′/2 − w)ϕ, π(z′/2 − w)ϕ⟩ dw.
+This concludes the argument.
+□
+4. A characterization of Schwartz operators
+In this section we introduce weighted versions of S0 and give an alternative de-
+scription of the class S. We use the polynomial weight
+(71)
+vs(z) := (1 + |z|2)
+s
+2,
+z ∈ R2d,
+where s ≥ 0. In order to avoid an extremely cumbersome notation, just for the
+weight functions vs we shall use the following:
+vs ⊗ vs(z, w) := Kvs⊗vs = vs(z)vs(w),
+∀z, w ∈ R2d.
+Definition 4.1. For s ≥ 0 we define the weighted class of Feichtinger operators as
+(72)
+M1
+s := {S : S′
+0(Rd) → S0(Rd) | S is linear, continuous with kernel KS ∈ M1
+vs⊗vs(R2d)}.
+For S in M1
+s we define the mapping
+(73)
+∥S∥M1s := ∥KS∥M1
+vs⊗vs .
+Remark 4.2.
+(i) For s = 0 we recover the Feichtinger operators S0;
+(ii) The mapping defined in (73) is a norm on M1
+s and it is easy to see that
+(M1
+s, ∥·∥M1s) is a Banach space and the following continuous inclusion holds
+true for every s ≥ 0:
+(74)
+M1
+s ֒→ S0.
+Lemma 4.3. For any S ∈ M1
+s there exist {fn}n, {gn}n ⊆ M1
+vs⊗vs(R2d) such that
+S =
+∞
+�
+n=1
+fn ⊗ gn,
+∞
+�
+n=1
+∥fn∥M1vs ∥gn∥M1vs ≤ +∞,
+KS =
+∞
+�
+n=1
+Kfn⊗gn.
+
+τ-QUANTIZATION AND τ-COHEN CLASSES OF FEICHTINGER OPERATORS
+29
+Proof. The proof follows from the fact that
+M1
+vs⊗vs(R2d) = M1
+vs(Rd)ˆ⊗M1
+vs(Rd).
+See also the proof of Lemma 3.4.
+□
+Theorem 4.4. For every τ ∈ [0, 1] the mapping Wτ : M1
+s → M1
+vs⊗vs(R2d) is a topo-
+logical isomorphism with inverse given by Opτ : M1
+vs⊗vs(R2d) → M1
+s.
+Proof. The proof follows the same pattern as the ones of Theorem 3.7 and Corollary
+3.8.
+□
+Corollary 4.5. An operator S belongs to M1
+s if and only if for some (hence every)
+τ ∈ [0, 1] WτS ∈ M1
+vs⊗vs(R2d).
+Theorem 4.6. The following is true:
+(75)
+S =
+�
+s≥0
+M1
+s.
+Proof. By Corollary 4.5, S belongs to the set on the right-hand side if and only if
+WτS ∈
+�
+s≥0
+M1
+vs⊗vs(R2d) = S(R2d).
+The claim follows since W1/2S is the Weyl symbol of S, i.e. aS
+1/2 = W1/2S.
+□
+We recall that a function F on R2d is called rapidly decaying if for every multiindex
+α, β ∈ Nd
+0 we have
+sup
+x,ω∈Rd
+��xαωβF(x, ω)
+�� < +∞,
+where, if x = (x1, . . . , xd) and α = (α1, . . . , αd), xα stands for xα1
+1 · . . . · xαd
+d .
+In [12, Theorem 1.1] a sufficient condition is given for a positive trace-class op-
+erator to be in S.
+Namely, if T ∈ B(L2), T ≥ 0, is such that WτT exists for
+some τ ∈ [0, 1] and it is rapidly decreasing, then T ∈ S and WτT exists for every
+τ ∈ [0, 1]. In this spirit, we provide the following sufficient condition for a generic
+S ∈ B(L2). Observe that we do not not require S to be positive.
+Corollary 4.7. Let S ∈ B(L2) and assume that for some τ ∈ [0, 1] WτS exists.
+Suppose also that, w.r.t. some non-zero window in L2(R2d), the STFT of WτS is
+rapidly decaying. Then WτS exists for every τ ∈ [0, 1] and S is in S.
+Proof. Let us pick G ∈ L2(R2d) ∖ {0}. If VGWτS is rapidly decaying then S ∈ M1
+s
+for every s ≥ 0. The claim follows from Theorem 4.6.
+□
+
+30
+FEDERICO BASTIANONI AND FRANZ LUEF
+Acknowledgments
+The first author would like to thank Eduard Ortega for the financial support to
+visit Trondheim which led to this work.
+References
+[1] F. Bastianoni, E. Cordero and F. Nicola. Decay and smoothness for eigenfunctions of local-
+ization operators. J. Math. Anal. Appl. 492, 124480, 2020.
+[2] O. Christensen. An introduction to frames and Riesz bases. Applied and Numerical Harmonic
+Analysis, Birkh¨auser Basel, Second Edition, 2016.
+[3] E. Cordero and K. Gr¨ochenig. Time-frequency analysis of localization operators. J. Funct.
+Anal., 205(1):107–131, 2003.
+[4] E. Cordero and F. Nicola. Sharp integral bounds for Wigner distributions. Int. Math. Res.
+Not. IMRN, (6):1779–1807, 2018.
+[5] E. Cordero and L. Rodino. Time-Frequency analysis of operators. De Gruyter Studies in
+Mathematics 75, Berlin/Boston, 2020.
+[6] M. D¨orfler, F. Luef, H. McNulty and E. Skrettingland. Time-Frequency Analysis and Coorbit
+Spaces of Operators. arXiv preprint arXiv:2210.04844, 2022.
+[7] C. de Gosson and M. de Gosson. On the Non-Uniqueness of Statistical Ensembles Defining a
+Density Operator and a Class of Mixed Quantum States with Integrable Wigner Distribution.
+Quantum Reports, 3(3):473-81, 2021.
+[8] J. De Vries. The local weight of an effective locally compact transformation group and the
+dimension og L2(G). Colloq. Math. 39(2): 319–3323, 1978.
+[9] H. G. Feichtinger. On a new Segal algebra. Monatshefte f¨ur Mathematik 92, 269–289, 1981.
+[10] H. G. Feichtinger and M. S. Jakobsen. The inner kernel theorem for a certain Segal algebra.
+Monatsh. Math., 2022.
+[11] K. Gr¨ochenig and T. Strohmer. Pseudodifferential operators on locally compact abelian groups
+and Sj¨ostrand’s symbol class. Journal f¨ur die reine und angewandte Mathematik, 2007(613),
+121–146, 2007.
+[12] F. Hern´andez and C. J. Riedel. Rapidly decaying Wigner functions are Schwartz functions. J.
+Math. Phys. 63, 022104, 2022.
+[13] M. S. Jakobsen. On a (no longer) new Segal algebra: a review of the Feichtinger algebra. J.
+Fourier Anal. Appl., 24:1579–1660, 2018.
+[14] M. Keyl, J. Kiukas and R. Werner. Schwartz operators. Rev. Math. Phys. 28(3), 1630001, 60,
+2016.
+[15] L. Lafleche. On Quantum Sobolev Inequalities. arXiv preprint arXiv:2210.03013, 2022.
+[16] F. Luef and E. Skrettingland. On accumulated Cohen’s class distributions and mixed-state
+localization operators. Constr. Approx. 52, 31–64, 2020.
+[17] F. Luef and E. Skrettingland. Mixed-state localization operators: Cohen’s class and trace class
+operators. J. Fourier Anal. Appl., 25(4):2064–2108, 2019.
+[18] R. Megginson. An Introduction to Banach Space Theory. Graduate Texts in Mathematics,
+vol.183, pp. xx+596. Springer, New York, 1998.
+[19] J. E. Moyal. Quantum mechanics as a statistical theory. Proc. Cambridge Phil. Soc., 45:99–
+124, 1949.
+[20] E. Skrettingland. Convolutions for Localization Operators. Master Thesis, NTNU, 2017.
+[21] B. Simon. Trace Ideal and Their Applications. Cambridge University Press, Cambridge, 1979.
+
+τ-QUANTIZATION AND τ-COHEN CLASSES OF FEICHTINGER OPERATORS
+31
+[22] J. Toft. Continuity and compactness for pseudo-differential operators with symbols in quasi-
+Banach spaces or H¨ormander classes. Anal. Appl. (Singap.), 15(3):353–389, 2017.
+[23] R. F. Werner. Quantum harmonic analysis on phase space. J. Math. Phys. 25(5), 1404–1411,
+1984.
+[24] E. P. Wigner. On the quantum correction for thermodynamic equilibrium. Phys. Rev. 40,
+749–759, 1932.
+Dipartimento di Scienze Matematiche, Politecnico di Torino, corso Duca degli
+Abruzzi 24, 10129 Torino, Italy
+Email address: [email protected]
+Department of Mathematics, NTNU Norwegian University of Science and Tech-
+nology, NO-7491 Trondheim, Norway
+Email address: [email protected]
+

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+arXiv:2301.04640v1  [math.GM]  2 Jan 2023
+Properties of the multi-index special function W(¯α,¯ν)(z)
+R. Drogheia
+aLiceo Scientifico Francesco Severi, Viale Europa,36, 03100 Frosinone (FR), ITALY
+ABSTRACT
+In this paper, we investigate some properties related to a multi-index special func-
+tion W(¯α,¯ν) that arose from an eigenvalue problem for a multi-order fractional hyper-
+Bessel operator, involving Caputo fractional derivatives. We show that for particular
+values of the parameters involved in this special function W(¯α,¯ν), this leads to the
+hyper-Bessel function of Delerue. The Laplace transform of the W(¯α,¯ν) is discussed
+obtaining, in particular cases, the well-known functional relation between hyper-
+Bessel function and multi-index Mittag-Leffler function, or, quite simply, between
+classical Wright and Mittag-Leffler functions. Moreover, it is shown that the multi-
+index special function satisfies the recurrence relation involving fractional deriva-
+tives. In a particular case, we derive, to the best of our knowledge, a new differential
+recurrence relation for the Mittag-Leffler function. We also provide derivatives of the
+3-parameters function Wα,β,ν with respect to parameters, leading to infinite power
+series with coefficients being quotients of digamma and gamma functions.
+KEYWORDS
+Special Function of Fractional Calculus; hyper-Bessel type operators; Wright and
+Mittag-Leffler functions; Caputo derivatives; recurrence relations of special
+functions; hyper-Bessel functions
+1. Introduction
+Nowadays, the interest in fractional differential equations is increasing because these
+are becoming more adequate than those of integer order to investigate various problems
+in different fields of physics, engineering and economics [1], [2], [3]. They have indeed
+the fundamental characteristic to describe memory and heredity properties of many
+materials. Some of them have been introduced within the framework of partition theory
+in solving number theory problems. This is the case of the Wright function, introduced
+by E. M. Wright in his articles on the asymptotic partition formulae[4], [5], [6] and [7],
+[8], [9].
+Recently, many authors are dealing with multi-indices special functions (SF) of
+fractional calculus (FC) appearing in solution of differential equations and systems of
+fractional multi-order type (e.g. hyper-Bessel and quasi-Bessel operators) [10], [11].
+Among them, the most general functions we just want to refer to are the Fox H-
+function and the Wright generalized hypergeometric function [12]. Indeed, one gets the
+classical SF setting their parameters with integer values.
+In the previous paper [13] the author investigated a hyper-Bessel-type operator in-
+volving Caputo derivatives. Solving the eigenvalue problem associated with this frac-
+tional operator, the author introduced a function, written in series expansion, that in
+specific cases is possible to refer to the well-known special function of the fractional
+CONTACT R. Droghei. Email: [email protected]
+
+calculus. According to the information we have, this special function was not studied
+by now. But as seen, it is reduced in particular cases to some known special func-
+tions, which on their side are cases of the Bessel and hyper-Bessel functions and more
+generally, of the multi-index Mittag-Leffer functions.
+This multi-index special function, called in the previous paper m-p generalized
+Wright function, plays an important role in nonlinear fractional differential equations,
+and in their isochronous ω-modified version[13],[14]. It is also a natural generalization
+of the applications of the Laguerre derivatives and the Laguerre-type exponentials [15],
+[16], [17], [18]. In this survey article, firstly, we want to examine several properties as-
+sociated with the multi-index special function investigated in [13].
+The outline of this work is as follows. In Section 2, we recall the definition of the
+multi-index function W(¯α,¯ν) introduced in [13] and its connection with the Hyper -
+Bessel function. Moreover, the simpler function in the only 3-parameters case Wα,β,ν
+is described. In Section 3 we computed the Laplace Transform of the function W(¯α,¯ν)
+and, using it, we derived some new functional relations between this function and
+other known special functions. The main result of this work is described in Section 4.
+Here we showed the recurrence relations of the function Wα,β,ν obtaining, we suppose,
+new differential recurrence relation for the Mittag-Leffler function. In Section 5 we
+investigated the derivatives of Wα,β,ν with respect to the parameters.
+2. Multi-index special function W(¯α,¯ν)(z)
+The multi-index special function W(¯α,¯ν)(z) investigated in [13], is defined by series
+representation as a function of the complex variable z and parameters αj, j = 1, ..., n+
+1 and νj, j = 1, ..., n:
+W(¯α,¯ν)(z) =
+∞
+�
+k=0
+k
+�
+i=1
+n
+�
+j=1
+Γ(αn+1i + aj)
+Γ(αn+1i + bj) ·
+zk
+Γ(αn+1k + bn+1)
+.
+(1)
+where
+aj = 1 +
+j
+�
+m=1
+(νm−1 − αm) ;
+bj = 1 +
+j
+�
+m=1
+(νm−1 − αm−1) .
+(2)
+and the relation aj = bj − αj with j = 1..n + 1.
+The W(¯α,¯ν)(z) is an entire function for αj > 0, j = 1..n + 1; νj ∈ C, j = 1..n and
+α0 = ν0 = 0.
+Theorem 2.1. The multi-index special function W(¯α,¯ν)(λxαn+1) with λ ∈ R, x ≥
+0, αj > 0, j = 1, ..., n + 1 and νj > 0, j = 1..n satisfy the following fractional differ-
+ential equation involving fractional hyper-Bessel-type operator.[see [13] for the proof]
+2
+
+ˆD(¯α,¯ν)
+nL
+W(¯α,¯ν)(λxαn+1) = λW(¯α,¯ν)(λxαn+1);
+(3)
+where
+ˆD(¯α,¯ν)
+nL
+= x
+�n
+s=1(αs−νs) dαn+1
+dxαn+1 xνn dαn
+dxαn xνn−1 dαn−1
+dxαn−1 · · · xν1 dα1
+dxα1 .
+(4)
+2.1. Hyper-Bessel function as a particular case
+The hyper-Bessel function of Delerue (or a multi-index analogue of Bessel function) of
+order d with indices µ1, ..., µd, introduced in 1953 by Delereu [19] as a generalization
+of the Bessel function of the first type (see also [20]) is defined by
+Jµd(z) = z−
+µ1+...+µd
+d+1
+Jµd((d + 1)
+d+1√z) =
+�
+k≥0
+(−1)kzk
+k! �d
+j=1 Γ(k + µj + 1)
+.
+(5)
+Setting αj = 1, j = 1, ..., n + 1 in the multi-index special function W(¯α,¯ν), we obtain
+the relation
+W(¯1,¯ν)(z) =
+n
+�
+j=1
+Γ(1 + aj)Jan(−z), .
+(6)
+with aj defined in (2). It is not surprising because the hyper-Bessel function satisfies the
+so-called hyper-Bessel differential operators of higher order, introduced by Dimovski
+and Kiryakova [21], [22], and obtained from (3) setting all parameters αj = 1 with
+j = 1..n + 1, i.e. derivatives of integer order.
+2.2. 3-parameters function Wα,β,ν
+In this section we analyse the simpler case of (1) with n = 1, α2 = β, α1 = α and
+ν1 = ν:
+Wα,β,ν(xβ) =
+∞
+�
+k=0
+k
+�
+i=1
+Γ(βi + 1 − α)
+Γ(βi + 1)
+xβk
+Γ(βk + 1 − α + ν).
+(7)
+Proposition 2.2. Obviously, the above function (7) satisfies the following fractional
+differential equation
+ˆDα,β,νf(x) = xα−ν dβ
+dxβ
+�
+xν dα
+dxα f(x)
+�
+= f(x),
+(8)
+involving two fractional derivatives in the sense of Caputo of orders α, β ∈ (0, 1).
+Where
+f(x) = Wα,β,ν(xβ)
+3
+
+.
+Remark 1. The Weinstein and Bessel-Clifford operators Setting α = β = 1
+and ν = k, k ≥ 1 the operator ˆDα,β,ν becomes
+ˆD1,1,k = xBk = x
+� d2
+dx2 + k
+x
+d
+dx
+�
+= x−k+1 d
+dxxk d
+dx
+where Bk is the well known Weinstein operator (or Bessel operator) from the so-
+called Darboux-Weinstein relation [23], [24]. In [25] Hayek studied in details exactly
+the operator ˆD1,1,k+1 calling its solution as Bessel-Clifford function of second order
+Cν(x) = x− ν−1
+2 Iν−1(2√x) =
+1
+Γ(ν+1) 0F1(ν + 1; −x), where Iν(x) is the modified Bessel
+function of the first kind. Later, in [26] he introduced the two indices Bessel-Clifford
+functions of the third order modifying the hyper-Bessel function J(2)
+µ,ν(x):
+Cµ,ν(x) = x− µ+ν
+3 J(2)
+µ,ν(3
+3√x) =
+1
+Γ(µ + 1)Γ(ν + 1) 0
+F2(µ + 1, ν + 1; −x);
+(9)
+satisfying the third-order Bessel-Clifford differential equation related to the operator
+ˆBµ,ν = x−ν d
+dxxµ−ν+1 d
+dxxν+1 d
+dx.
+(10)
+As it is simple to see, the two-parameter operator ˆBµ,ν is equivalent to the operator
+(4), ˆD({α1,α2,α3},{ν1,ν2})
+2L
+with α1 = α2 = α3 = 1; ν1 = ν + 1 and ν2 = µ − ν + 1; and
+then the Bessel-Clifford of the third order function (9) is equal to
+Cµ,ν(x) =
+1
+Γ(ν + 1)W({1,1,1},{ν+1,µ−ν+1})(x).
+These differential operators appear very often in the PDEs of mathematical physics
+(especially in fluid mechanics, elasticity, and transonic flow), for instance in the gen-
+eralized Bessel heat equation and other equations of generalized axially symmetric
+potentials (GASP) theory [27].
+2.2.1. Particular cases of Wα,β,ν
+For α = 1, β = λ and ν = µ the function corresponds to the Classical Wright function
+W1,λ,µ(xλ) = Wλ,µ
+�xλ
+λ
+�
+=
+∞
+�
+k=0
+�
+xλ
+λ
+�k
+k!Γ(λk + µ).
+(11)
+For α = 0, β → α, ν → β − 1 the function corresponds to the generalized Mittag-
+Leffler function
+W0,α,β−1(z) = Eα,β(z) =
+∞
+�
+k=0
+zk
+Γ(αk + β).
+(12)
+4
+
+In case of α = β = ν holds the relation
+Wν,ν,ν(xν) = E1;ν,1(xν)
+where Eα;ν,γ(x) = �∞
+k=0
+xk
+Γα+1(νk+γ) is the α-Mittag-Leffler function.
+In Addition, we present some examples of the 3-parameters function Wα,β,ν in the
+following table, and in Figure 1 we represent the behavior of this function for different
+values of the parameters α, β, ν;
+Integer order derivatives
+Fractional order derivatives
+W0,1,0(x) = ex
+W 1
+2 , 1
+2, 1
+2 (√x) = +I0(2√x) + L0(2√x)
+W0,1,n(x) = ex
+xn − �n−1
+i=0
+xi−n
+i!
+with n ∈ N
+W 1
+2, 1
+2, 3
+2 (√x) = +I1(2√x) + L1(√x)
+W1,1,0(x) = √xI1(2√x)
+W 1
+2, 1
+2,1(√x) = sinh(2√x)+cosh(2√x)−1
+√πx
+W1,1,ν(x) = x− ν−1
+2 Iν−1(2√x)
+W 1
+2, 1
+2,2(√x) = (2√x−1)e2√x−2x+1
+2x√πx
+where Iα(x) = i−αJα(ix) = �∞
+m=0
+1
+m!Γ(m+α+1)(x
+2)2m+α is the modified Bessel func-
+tion of the first kind and Lα(x) =
+� x
+2
+�ν+1 �∞
+m=0
+( x
+2)
+2m
+Γ(m+ 3
+2)Γ(m+ν+ 3
+2 ) is the modified Struve
+function.
+(a)
+Plot
+of
+the
+function
+W0,1,ν(x)
+for
+ν
+=
+0; 0.25; 0.5; 0.75; 1; 1.25; 1.5; 1.75; 2.
+(b)
+Plot
+of
+the
+function
+W1,1,ν(x)
+for
+ν
+=
+0; 0.25; 0.5; 0.75; 1; 1.25; 1.5; 1.75; 2.
+(c)
+Plot
+of
+the
+function
+W 1
+2 , 1
+2 ,ν(√x)
+for
+ν
+=
+0; 0.25; 0.5; 0.75; 1; 1.25; 1.5; 1.75; 2.
+(d)
+Plot
+of
+the
+function
+W 1
+2 ,1,ν(x)
+for
+ν
+=
+0; 0.25; 0.5; 0.75; 1; 1.25; 1.5; 1.75; 2.
+Figure 1.
+3. Laplace Transform
+Let us compute the Laplace transform of the W(¯α,¯ν)(λx)
+5
+
+L
+�
+W(¯α,¯ν)(λxαn+1), s
+�
+=
+� ∞
+0
+e−sx
+∞
+�
+k=0
+k
+�
+i=1
+n
+�
+j=1
+Γ(αn+1i + aj)
+Γ(αn+1i + bj)
+λkxαn+1k
+Γ(αn+1k + bn+1)dx
+=
+∞
+�
+k=0
+k
+�
+i=1
+n
+�
+j=1
+λkΓ(αn+1i + aj)
+Γ(αn+1i + bj)Γ(αn+1k + bn+1)
+� ∞
+0
+e−sxxαn+1kdx
+=
+1
+s
+∞
+�
+k=0
+k
+�
+i=1
+n
+�
+j=1
+Γ(αn+1i + aj)Γ(αn+1k + 1)
+Γ(αn+1i + bj)Γ(αn+1k + bn+1)
+�
+λ
+sαn+1
+�k
+. (13)
+th analytical properties of the W(¯α,¯ν) provides that the resulting Laplace transform
+turns out to be an analytic function, vanishing at infinity and exhibiting an essential
+singularity at s = 0.
+Remark 2. In case we set αj = 1, j = 1, ..., n + 1, the multi-index special functions
+W(¯α,¯ν) will be related to the hyper-Bessel functions as is showed in (6). After some
+calculations, we obtain the following functional relation between the Laplace transform
+of the Hyper-Bessel function and the multi-index Mittag-Leffler function. A more
+general relation between these two functions can be found in the article of Kiryakova
+and Luchko [28].
+L
+�
+W(¯1,¯ν)(λx), s
+�
+=
+n
+�
+j=1
+Γ(1 + aj)1
+s
+∞
+�
+k=0
+1
+�n
+j=1 Γ(k + aj+1 + 1)
+�λ
+s
+�k
+=
+n
+�
+j=1
+Γ(1 + aj)1
+sE(n)
+(1,1,...,1),(aj+1+1)
+�λ
+s
+�
+.
+(14)
+Remark 3. The Laplace transform of Wα,β,ν(x) can be obtained as a special case of
+the (13) as follows:
+L (Wα,β,ν(λxρ), s) = 1
+s
+∞
+�
+k=0
+k
+�
+i=1
+βΓ(βi + 1 − α)
+Γ(βi + 1)
+Γ(ρk + 1)
+Γ(βk + 1 − α + ν)
+� λ
+sρ
+�k
+;
+(15)
+and, in the case of the parameter α = 1, we obtain the well-known Laplace transform
+of the Wright function which can be expressed in terms of the two-parameter Mittag-
+Leffler function.
+L (W1,β,ν(λx), s) = L (Wβ,ν(λx), s) = 1
+sEβ,ν
+� λ
+β s
+�
+.
+(16)
+•
+6
+
+4. Recurrence relations of Wα,β,ν
+A recurrence relation is an equation that recursively defines a sequence of values; given
+one or more initial terms, each further term of the sequence is defined as a function of
+the previous terms. Differential recurrence relation of the generalized Wright function
+can be used in the study of fractional differential equations, and it is obtained directly
+from series representation.
+xα+β dβ
+dxβ
+�
+xν−α+βWα,β,ν+β(xβ)
+�
+− 2xν+βWα,β,ν(xβ) + xα+ν dα
+dxα Wα,β,ν−β(xβ) = 0.
+(17)
+Remark 4. In case α = β = 1; ν = n + 1 with n ∈ N0 and
+d
+dxW1,1,n(x) = W1,1,n+1(x) = Cn(x);
+we obtain the well known three-term recurrence relation for the Bessel-Clifford func-
+tion Cn(x)
+xCn+2(x) + (n + 1)Cn+1(x) = Cn(x).
+(18)
+Remark 5. Recurrence fractional derivatives relation for the Wright and
+Mittag-Leffler functions. From the relation (11) between the generalized Wright
+function and the classical Wright function the relation (17) becomes
+dλ
+dzλ
+�
+zλ+ν−1Wλ,λ+ν
+�zλ
+λ
+��
+= zν−1Wλ,ν
+�zλ
+λ
+�
+;
+(19)
+using the formula
+d
+dz Wλ,ν−λ
+�zλ
+λ
+�
+= zλ−1Wλ,ν
+�zλ
+λ
+�
+.
+(20)
+In case α = 0, and β → α, ν → β − 1 the generalized Wright function is related to
+the Mittag-Leffler function by the following relation:
+W0,α,β−1(z) = Eα,β(z).
+(21)
+In particular, from the recurrence relation (17), we obtain the new recurrence relation
+involving fractional derivatives for M-L functions.
+zα dα
+dzα
+�
+zα+β−1Eα,α+β(zα)
+�
+− 2zα+β−1Eα,β(zα) + zβ−1Eα,β−α(zα) = 0.
+(22)
+7
+
+5. Partial derivatives of Wα,β,ν with respect to the parameters
+In this section, taking inspiration from the works of Apelblat and Mainardi [29], [30] we
+analyse the derivatives of Wα,β,ν respect the three parameters included in the function.
+We can treat parameters as variables and hence the derivatives with respect to them
+can be obtained. These derivatives lead to infinite power series involving digamma (ψ)
+and gamma functions.
+∂
+∂ν Wα,β,ν(z) = −
+∞
+�
+k=0
+k
+�
+i=1
+Γ(βi + 1 − α)
+Γ(βi + 1)
+ψ(βk + 1 − α + ν)
+Γ(βk + 1 − α + ν)zk;
+(23)
+∂
+∂β Wα,β,ν(z) =
+∞
+�
+k=0
+k
+�
+i=1
+Γ(βi + 1 − α)
+Γ(βi + 1)
+zk
+Γ(βk + 1 − α + ν) ·
+·
+
+
+k
+�
+j=1
+j [ψ(βj + 1 − α) − ψ(βj + 1)] − kψ(βk + 1 − α + ν)
+
+ ;
+(24)
+∂
+∂αWα,β,ν(z) =
+∞
+�
+k=0
+k
+�
+i=1
+Γ(βi + 1 − α)
+Γ(βi + 1)
+zk
+Γ(βk + 1 − α + ν)
+
+−
+k
+�
+j=1
+ψ(βj + 1 − α) + ψ(βk + 1 − α + ν)
+
+ ;
+(25)
+where ψ(z) = Γ′(z)
+Γ(z) denotes the digamma function.
+Remark 6. In the case α = 1 and considering the property of the digamma function
+ψ(z+1) = ψ(z)+ 1
+z; we obtain the formula (5) and (6) of the Apelblat-Mainardi article
+([30]) for the classical Wright function
+∂
+∂β W1,β,ν(z) =
+� ∂
+∂β Wβ,ν
+�
+(βz) = −
+∞
+�
+k=0
+� ψ(βk + ν)
+k!Γ(βk + ν)
+�
+kzk;
+∂
+∂ν W1,β,ν(z) =
+� ∂
+∂ν Wβ,ν
+�
+(βz) = −
+∞
+�
+k=0
+� ψ(βk + ν)
+k!Γ(βk + ν)
+�
+zk.
+By setting the parameters, α = 0, β → α and ν → β − 1, we obtain the formulas
+(95) and (96) of the Apelblat paper ([29])
+∂
+∂αW0,α,β−1(z) = ∂
+∂αEα,β(z) = −
+∞
+�
+k=0
+�kψ(αk + β)
+Γ(αk + β)
+�
+zk;
+∂
+∂β W0,α,β−1(z) = ∂
+∂β Eα,β(z) = −
+∞
+�
+k=0
+�ψ(αk + β)
+Γ(αk + β)
+�
+zk.
+8
+
+6. Conclusion
+The aim of this paper is to investigate several properties related to the multi-index
+special function W(¯α,¯ν) and its 3-parameters version. An important result was finding
+the connection between the W(¯α,¯ν) and the hyper-Bessel function of Delerue. Here we
+analyzed the Laplace transform, recurrence relation and derivatives of the function
+with respect to the parameters. In particular, we found new findings that, for special
+values of the parameters, retrieve some well-known relations. Indeed, a simple func-
+tional relation is obtained between the Laplace transform of the hyper-Bessel function
+and the multi-index Mittag-Leffler.
+Disclosure statement
+No potential conflict of interest was reported by the author.
+Acknowledgements
+The author is grateful to Dr Roberto Garra for providing essential information, help
+and advice.
+References
+[1] Podlubny I. Fractional Differential Equations. Academic Press, San Diego; 1999.
+[2] Mainardi F. Fractional calculus and waves in linear viscoelasticity: an introduction to
+mathematical models. World Scientific; 2010.
+[3] Gorenflo R, Kilbas AA, Mainardi F, Rogosin SV. Mittag-Leffler functions, related topics
+and applications (p. 540). New York, NY, USA: Springer; 2020.
+[4] Wright E. M. Asymptotic partition formulae: I. plane partitions. The Quarterly Journal of
+Mathematics, Volume os-2, Issue 1; 1931; p. 177–189. https://doi.org/10.1093/qmath/os-
+2.1.177
+[5] Wright E. M. Asymptotic partition formulae:(II) weighted partitions. Proceedings of the
+London Mathematical Society, 2(1); 1934; p. 117-141. https://doi.org/10.1112/plms/s2-
+36.1.117
+[6] Wright E. M. Asymptotic partition formulae. III. Partitions intok-th powers. Acta Math-
+ematica, 63(1); 1934; p.143-191. https://doi.org/10.1007/BF02547353
+[7] Wright EM. On the coefficients of power series having exponential singularities. Journal
+London Math. Soc. 8; 1933; p. 71–79.
+[8] Wright EM. The asymptotic expansion of the generalized Bessel function. Proc. London
+Math. Soc. (Ser. II) 38; 1935; p. 257–270.
+[9] Wright EM. The asymptotic expansion of the generalized hypergeometric function. Jour-
+nal London Math. Soc. 10; 1935; p. 287–293.
+[10] Garra R, Polito F. On some operators involving Hadamard derivatives. Integral Trans-
+forms and Special Functions; 2013. https://doi.org/10.1080/10652469.2012.756875.
+[11] Dubovski PB, Slepoi JA. Construction and analysis of series solutions for frac-
+tional
+quasi-Bessel
+equations.
+Fract
+Calc
+Appl
+Anal
+25;
+2022;
+p.1229–1249.
+https://doi.org/10.1007/s13540-022-00045-z
+[12] Kiryakova V. Fractional calculus of some ”new” but not new special function: K-, multi-
+index-, and S-analogues. AIP Conference Proceedings. 2172, 050008; 2019.
+9
+
+[13] Droghei R. On a Solution of a Fractional Hyper-Bessel Differential Equation by Means
+of a Multi-Index Special Function. Fract Calc Appl Anal 24; 2021; p. 1559–1570.
+https://doi.org/10.1515/fca-2021-0065
+[14] Droghei R, Garra R. Isochronous fractional PDEs. Lecture Notes of TICMI 21; 2020; p.
+43–51.
+[15] Dattoli G, Ricci PE. Laguerre-type exponentials, and the relevant-circular and-hyperbolic
+functions. Georgian Mathematical Journal, 10(3); 2003; p. 481-494.
+[16] Bretti G, Ricci PE. Laguerre-type special functions and population dynamics. Applied
+mathematics and computation, 187(1); 2007; p. 89-100.
+[17] Ricci PE. Laguerre-Type Exponentials, Laguerre Derivatives and Applications. A Survey.
+Mathematics 8, 2054; 2020.
+[18] Garra R, Tomovski Z. Exact results on some nonlinear Laguerre-type diffusion equations.
+Mathematical Modelling and Analysis, 26(1); 2021; p. 72-81.
+[19] Delerue P., Sur le calcul symbolique `a n variables et fonctions hyperbesseliennes (II). Ann.
+Soc. Sci. Brux. 3; 1953; p. 229–274.
+[20] Kiryakova V. Generalized Fractional Calculus and Applications. Longman – J. Wiley,
+Harlow, N. York; 1994.
+[21] Dimowski I, Kiryakova V. Generalized Poisson transmutations and corresponding repre-
+sentations of hyper-Bessel functions. C. R. Acad. Bulg. Sci. 39, N. 10; 1986; p. 20-32.
+[22] Dimowski I, Kiryakova V. Generalized Poisson representations of hyper-geometric func-
+tions pFq , p < q using fractional integrals. In: Proc.16th Spring Conf Union Bulg. Math.
+Sofia; 1987; p. 205-212.
+[23] Weinstein A. The generalized radiation problem and the Euler-Poisson-Darboux equation.
+Summa Brazil Math. 3; 1955; p. 125-147.
+[24] Kiryakova V, Hernandez-Suarez V. Bessel-Clifford third order differential operator and
+corresponding Laplace type integral transform. Dissertationes Mathematicae 340; 1995);
+p. 143-161.
+[25] Hayek N. Estudio de la ecuaci`on diferencial xy′′ + (ν + 1)y′ + y = 0 y de sus aplicaciones.
+Collect. Math. 18, No 1-2; 1967; p. 57-174.
+[26] Hayek N. Funciones de Bessel-Cliff`ord de tercer orden. Actas XII Jornadas Luso-Esp. de
+Mat. (Braga); 1987; p. 346-351.
+[27] Weinstein A. Generalized axially symmetric potential theory. Bull.AMS 59, 20; 1955.
+[28] Kiryakova V, Luchko Yu. The Multiindex MittagLeffler Functions and Their Applications
+for Solving Fractional Order Problems in Applied Analysis. AIP Conf. Proc. 1301, 597;
+2010; doi: 10.1063/1.3526661.
+[29] Apelblat A. Differentiation of the Mittag-Leffler functions with respect to parameters in
+the Laplace transform approach. Mathematics, 8(5), 657; 2020.
+[30] Apelblat A, Mainardi F. Differentiation of the Wright functions with respect to parameters
+and other results. arXiv e-prints, arXiv-2009; 2020.
+Appendix A. Fractional calculus
+In order to make the papar self-contained, we briefly recall main definitions and prop-
+erties of fractional calculus operators.
+Let γ ∈ R+. The Riemann-Liouville fractional integral is defined by
+Jγ
+x f(x) =
+1
+Γ(γ)
+� x
+0
+(x − x′)γ−1f(x′)dx′,
+(A1)
+10
+
+where
+Γ(γ) =
+� +∞
+0
+xγ−1e−xdx,
+is the Euler Gamma function.
+Note that, by definition, J0
+xf(x) = f(x).
+Moreover it satisfies the semigroup property, i.e. Jα
+x Jβ
+x f(x) = Jα+β
+x
+f(x).
+There are different definitions of fractional derivative (see e.g. [1]). In this paper we
+used the fractional derivatives in the sense of Caputo, that is
+Dγ
+xf(x) = Jm−γ
+x
+Dm
+x f(x) =
+1
+Γ(m − γ)
+� x
+0
+(x−x′)m−γ−1
+dm
+d(x′)m f(x′) dx′, γ ̸= m. (A2)
+It is simple to prove the following properties of fractional derivatives and integrals
+(see e.g. [1]) that will be used in the analysis:
+Dγ
+xJγ
+x f(x) = f(x),
+γ > 0,
+(A3)
+Jγ
+x Dγ
+xf(x) = f(x) −
+m−1
+�
+k=0
+f (k)(0)xk
+k! ,
+γ > 0, x > 0,
+(A4)
+Jγ
+x xδ =
+Γ(δ + 1)
+Γ(δ + γ + 1)xδ+γ
+γ > 0, δ > −1, t > 0,
+(A5)
+Dγ
+xxδ =
+Γ(δ + 1)
+Γ(δ − γ + 1)xδ−γ
+γ > 0, δ > −1, t > 0.
+(A6)
+11
+

5dE3T4oBgHgl3EQfpQo9/content/tmp_files/load_file.txt

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+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='04640v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='GM]  2 Jan 2023  Properties of the multi-index special function W(¯α,¯ν)(z)  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Drogheia  aLiceo Scientifico Francesco Severi, Viale Europa,36, 03100 Frosinone (FR), ITALY  ABSTRACT  In this paper, we investigate some properties related to a multi-index special func-  tion W(¯α,¯ν) that arose from an eigenvalue problem for a multi-order fractional hyper-  Bessel operator, involving Caputo fractional derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' We show that for particular  values of the parameters involved in this special function W(¯α,¯ν), this leads to the  hyper-Bessel function of Delerue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' The Laplace transform of the W(¯α,¯ν) is discussed  obtaining, in particular cases, the well-known functional relation between hyper-  Bessel function and multi-index Mittag-Leffler function, or, quite simply, between  classical Wright and Mittag-Leffler functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Moreover, it is shown that the multi-  index special function satisfies the recurrence relation involving fractional deriva-  tives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' In a particular case, we derive, to the best of our knowledge, a new differential  recurrence relation for the Mittag-Leffler function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' We also provide derivatives of the  3-parameters function Wα,β,ν with respect to parameters, leading to infinite power  series with coefficients being quotients of digamma and gamma functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  KEYWORDS  Special Function of Fractional Calculus;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' hyper-Bessel type operators;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Wright and  Mittag-Leffler functions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Caputo derivatives;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' recurrence relations of special  functions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' hyper-Bessel functions  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Introduction  Nowadays, the interest in fractional differential equations is increasing because these  are becoming more adequate than those of integer order to investigate various problems  in different fields of physics, engineering and economics [1], [2], [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' They have indeed  the fundamental characteristic to describe memory and heredity properties of many  materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Some of them have been introduced within the framework of partition theory  in solving number theory problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' This is the case of the Wright function, introduced  by E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Wright in his articles on the asymptotic partition formulae[4], [5], [6] and [7],  [8], [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  Recently, many authors are dealing with multi-indices special functions (SF) of  fractional calculus (FC) appearing in solution of differential equations and systems of  fractional multi-order type (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' hyper-Bessel and quasi-Bessel operators) [10], [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  Among them, the most general functions we just want to refer to are the Fox H-  function and the Wright generalized hypergeometric function [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Indeed, one gets the  classical SF setting their parameters with integer values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  In the previous paper [13] the author investigated a hyper-Bessel-type operator in-  volving Caputo derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Solving the eigenvalue problem associated with this frac-  tional operator, the author introduced a function, written in series expansion, that in  specific cases is possible to refer to the well-known special function of the fractional  CONTACT R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Droghei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Email: riccardo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='droghei@francescoseveri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='org  calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' According to the information we have, this special function was not studied  by now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' But as seen, it is reduced in particular cases to some known special func-  tions, which on their side are cases of the Bessel and hyper-Bessel functions and more  generally, of the multi-index Mittag-Leffer functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  This multi-index special function, called in the previous paper m-p generalized  Wright function, plays an important role in nonlinear fractional differential equations,  and in their isochronous ω-modified version[13],[14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' It is also a natural generalization  of the applications of the Laguerre derivatives and the Laguerre-type exponentials [15],  [16], [17], [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' In this survey article, firstly, we want to examine several properties as-  sociated with the multi-index special function investigated in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  The outline of this work is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' In Section 2, we recall the definition of the  multi-index function W(¯α,¯ν) introduced in [13] and its connection with the Hyper -  Bessel function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Moreover, the simpler function in the only 3-parameters case Wα,β,ν  is described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' In Section 3 we computed the Laplace Transform of the function W(¯α,¯ν)  and, using it, we derived some new functional relations between this function and  other known special functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' The main result of this work is described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  Here we showed the recurrence relations of the function Wα,β,ν obtaining, we suppose,  new differential recurrence relation for the Mittag-Leffler function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' In Section 5 we  investigated the derivatives of Wα,β,ν with respect to the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Multi-index special function W(¯α,¯ν)(z)  The multi-index special function W(¯α,¯ν)(z) investigated in [13], is defined by series  representation as a function of the complex variable z and parameters αj, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=', n+  1 and νj, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=', n:  W(¯α,¯ν)(z) =  ∞  �  k=0  k  �  i=1  n  �  j=1  Γ(αn+1i + aj)  Γ(αn+1i + bj) ·  zk  Γ(αn+1k + bn+1)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  (1)  where  aj = 1 +  j  �  m=1  (νm−1 − αm) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  bj = 1 +  j  �  m=1  (νm−1 − αm−1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  (2)  and the relation aj = bj − αj with j = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='.n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  The W(¯α,¯ν)(z) is an entire function for αj > 0, j = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='.n + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' νj ∈ C, j = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='.n and  α0 = ν0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' The multi-index special function W(¯α,¯ν)(λxαn+1) with λ ∈ R, x ≥  0, αj > 0, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=', n + 1 and νj > 0, j = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='.n satisfy the following fractional differ-  ential equation involving fractional hyper-Bessel-type operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' [see [13] for the proof]  2  ˆD(¯α,¯ν)  nL  W(¯α,¯ν)(λxαn+1) = λW(¯α,¯ν)(λxαn+1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  (3)  where  ˆD(¯α,¯ν)  nL  = x  �n  s=1(αs−νs) dαn+1  dxαn+1 xνn dαn  dxαn xνn−1 dαn−1  dxαn−1 · · · xν1 dα1  dxα1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  (4)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Hyper-Bessel function as a particular case  The hyper-Bessel function of Delerue (or a multi-index analogue of Bessel function) of  order d with indices µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=', µd, introduced in 1953 by Delereu [19] as a generalization  of the Bessel function of the first type (see also [20]) is defined by  Jµd(z) = z−  µ1+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='+µd  d+1  Jµd((d + 1)  d+1√z) =  �  k≥0  (−1)kzk  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' �d  j=1 Γ(k + µj + 1)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  (5)  Setting αj = 1, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=', n + 1 in the multi-index special function W(¯α,¯ν), we obtain  the relation  W(¯1,¯ν)(z) =  n  �  j=1  Γ(1 + aj)Jan(−z), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  (6)  with aj defined in (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' It is not surprising because the hyper-Bessel function satisfies the  so-called hyper-Bessel differential operators of higher order, introduced by Dimovski  and Kiryakova [21], [22], and obtained from (3) setting all parameters αj = 1 with  j = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='.n + 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' derivatives of integer order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 3-parameters function Wα,β,ν  In this section we analyse the simpler case of (1) with n = 1, α2 = β, α1 = α and  ν1 = ν:  Wα,β,ν(xβ) =  ∞  �  k=0  k  �  i=1  Γ(βi + 1 − α)  Γ(βi + 1)  xβk  Γ(βk + 1 − α + ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  (7)  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Obviously, the above function (7) satisfies the following fractional  differential equation  ˆDα,β,νf(x) = xα−ν dβ  dxβ  �  xν dα  dxα f(x)  �  = f(x),  (8)  involving two fractional derivatives in the sense of Caputo of orders α, β ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  Where  f(x) = Wα,β,ν(xβ)  3  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' The Weinstein and Bessel-Clifford operators Setting α = β = 1  and ν = k, k ≥ 1 the operator ˆDα,β,ν becomes  ˆD1,1,k = xBk = x  � d2  dx2 + k  x  d  dx  �  = x−k+1 d  dxxk d  dx  where Bk is the well known Weinstein operator (or Bessel operator) from the so-  called Darboux-Weinstein relation [23], [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' In [25] Hayek studied in details exactly  the operator ˆD1,1,k+1 calling its solution as Bessel-Clifford function of second order  Cν(x) = x− ν−1  2 Iν−1(2√x) =  1  Γ(ν+1) 0F1(ν + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' −x), where Iν(x) is the modified Bessel  function of the first kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Later, in [26] he introduced the two indices Bessel-Clifford  functions of the third order modifying the hyper-Bessel function J(2)  µ,ν(x):  Cµ,ν(x) = x− µ+ν  3 J(2)  µ,ν(3  3√x) =  1  Γ(µ + 1)Γ(ν + 1) 0  F2(µ + 1, ν + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' −x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  (9)  satisfying the third-order Bessel-Clifford differential equation related to the operator  ˆBµ,ν = x−ν d  dxxµ−ν+1 d  dxxν+1 d  dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  (10)  As it is simple to see, the two-parameter operator ˆBµ,ν is equivalent to the operator  (4), ˆD({α1,α2,α3},{ν1,ν2})  2L  with α1 = α2 = α3 = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' ν1 = ν + 1 and ν2 = µ − ν + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' and  then the Bessel-Clifford of the third order function (9) is equal to  Cµ,ν(x) =  1  Γ(ν + 1)W({1,1,1},{ν+1,µ−ν+1})(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  These differential operators appear very often in the PDEs of mathematical physics  (especially in fluid mechanics, elasticity, and transonic flow), for instance in the gen-  eralized Bessel heat equation and other equations of generalized axially symmetric  potentials (GASP) theory [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Particular cases of Wα,β,ν  For α = 1, β = λ and ν = µ the function corresponds to the Classical Wright function  W1,λ,µ(xλ) = Wλ,µ  �xλ  λ  �  =  ∞  �  k=0  �  xλ  λ  �k  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='Γ(λk + µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  (11)  For α = 0, β → α, ν → β − 1 the function corresponds to the generalized Mittag-  Leffler function  W0,α,β−1(z) = Eα,β(z) =  ∞  �  k=0  zk  Γ(αk + β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  (12)  4  In case of α = β = ν holds the relation  Wν,ν,ν(xν) = E1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='ν,1(xν)  where Eα;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='ν,γ(x) = �∞  k=0  xk  Γα+1(νk+γ) is the α-Mittag-Leffler function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  In Addition, we present some examples of the 3-parameters function Wα,β,ν in the  following table, and in Figure 1 we represent the behavior of this function for different  values of the parameters α, β, ν;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  Integer order derivatives  Fractional order derivatives  W0,1,0(x) = ex  W 1  2 , 1  2, 1  2 (√x) = +I0(2√x) + L0(2√x)  W0,1,n(x) = ex  xn − �n−1  i=0  xi−n  i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  with n ∈ N  W 1  2, 1  2, 3  2 (√x) = +I1(2√x) + L1(√x)  W1,1,0(x) = √xI1(2√x)  W 1  2, 1  2,1(√x) = sinh(2√x)+cosh(2√x)−1  √πx  W1,1,ν(x) = x− ν−1  2 Iν−1(2√x)  W 1  2, 1  2,2(√x) = (2√x−1)e2√x−2x+1  2x√πx  where Iα(x) = i−αJα(ix) = �∞  m=0  1  m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='Γ(m+α+1)(x  2)2m+α is the modified Bessel func-  tion of the first kind and Lα(x) =  � x  2  �ν+1 �∞  m=0  ( x  2)  2m  Γ(m+ 3  2)Γ(m+ν+ 3  2 ) is the modified Struve  function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  (a)  Plot  of  the  function  W0,1,ν(x)  for  ν  =  0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='25;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='75;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='25;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='75;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  (b)  Plot  of  the  function  W1,1,ν(x)  for  ν  =  0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='25;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='75;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='25;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='75;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  (c)  Plot  of  the  function  W 1  2 , 1  2 ,ν(√x)  for  ν  =  0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='25;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='75;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='25;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='75;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  (d)  Plot  of  the  function  W 1  2 ,1,ν(x)  for  ν  =  0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='25;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='75;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='25;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='75;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Laplace Transform  Let us compute the Laplace transform of the W(¯α,¯ν)(λx)  5  L  �  W(¯α,¯ν)(λxαn+1), s  �  =  � ∞  0  e−sx  ∞  �  k=0  k  �  i=1  n  �  j=1  Γ(αn+1i + aj)  Γ(αn+1i + bj)  λkxαn+1k  Γ(αn+1k + bn+1)dx  =  ∞  �  k=0  k  �  i=1  n  �  j=1  λkΓ(αn+1i + aj)  Γ(αn+1i + bj)Γ(αn+1k + bn+1)  � ∞  0  e−sxxαn+1kdx  =  1  s  ∞  �  k=0  k  �  i=1  n  �  j=1  Γ(αn+1i + aj)Γ(αn+1k + 1)  Γ(αn+1i + bj)Γ(αn+1k + bn+1)  �  λ  sαn+1  �k  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' (13)  th analytical properties of the W(¯α,¯ν) provides that the resulting Laplace transform  turns out to be an analytic function, vanishing at infinity and exhibiting an essential  singularity at s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' In case we set αj = 1, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=', n + 1, the multi-index special functions  W(¯α,¯ν) will be related to the hyper-Bessel functions as is showed in (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' After some  calculations, we obtain the following functional relation between the Laplace transform  of the Hyper-Bessel function and the multi-index Mittag-Leffler function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' A more  general relation between these two functions can be found in the article of Kiryakova  and Luchko [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  L  �  W(¯1,¯ν)(λx), s  �  =  n  �  j=1  Γ(1 + aj)1  s  ∞  �  k=0  1  �n  j=1 Γ(k + aj+1 + 1)  �λ  s  �k  =  n  �  j=1  Γ(1 + aj)1  sE(n)  (1,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=',1),(aj+1+1)  �λ  s  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  (14)  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' The Laplace transform of Wα,β,ν(x) can be obtained as a special case of  the (13) as follows:  L (Wα,β,ν(λxρ), s) = 1  s  ∞  �  k=0  k  �  i=1  βΓ(βi + 1 − α)  Γ(βi + 1)  Γ(ρk + 1)  Γ(βk + 1 − α + ν)  � λ  sρ  �k  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  (15)  and, in the case of the parameter α = 1, we obtain the well-known Laplace transform  of the Wright function which can be expressed in terms of the two-parameter Mittag-  Leffler function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  L (W1,β,ν(λx), s) = L (Wβ,ν(λx), s) = 1  sEβ,ν  � λ  β s  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  (16) 6  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Recurrence relations of Wα,β,ν  A recurrence relation is an equation that recursively defines a sequence of values;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' given  one or more initial terms, each further term of the sequence is defined as a function of  the previous terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Differential recurrence relation of the generalized Wright function  can be used in the study of fractional differential equations, and it is obtained directly  from series representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  xα+β dβ  dxβ  �  xν−α+βWα,β,ν+β(xβ)  �  − 2xν+βWα,β,ν(xβ) + xα+ν dα  dxα Wα,β,ν−β(xβ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  (17)  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' In case α = β = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' ν = n + 1 with n ∈ N0 and  d  dxW1,1,n(x) = W1,1,n+1(x) = Cn(x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  we obtain the well known three-term recurrence relation for the Bessel-Clifford func-  tion Cn(x)  xCn+2(x) + (n + 1)Cn+1(x) = Cn(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  (18)  Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Recurrence fractional derivatives relation for the Wright and  Mittag-Leffler functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' From the relation (11) between the generalized Wright  function and the classical Wright function the relation (17) becomes  dλ  dzλ  �  zλ+ν−1Wλ,λ+ν  �zλ  λ  ��  = zν−1Wλ,ν  �zλ  λ  �  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  (19)  using the formula  d  dz Wλ,ν−λ  �zλ  λ  �  = zλ−1Wλ,ν  �zλ  λ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  (20)  In case α = 0, and β → α, ν → β − 1 the generalized Wright function is related to  the Mittag-Leffler function by the following relation:  W0,α,β−1(z) = Eα,β(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  (21)  In particular, from the recurrence relation (17), we obtain the new recurrence relation  involving fractional derivatives for M-L functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  zα dα  dzα  �  zα+β−1Eα,α+β(zα)  �  − 2zα+β−1Eα,β(zα) + zβ−1Eα,β−α(zα) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  (22)  7  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Partial derivatives of Wα,β,ν with respect to the parameters  In this section, taking inspiration from the works of Apelblat and Mainardi [29], [30] we  analyse the derivatives of Wα,β,ν respect the three parameters included in the function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  We can treat parameters as variables and hence the derivatives with respect to them  can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' These derivatives lead to infinite power series involving digamma (ψ)  and gamma functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  ∂  ∂ν Wα,β,ν(z) = −  ∞  �  k=0  k  �  i=1  Γ(βi + 1 − α)  Γ(βi + 1)  ψ(βk + 1 − α + ν)  Γ(βk + 1 − α + ν)zk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  (23)  ∂  ∂β Wα,β,ν(z) =  ∞  �  k=0  k  �  i=1  Γ(βi + 1 − α)  Γ(βi + 1)  zk  Γ(βk + 1 − α + ν) · \uf8ee  \uf8f0  k  �  j=1  j [ψ(βj + 1 − α) − ψ(βj + 1)] − kψ(βk + 1 − α + ν)  \uf8f9  \uf8fb ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  (24)  ∂  ∂αWα,β,ν(z) =  ∞  �  k=0  k  �  i=1  Γ(βi + 1 − α)  Γ(βi + 1)  zk  Γ(βk + 1 − α + ν)  \uf8ee  \uf8f0−  k  �  j=1  ψ(βj + 1 − α) + ψ(βk + 1 − α + ν)  \uf8f9  \uf8fb ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  (25)  where ψ(z) = Γ′(z)  Γ(z) denotes the digamma function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' In the case α = 1 and considering the property of the digamma function  ψ(z+1) = ψ(z)+ 1  z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' we obtain the formula (5) and (6) of the Apelblat-Mainardi article  ([30]) for the classical Wright function  ∂  ∂β W1,β,ν(z) =  � ∂  ∂β Wβ,ν  �  (βz) = −  ∞  �  k=0  � ψ(βk + ν)  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='Γ(βk + ν)  �  kzk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  ∂  ∂ν W1,β,ν(z) =  � ∂  ∂ν Wβ,ν  �  (βz) = −  ∞  �  k=0  � ψ(βk + ν)  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='Γ(βk + ν)  �  zk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  By setting the parameters, α = 0, β → α and ν → β − 1, we obtain the formulas  (95) and (96) of the Apelblat paper ([29])  ∂  ∂αW0,α,β−1(z) = ∂  ∂αEα,β(z) = −  ∞  �  k=0  �kψ(αk + β)  Γ(αk + β)  �  zk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  ∂  ∂β W0,α,β−1(z) = ∂  ∂β Eα,β(z) = −  ∞  �  k=0  �ψ(αk + β)  Γ(αk + β)  �  zk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  8  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Conclusion  The aim of this paper is to investigate several properties related to the multi-index  special function W(¯α,¯ν) and its 3-parameters version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' An important result was finding  the connection between the W(¯α,¯ν) and the hyper-Bessel function of Delerue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Here we  analyzed the Laplace transform, recurrence relation and derivatives of the function  with respect to the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' In particular, we found new findings that, for special  values of the parameters, retrieve some well-known relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Indeed, a simple func-  tional relation is obtained between the Laplace transform of the hyper-Bessel function  and the multi-index Mittag-Leffler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  Disclosure statement  No potential conflict of interest was reported by the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  Acknowledgements  The author is grateful to Dr Roberto Garra for providing essential information, help  and advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  References  [1] Podlubny I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
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+page_content=' New York, NY, USA: Springer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
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+page_content=' plane partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
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+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
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+page_content='1093/qmath/os-  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='177  [5] Wright E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
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+page_content=' Asymptotic partition formulae:(II) weighted partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Proceedings of the  London Mathematical Society, 2(1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
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+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='1112/plms/s2-  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='117  [6] Wright E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
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+page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Partitions intok-th powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Acta Math-  ematica, 63(1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
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+page_content=' Jour-  nal London Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
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+page_content=' Integral Trans-  forms and Special Functions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
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+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='1080/10652469.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='756875.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  [11] Dubovski PB, Slepoi JA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Construction and analysis of series solutions for frac-  tional  quasi-Bessel  equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  Fract  Calc  Appl  Anal  25;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='1229–1249.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='1007/s13540-022-00045-z  [12] Kiryakova V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Fractional calculus of some ”new” but not new special function: K-, multi-  index-, and S-analogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' AIP Conference Proceedings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 2172, 050008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  9  [13] Droghei R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' On a Solution of a Fractional Hyper-Bessel Differential Equation by Means  of a Multi-Index Special Function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Fract Calc Appl Anal 24;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 1559–1570.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='1515/fca-2021-0065  [14] Droghei R, Garra R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Isochronous fractional PDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Lecture Notes of TICMI 21;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  43–51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  [15] Dattoli G, Ricci PE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Laguerre-type exponentials, and the relevant-circular and-hyperbolic  functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Georgian Mathematical Journal, 10(3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 481-494.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  [16] Bretti G, Ricci PE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Laguerre-type special functions and population dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Applied  mathematics and computation, 187(1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 89-100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  [17] Ricci PE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Laguerre-Type Exponentials, Laguerre Derivatives and Applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' A Survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  Mathematics 8, 2054;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  [18] Garra R, Tomovski Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Exact results on some nonlinear Laguerre-type diffusion equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  Mathematical Modelling and Analysis, 26(1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 72-81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  [19] Delerue P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=', Sur le calcul symbolique `a n variables et fonctions hyperbesseliennes (II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Brux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 1953;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 229–274.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  [20] Kiryakova V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Generalized Fractional Calculus and Applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Longman – J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Wiley,  Harlow, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' York;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  [21] Dimowski I, Kiryakova V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Generalized Poisson transmutations and corresponding repre-  sentations of hyper-Bessel functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Bulg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 39, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 1986;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 20-32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  [22] Dimowski I, Kiryakova V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Generalized Poisson representations of hyper-geometric func-  tions pFq , p < q using fractional integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' In: Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='16th Spring Conf Union Bulg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  Sofia;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 1987;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 205-212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  [23] Weinstein A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' The generalized radiation problem and the Euler-Poisson-Darboux equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  Summa Brazil Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 1955;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 125-147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  [24] Kiryakova V, Hernandez-Suarez V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Bessel-Clifford third order differential operator and  corresponding Laplace type integral transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Dissertationes Mathematicae 340;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 1995);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 143-161.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  [25] Hayek N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Estudio de la ecuaci`on diferencial xy′′ + (ν + 1)y′ + y = 0 y de sus aplicaciones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  Collect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 18, No 1-2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 1967;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 57-174.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  [26] Hayek N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Funciones de Bessel-Cliff`ord de tercer orden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Actas XII Jornadas Luso-Esp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' de  Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' (Braga);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 1987;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 346-351.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  [27] Weinstein A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Generalized axially symmetric potential theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='AMS 59, 20;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 1955.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  [28] Kiryakova V, Luchko Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' The Multiindex MittagLeffler Functions and Their Applications  for Solving Fractional Order Problems in Applied Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' AIP Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 1301, 597;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='1063/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='3526661.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  [29] Apelblat A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Differentiation of the Mittag-Leffler functions with respect to parameters in  the Laplace transform approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Mathematics, 8(5), 657;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  [30] Apelblat A, Mainardi F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Differentiation of the Wright functions with respect to parameters  and other results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' arXiv e-prints, arXiv-2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Fractional calculus  In order to make the papar self-contained, we briefly recall main definitions and prop-  erties of fractional calculus operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  Let γ ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' The Riemann-Liouville fractional integral is defined by  Jγ  x f(x) =  1  Γ(γ)  � x  0  (x − x′)γ−1f(x′)dx′,  (A1)  10  where  Γ(γ) =  � +∞  0  xγ−1e−xdx,  is the Euler Gamma function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  Note that, by definition, J0  xf(x) = f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  Moreover it satisfies the semigroup property, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' Jα  x Jβ  x f(x) = Jα+β  x  f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  There are different definitions of fractional derivative (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' In this paper we  used the fractional derivatives in the sense of Caputo, that is  Dγ  xf(x) = Jm−γ  x  Dm  x f(x) =  1  Γ(m − γ)  � x  0  (x−x′)m−γ−1  dm  d(x′)m f(x′) dx′, γ ̸= m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' (A2)  It is simple to prove the following properties of fractional derivatives and integrals  (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' [1]) that will be used in the analysis:  Dγ  xJγ  x f(x) = f(x),  γ > 0,  (A3)  Jγ  x Dγ  xf(x) = f(x) −  m−1  �  k=0  f (k)(0)xk  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content=' ,  γ > 0, x > 0,  (A4)  Jγ  x xδ =  Γ(δ + 1)  Γ(δ + γ + 1)xδ+γ  γ > 0, δ > −1, t > 0,  (A5)  Dγ  xxδ =  Γ(δ + 1)  Γ(δ − γ + 1)xδ−γ  γ > 0, δ > −1, t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}
+page_content='  (A6)  11' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfpQo9/content/2301.04640v1.pdf'}

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+Beckman Defense
+A V Subramanyam
+IIITD
+[email protected]
+Abstract
+Optimal transport (OT) based distributional robust
+optimisation (DRO) has received some traction in
+the recent past. However, it is at a nascent stage but
+has a sound potential in robustifying the deep learn-
+ing models. Interestingly, OT barycenters demon-
+strate a good robustness against adversarial attacks.
+Owing to the computationally expensive nature of
+OT barycenters, they have not been investigated
+under DRO framework.
+In this work, we pro-
+pose a new barycenter, namely Beckman barycen-
+ter, which can be computed efficiently and used
+for training the network to defend against adver-
+sarial attacks in conjunction with adversarial train-
+ing. We propose a novel formulation of Beckman
+barycenter and analytically obtain the barycenter
+using the marginals of the input image. We show
+that the Beckman barycenter can be used to train
+adversarially trained networks to improve the ro-
+bustness. Our training is extremely efficient as it re-
+quires only a single epoch of training. Elaborate ex-
+periments on CIFAR-10, CIFAR-100 and Tiny Im-
+ageNet demonstrate that training an adversarially
+robust network with Beckman barycenter can sig-
+nificantly increase the performance. Under auto at-
+tack, we get a a maximum boost of 10% in CIFAR-
+10, 8.34% in CIFAR-100 and 11.51% in Tiny Ima-
+geNet. Our code is available at http://bitly.ws/yvgh.
+1
+Introduction
+Optimal mass transport (OT), originally proposed by Monge
+in his seminal work [Monge,
+1781],
+has gathered a
+widespread interest in the field of learning representations.
+The original deterministic OT problem was later relaxed by
+Kantorovich [Kantorovich, 1942] and considered a proba-
+bilistic transport problem. This formulation seeks solution
+for the optimal transport plan which can transport mass be-
+tween two measures by incurring the minimum cost and is
+solved using a linear program. The modern day OT is also at-
+tributed to the phenomenal work of Kantorovich. Following
+the OT theory, barycenters in Wasserstein space was proposed
+by Agueh and Carlier in their remarkable work [Agueh and
+Carlier, 2011]. Further, using entropic regularization [Cuturi,
+2013], a fast method of computing barycenters was proposed
+by Cuturi and Doucet [Cuturi and Doucet, 2014]. Recent
+works addresses the challenge of computational complexity
+of barycenters using neural networks [Lacombe et al., 2021].
+In this work, we investigate the barycenters towards robust
+learning of deep learning models.
+Deep learning systems have shown impressive perfor-
+mance in various applications. However, these systems are
+vulnerable to adversarial perturbations [Wong et al., 2020],
+[Croce and Hein, 2020], [Xie et al., 2019].
+In order to
+counter these attacks, several defense mechanisms have also
+been proposed.
+In one of the early works, Szegedy et
+al. [Szegedy et al., 2013] formulated the adversarial attack
+as an optimization problem and obtained the adversarial sam-
+ple using L-BFGS. Several adversarial attacks have been
+proposed since Szegedy’ work [Goodfellow et al., 2014;
+Kurakin et al., 2016]. On the other hand, strong defense mea-
+sures have been studied in [Madry et al., 2017], [Theagarajan
+et al., 2019], [Wong et al., 2020], [Rebuffi et al., 2021].
+Rotated  
+samples of
+Classical AT
+Barycentric
+Training 
+ 
+Barycenter of  
+adversarial sample
+Inference
+Figure 1: Illustration: Classical defense methods use Adversarial
+Training (AT) as a major defense technique. Our method obtains
+barycenter from rotated inputs and uses them for training the model
+using a cross-entropy loss. During inference time also we compute
+barycenter of the given sample. The dashed boundary of barycenter
+indicates that the barycenter is close to input samples in terms of
+appearance but there are some differences. In the computation of
+barycenter of adversarial sample, the barycenter shows the changes
+in same color as that of the background to imply that barycenter
+suppresses the adversarial noise.
+In the field of adversarial attacks and defense, lP space has
+arXiv:2301.01495v1  [cs.LG]  4 Jan 2023
+
+been extensively studied. However, only a few works investi-
+gate attacks under OT framework [Wong et al., 2019], [Li et
+al., 2021]. There are even fewer works which investigate ro-
+bustness using OT theory [Kwon et al., 2020], [Subramanyam
+and Raj, 2022]. Distinct from these works, we first intro-
+duce Beckman barycenter, a concept analogous to Wasser-
+stein barycenter. We use proximal operator methods to solve
+for the barycenter. The barycenters obtained from the clean
+samples are used to train a pretrained adversarially robust net-
+work. We note that in the absence of adversarial samples in
+the training, the model would give a better clean accuracy but
+will suffer in terms of adversarial accuracy. Therefore, we
+use a pre-trained adversarially robust network to overcome
+this challenge. An abstract illustration of our method is given
+in Figure 1.
+Beckman barycenter is obtained from input marginals via
+a non-linear interpolation. The input marginals are linearly
+transformed versions of the input and thus interfere with the
+adversarial noise. Using these marginals the barycenter gen-
+erates a sample which is similar in appearance to the input
+and is closer in terms of class label. Thus, the class label
+is preserved when the input is a clean sample, whereas, the
+adversarial noise gets suppressed when the input is an ad-
+versarial sample. Further, the network needs to be trained
+with barycenter of clean samples so as to correctly classify
+them. However, this training is cheap as a single epoch is
+sufficient. We prove our hypothesis using extensive qualita-
+tive and quantitative experiments.
+2
+Related Works
+Adversarial Attacks Given an adversarial sample x with la-
+bel y, a target network f parameterized by θ, the adversary
+tries to find xadv by adding an adversarial noise such that
+the prediction fθ(xadv) ̸= fθ(x) = y. Some of the robust at-
+tacks are iterative FGSM [Kurakin et al., 2016], PGD [Madry
+et al., 2017], Carlini and Wagner attacks [Carlini and Wagner,
+2017], Jacobian based attack [Papernot et al., 2016], physical
+attack Athalye [Athalye et al., 2018], and Autoattack [Croce
+and Hein, 2020]. These attacks are primarily focused in lp
+domain.
+Adversarial Defense In response to adversarial attacks, sev-
+eral defenses been proposed. One of the best defense ap-
+proach is adversarial training [Szegedy et al., 2013], [Good-
+fellow et al., 2014], [Moosavi-Dezfooli et al., 2016]. Madry
+et al. [Madry et al., 2017] formally studied adversarial train-
+ing and proposed that such training allows network to de-
+fend well against first order adversary. Adversarial logit pair-
+ing uses a pair of logits from clean and adversarial examples
+to defend against adversarial samples [Kannan et al., 2018].
+TRADES [Zhang et al., 2019] prove the bounds based on
+regularization term which minimizes the difference in pre-
+diction between clean and adversarial examples. In [Wong et
+al., 2020], authors proposed to effectively combine FGSM
+and random initialization to demonstrate better adversarial
+training. RST [Carmon et al., 2019] propose a self-training
+technique using unlabelled samples to improve the robust-
+ness. Observing the correlation between flatness of weight
+loss landscape and adversarial robustness, Wu et al. proposed
+adversarial weight perturbation (AWP) to regularize the flat-
+ness of weight loss [Wu et al., 2020]. On similar lines, [Yu et
+al., 2022] propose a criterion called Loss Stationary Condi-
+tion (LSC) for constrained perturbation, which regulates the
+weight perturbation to prevent overfitting. LBGAT [Cui et al.,
+2021] constrains the logits of a robust model, trained with ad-
+versarial examples, to be similar to the logits of a clean model
+trained on natural data.
+While adversarial training uses all the samples, many tech-
+niques propose that naively using adversarial samples in ad-
+versarial training is not efficient.
+This primarily involves
+training the model with a weak attack first, and then grad-
+ually increasing the strength of the adversary - CAT [Cai et
+al., 2018], DART [Wang et al., 2019a], MART [Wang et al.,
+2019b], FAT [Zhang et al., 2020]. Aforementioned methods
+rely on pre-determined attack parameters for adversarial sam-
+ple generation. However, this restricts the model’s robust-
+ness. To address this issue, LAS-AT [Jia et al., 2022] propose
+a framework for adversarial training that introduces the no-
+tion of learnable attack strategy. It is composed of two com-
+ponents: a target network that uses adversarial examples for
+training to improve robustness, and a strategy network that
+produces attack strategies to control adversarial sample gen-
+eration. In similar spirit, A2 [Xu et al., 2022] and [Cheng
+et al., 2022] have also been proposed. A classical review of
+defense methods can be obtained in [Bai et al., 2021].
+In a parallel line of defense works, input purification has
+also been explored. At the test time, these techniques try to
+remove the adversarial noise [Shi et al., 2021], TRADESSSL
+[Mao et al., 2021], HedgeRST [Wu et al., 2021]. Score based
+generative models such as [Yoon et al., 2021] and [Nie et al.,
+2022] have also been used to purify the images before sending
+them for classification.
+Our work is inspired from two different theories, namely,
+OT barycenters and distributional robust optimization. We
+discuss these theories in the following.
+Wasserstein Barycenter In the following we discuss Wasser-
+stein distance and barycenter. Given probability distributions,
+µ1, µ2 ∈ Ω, the Wasserstein distance is defined as,
+W(µ1, µ2) = inf
+Ω×Ω c(x, y)π(x, y)dxdy,
+(1)
+s.t.
+�
+Ω
+π(x, y)dx = µ1(x),
+�
+Ω
+π(x, y)dy = µ2(y),
+where the cost matrix c(x, y) = ∥x − y∥1 and π denotes the
+transport plan. This is also known as Earth Mover’ Distance
+(EMD). This form is also used to compute barycenter [Cuturi
+and Peyr´e, 2016] wherein the summation of Wasserstein dis-
+tance between the barycenter and each input marginal is con-
+sidered. However, barycenters are costly to compute and the
+best known complexity scales exponentially with the number
+of marginals [Fan et al., 2022].
+EMD can also be represented as dual of the dual of Eq 1
+in variational form popularly introduced by Beckman [Beck-
+
+mann, 1952], [Li et al., 2018], [Lee et al., 2020],
+W(µ1, µ2) = inf
+M
+�
+Ω
+∥M∥
+(2)
+s.t. div(M) + µ1 − µ2 = 0
+M.n = 0 ∀x ∈ ∂Ω; n is normal to ∂Ω
+Under appropriate discretisation, M = (Mx, My), M ∈
+Rn×2 is flux vector satisfying zero flux boundary conditions.
+µ1, µ2 ∈ Rn, and,
+div(M) = (Mx[i, j]−Mx[i−1, j])+(My[i, j]−My[i, j−1])
+and the zero-flux boundary conditions mean that Mx[i, j] =
+My[i, j] = 0 outside the boundary. Eq 2 is favorable com-
+pared to Eq 1 as it reduces the complexity from O(n2) to
+O(n) [Li et al., 2018]. Motivated by the recent developments
+of OT barycenters, we make use of Eq 2 to propose Beckman
+barycenter as they can be efficiently solved using well known
+techniques like [Goldstein and Osher, 2009], [Chambolle and
+Pock, 2011].
+DRO One of the influential works in DRO was proposed by
+Scarf [Scarf, 1957]. Following this work, significant research
+has been done in this field [Ben-Tal et al., 2009], [Duchi et
+al., 2021], [Staib and Jegelka, 2017]. DRO aims to address
+the problem of uncertainty or shift in the data distribution that
+can arise due to measurement errors and admits a solution
+for the worst case scenario. Let L(θ, x) be the loss function
+where θ are network parameters. Then, DRO solves for,
+inf
+θ sup
+Q∈Q
+EQL(θ, x)
+(3)
+Here, Q is the distribution against which DRO minimizes
+the loss. For instance, Q can be considered as a distribu-
+tion set which contains perturbations of input samples x.
+Here we note that adversarial training can be considered to
+be a specific instance of DRO wherein the distribution Q is
+drawn from adversarial samples. In our case, we consider the
+barycenters as the samples drawn from the distribution Q and
+thus provide robustness against perturbed samples.
+2.1
+Proposed Algorithm
+In this work, we propose a novel Beckman Barycenter for-
+mulation and derive the barycenter analytically. We use the
+barycenter to demonstrate that it can be applied for adversar-
+ial defense. We first obtain the barycenter using the marginals
+from the given input image and then train the network using
+barycenter.
+While OT barycenters are a good choice for the distri-
+bution Q in Eq 3, computing OT barycenter suffers from
+high complexity and exponentially increases with the number
+marginals [Fan et al., 2022]. To counter this high complexity
+challenge, we first discuss an analogous barycenter problem
+by building upon the formulation given in Eq 2.
+inf
+M1,M2
+r1,r2,µ
+∥M1∥2,1 + ∥M2∥2,1 + α(∥r1∥1
+(4)
++∥r2∥1) + β∥µ∥1
+s.t. div(M1) + µ1 − µ = r1
+div(M2) + µ2 − µ = r2
+where, r1, r2, µ ∈ Rn. Our formulation is loosely inspired
+from the Beckman OT formulation that are given in [Li et al.,
+2018], [Lee et al., 2020]. There are notable changes in Eq 4
+from Eq 2. First we solve for Beckman barycenter µ in ad-
+dition to other variables. Similar to Wasserstein barycenter
+which acts as a representative of marginals using Wasserstein
+metric, the Beckman barycenter µ minimizes the flux with
+respect to input marginals µ1 and µ2. In our experiments,
+these marginals are obtained by rotating the input image with
+±4◦. Second, the variables r1 and r2 allow the mass to be
+created or destroyed [Lee et al., 2020] and the regularization
+over r1, r2 and µ ensure that these variable do not take ar-
+bitrarily large values. Third, Eq 4 can be easily converted to
+Lagrange formulation and solved in linear time using primal-
+dual method of Chambolle and Pock [Chambolle and Pock,
+2011].
+In order to make the objective strongly convex, we first
+apply proximal operators. The l2 regularizer makes the ob-
+jective strongly convex. Using the proximal operator,
+inf
+M1,M2,r1
+r2,µ′
+1,µ′
+2,µ
+∥M1∥2,1 + ∥M2∥2,1 + α(∥r1∥1 (5)
++∥r2∥1) + 1
+2ρ(∥µ′
+1 − µ1∥2 + ∥µ′
+2 − µ2∥2) + β∥µ∥1
+s.t. div(M1) + µ′
+1 − µ = r1
+div(M2) + µ′
+2 − µ = r2
+The Lagrangian of Eq 5 is given as,
+inf
+M1,M2,r1
+r2,µ′
+1,µ′
+2,µ
+∥M1∥2,1 + ∥M2∥2,1 + α(∥r1∥1 (6)
++∥r2∥1) + 1
+2ρ(∥µ′
+1 − µ1∥2 + ∥µ′
+2 − µ2∥2) + β∥µ∥1
++
+�
+i
+⟨λi, div(Mi) + µ′
+i − µ − ri⟩
+Eq 6 can be solved using first-order primal dual method of
+Chambolle and Pock [Chambolle and Pock, 2011]1.
+Mt+1
+i
+← arg min
+Mi
+∥Mi∥2,1 + ⟨λi, div(Mi) + µ′
+i−
+µ − ri⟩ + 1
+2τ1
+∥Mi − Mt
+i∥2
+∀i = {1, 2}
+µ′
+i
+t+1 ← arg min
+µ′
+i
+1
+2τ1
+(∥µ′
+i − µi∥2) + ⟨λi, µ′
+i⟩
++ 1
+2τ1
+∥µ′
+i − µ′
+i
+t∥2
+rt+1
+i
+← arg min
+ri
+α∥ri∥1 + ⟨λt
+i, ri⟩ + 1
+2τ1
+∥ri − rt
+i∥2
+µt+1 ← arg min
+µ
+∥µ∥1 + ⟨λt
+i, µ⟩ + 1
+2τ1
+∥µ − µt∥2
+λt+1
+i
+← arg max
+λi
+⟨λi, κt+1⟩ − 1
+2τ2
+∥λi − λt
+i∥2,
+1We use similar notations to that of [Li et al., 2018], [Chambolle
+and Pock, 2011] for consistency and simplicity.
+
+where, κt+1 = 2(div(Mi)t+1+µ′
+i
+t+1−rt+1
+i
+)−(div(Mi)t+
+µ′
+i
+t − rt
+i)
+We now discuss the solution of each individual optimiza-
+tion.
+Solving for Mi: The rows mij of Mi can be expressed
+and solved using l21 norm shrinkage operator,
+mt+1
+ij
+← shrinkl2
+τ1(mt
+ij − τ1div∗(λt
+i)j)
+(7)
+Here,
+div∗
+denotes the adjoint of div operator,
+and
+shrinkl2
+τ1η = max(∥η∥2 − τ1, 0) ⊙
+η
+(∥η∥2). “⊙” denotes the
+Hadamard product.
+Solving for µ′
+i:
+µ′
+i
+t+1 ← max{0,
+ρτ1
+1 + ρτ1
+µ′
+i +
+1
+1 + ρτ1
+(µ′
+i
+t − τ1λt
+i)}, (8)
+Solving for ri: We use an l1 shrinkage operator.
+rt+1
+i
+← shrinkl1
+ατ1(rt
+i + τ1λt
+i)
+(9)
+Here, shrinkl1
+ατ1(η) = sign(η) ⊙ max(∥η∥ − ατ1, 0).
+Solving for barycenter µ:
+µt+1 ← shrinkl1
+βτ1(µt + τ1(λt
+1 + λt
+2))
+(10)
+Solving for λ:
+λt+1
+i
+← λt
+i + τ2κt+1
+(11)
+2.2
+Toy example
+We demonstrate the barycenter computation using a Gaussian
+image in Figure 2. The barycenter of clean samples, sample
+with random noise and adversarial sample are shown. As we
+see, for the clean case the barycenter is very similar to that of
+the original image. In the second column where random noise
+is added, the barycenter reduces the noise.
+Similar effect
+is also seen for the case where adversarial noise is present.
+This indicates that non-linear interpolation of Barycenter sup-
+presses the adversarial noise.
+(a) Clean
+(b) Random
+(c) Adversarial
+Figure 2: Top: Clean image, noisy image, adversarial image. Bot-
+tom: Barycenter of clean image, noisy image, adversarial image.
+2.3
+Training
+Let a model be given by fθ, the barycenter of clean samples
+be denoted by x and its labels as y. We then optimize the
+following loss
+arg min
+θ
+1
+n
+n
+�
+i=1
+LCE(fθ(xi), yi)
+where LCE is the cross-entropy loss. We would like to em-
+phasize that we do not perform adversarial training. Instead
+we use an adversarially pretrained model. Thus, fθ is an ad-
+versarial robust model and our training further enhances the
+robustness. We also note that this optimization falls under
+DRO as the samples used are barycenters which belong to the
+distribution Q.
+2.4
+Theoretical analysis
+We first present a convergence analysis of Eq 6.
+Theorem 1.
+Let τ1τ2(λmax(∇2) + 3)
+<
+1, where
+λmax(∇2) denotes the largest eigenvalue of discrete Lapla-
+cian operator ∇2 = DD⊤, where D is the matrix repre-
+senting div operator. Then, the iterations Mt
+i, µ′
+i
+t
+i, µt, rt
+i, λt
+converge to the saddle point solution of the Lagrangian
+M∗
+i , µ∗
+i , µ∗, r∗
+i , λ∗.
+Proof: Let u = {M1, M2, µ2, µ2, µ, r}. Then, we write
+Eq 6 as
+L(u, λ) = G(u) + ⟨λ, ˜Kb⟩
+where λ
+=
+[λ1; λ2],
+K
+=
+[D, I, −I, −I],
+˜K
+=
+[K, 0; 0, K], b = [b1; b2], b1 = [vec(M1); µ′
+1; µ; r1]; b2 =
+[vec(M2); µ′
+2; µ; r2].
+The function G
+=
+∥M1∥2,1 +
+∥M2∥2,1 + α(∥r1∥1 + ∥r2∥1) +
+1
+2ρ(∥µ′
+1 − µ1∥2 + ∥µ′
+2 −
+µ2∥2) + β∥µ∥1 is convex and ˜K is a linear operator. These
+conditions satisfy Theorem 1 of [Chambolle and Pock, 2011].
+If λmax(∇2) is the max eigenvalue of DD⊤, then the max
+eigenvalue of [D, ±I][D, ±I]⊤ is λmax(∇2) + 1. Similarly,
+for KK⊤, it is λmax(∇2) + 3. Since ˜K is obtained from
+K by padding zeros only, ˜K has the same max eigenvalue
+as that of K. Further, since ∥ ˜K ˜K⊤∥2
+2 ≥ λmax( ˜K ˜K⊤) =
+λmax(∇2) + 3, we can also write the convergence criteria as
+τ1τ2∥ ˜K ˜K⊤∥2
+2 < 1.
+Since we solve for the Lagrangian dual function, we anal-
+yse the primal dual gap which is given as [Jacobs et al., 2019]
+G(u, λ) =
+sup
+∥λ′−λ0∥≤R1
+L(u, λ′) −
+inf
+∥u′−u0∥≤R2L(u′, λ)
+Theorem 2.
+Suppose the step sizes τ1 and τ2 satisfy
+τ1τ2∥ ˜K ˜K⊤∥2
+2 < 1. Let uN =
+1
+N
+�N
+n=1 un and λN =
+1
+N
+�N
+n=1 λn, where un and λn are sequences generated from
+Eqns 7 - 11. Then after N iterations, we have,
+G(u, λ) ≤ sup
+u,λ
+1
+2N
+�
+∥u − u0∥2
+τ1
++ ∥λ − λ0∥2
+τ2
+�
+This rate is similar to convergence rates in gradient descent
+and shows that the gap converges with rate O(1/N). For
+brevity, we omit the proof and it can be derived as an exten-
+sion of Theorem 1 [Chambolle and Pock, 2011].
+
+2.5
+Mutual Information
+In order to understand the underlying reason behind the per-
+formance of our method, we provide more insights using
+mutual information (MI). We first note that the MI between
+two random variables is given by I(X, y) = H(P(y)) −
+E
+P (x)[H(P(y|X))]. In our case, we take the random variables
+as model parameters θ and softmax output y. Then, given a
+sample x and dataset D,
+I(θ, y|D, x) = H(p(y|x, D)) −
+E
+p(θ|D)
+H(p(y|x, θ)) (12)
+Eq 12 measures the information shared between θ and y.
+A tractable way of computing I(θ, y|D, x) is given in [Smith
+and Gal, 2018], [Houlsby et al., 2011].
+I(θ, y|D, x) = 1
+C
+C
+�
+j=1
+1
+n
+n
+�
+i=1
+(pij − ˆp)2
+(13)
+where, ˆp ∈ [0, 1]C is computed as the mean of all softmax
+probabilities, C is the number of classes, pi ∈ [0, 1]C, pij ∈
+[0, 1] denotes the softmax probability for a particular class j.
+A higher I indicates that knowing θ (or y) gives a higher
+information about y (or θ). In other words, the model will
+perform better if the mutual information is high.
+In addition, we also compute MI between the predictions
+for the following two cases - (i) clean test set and adversarial
+test set, and (ii) barycenter of clean test set and barycenter of
+adversarial test set using [Ji et al., 2019]. Given a model f
+paramterised by θ, clean sample xi and its adversarial coun-
+terpart x′
+i, the joint probability distribution between natural
+and adversarial samples is given by the following C × C ma-
+trix,
+I(f(xi, θ), f(x′, θ)) =
+C
+�
+y=1
+C
+�
+y′=1
+Pyy′ ln Pyy′
+PyPy′
+(14)
+where, Pyy′ is given as,
+Pyy′ = 1
+n
+n
+�
+i=1
+f(xi, θ)f(x′
+i, θ)⊤
+(15)
+and the marginals Py, Py′ are obtained by row and column
+sum of Pyy′. A higher value of I(., .) indicates that know-
+ing about clean samples gives a higher amount of information
+about the adversarial samples.
+3
+Experiments
+We present elaborate experimental results on CIFAR-10,
+CIFAR-100 and Tiny ImageNet. We use strong baseline of
+LAS [Jia et al., 2022]. We also show improvements over
+other baselines - LBGAT [Cui et al., 2021], PGD-AT [Madry
+et al., 2017], TRADES [Zhang et al., 2019], RST [Carmon
+et al., 2019]. We compare against several popular adversarial
+training models, MART [Wang et al., 2019b], AWP-A2 [Xu
+et al., 2022], RST-RWT [Yu et al., 2022], TRADESAWP [Wu
+et al., 2020], AWP [Wu et al., 2020], LASAT, LASTRADES,
+LASAWP [Jia et al., 2022]. We also compare with adaptive test
+time defenses HedgeRST [Wu et al., 2021] and TRADESSSL
+[Mao et al., 2021]. In the Tables, we use “+B” to indicate the
+results obtained using our approach.
+3.1
+Implementation details
+In case of CIFAR10 and CIFAR100, WideResNet34-10 is
+used and for Tiny ImageNet PreActResnet18 is used. Ad-
+ditionally, we evaluate on CIFAR-10 with WideResNet28-
+10, WideResNet32-10, WideResNet70-16 and on CIFAR-
+100 with WideResNet34-20. We use these models for a fair
+comparison with existing works as these models are widely
+used for adversarial defense evaluation. We evaluate against
+different attacks namely FGSM, PGD-10, PGD-20, CW, and
+AA using l∞ attack with ϵ = 8/255. Our evaluation proto-
+cols are similar to the protocols given in [Zhang et al., 2019],
+[Jia et al., 2022]. We would like to emphasize that we use the
+checkpoints from the baseline models and perform a single
+epoch training using clean barycenters. Upon increasing the
+number of epochs, the clean accuracy improves, however, the
+adversarial accuracy becomes comparable to that of baseline
+and further increasing epochs leads to subsequent drop in ac-
+curacy against adversarial samples. We use SGD optimizer
+with a learning rate of 1e-4, momentum = 0.9 without any
+weight decay.
+In order to compute the barycenter, we set ρ = 5e−1, τ1 =
+1e − 1, τ2 = α = β = 1 and iterations is set to 200. While
+one can also attack the barycenter, we give experiments for
+the case where the clean image is attacked. This is because
+the barycenter itself lies at an ϵ which is greater than attacker’
+budget. Thus attacking barycenter has little incentive as in
+that case the attacked image will lie at an ϵ outside the given
+ϵ = 8/255 for the l∞ attack.
+3.2
+Comparison on CIFAR-10
+In Table 1, we observe that clean performance is better
+for the models trained with barycenters - TRADESAWP+B,
+LBGAT+B, LASTRADES+B and RST+B. Amongst WRN-
+28-10 models, RST has the best clean performance and our
+method enhances it by 1%. In PGD-10, there is a rise of
+2.57%. In case of AA, there is a boost of 6.49%.
+In case of WRN-34-10, LASAT+B shows a huge boost
+of 10% under AA. Further, LASAWP+B shows the best per-
+formance under PGD-10, PGD-20 and CW attack amongst
+WRN-34-10 models. Under AA it shows an improvement of
+7.71%.
+Comparison with Adversarial Purification models Our
+RST+B model outperforms Hedge∗
+RST under all the cases.
+Against AA, our approach gives 3.1% higher accuracy
+compared to Hedge∗
+RST.
+We also see that compared to
+TRADESSSL, TRADESAWP has a better performance.
+3.3
+Comparison on CIFAR-100
+In Table 2, we observe that our method gives a significant
+boost under all the cases. In case of strong baseline LASAWP,
+our method increases the performance by 0.85% under clean
+accuracy. For PGD-20, there is a rise of 0.91%. In case of
+CW, there is an increase of 18.35%. In other models such as
+LBGAT, we see a rise of 5.4% in clean accuracy.
+3.4
+Comparison with Curriculum based AT
+In Table 3, we compare against curriculum based AT meth-
+ods like CAT [Cai et al., 2018], FAT [Zhang et al., 2020] and
+
+Table 1: CIFAR-10.
+∗ indicates that the model uses WRN-28-10.
+Bold font is used to indicate the best performance amongst WRN-
+34-10 and Red color font is used to indicate the best performance
+amongst WRN-28-10.
+Method
+Clean
+PGD10
+PGD20
+CW
+AA
+Adversarial Training
+PGD-AT
+85.17
+56.07
+55.08
+53.91
+51.69
+TRADES
+85.72
+56.75
+56.10
+53.87
+53.40
+MART
+84.17
+58.98
+58.56
+54.58
+51.10
+AWP-A2
+87.54
+-
+59.50
+57.42
+54.86
+RST-RWT∗
+88.87
+-
+64.11
+62.03
+60.36
+Adversarial Purification
+TRADESSSL
+82.12
+-
+-
+-
+60.67
+Hedge∗
+RST
+88.64
+-
+-
+73.89
+63.10
+Adversarial and Barycentric Training
+TRADESAWP
+85.36
+59.58
+59.25
+57.07
+56.17
++B
+87.32
+62.60
+62.32
+75.85
+65.32
+LBGAT
+88.22
+56.25
+54.60
+54.29
+52.23
++B
+88.38
+59.28
+58.43
+74.61
+61.22
+LASAT
+86.23
+57.11
+56.41
+55.54
+53.58
++B
+86.21
+61.08
+60.64
+74.09
+63.59
+LASTRADES
+85.24
+57.66
+57.07
+55.45
+54.15
++B
+86.15
+60.32
+60.03
+73.75
+63.43
+LASAWP
+87.74
+61.09
+60.16
+58.22
+55.52
++B
+87.45
+63.66
+61.16
+74.81
+63.23
+RST∗
+89.69
+63.48
+62.51
+61.06
+59.71
++B∗
+90.68
+65.12
+64.38
+77.08
+66.20
+DART [Wang et al., 2019a]. Under FGSM, PGD-20 and CW,
+our model shows a huge improvement. In case of clean sam-
+ples, we see that the accuracy of FAT+B compared to FAT is
+less. This may be due to the fact that FAT employs curricu-
+lum learning in the training whereas our method does not use
+curriculum learning.
+3.5
+Comparison on Tiny ImageNet
+We present the results in Table 4. In comparison to baselines,
+our method shows significant improvement in all cases. For
+LASAWP, our method improves the performance under clean
+samples by 1.65%. In case of PGD-50, our method shows
+a rise of 1.28%, and in case of CW attack, our method al-
+most doubles the accuracy. Under AA, LASTRADES observes
+a maximum performance rise by 11.51%.
+3.6
+Analysis using Deeper and Wider Models
+We use WRN-70-16 and WRN-34-20 to analyse the effect
+when the models get deeper and wider. In particular, for clean
+samples, we can observe that the deeper and wider models
+give a better boost. In CIFAR-10, WRN-70-16 gives 88.87%
+for clean samples which is 3.21% better than LASAWP model’
+85.66%. In contrast, for WRN-34-10, our method gives accu-
+racy similar to that of LASAWP. In CIFAR-100, our method
+boosts the performance by 8.34% under AA. In other cases
+also we see that the barycenters improve the performance by
+a significant margin.
+Table 2: CIFAR-100 WRN-34-10.
+Method
+Clean
+PGD-10
+PGD-20
+CW
+PGD-AT
+60.89
+32.19
+31.69
+30.10
+TRADES
+58.61
+29.20
+28.66
+27.05
+TRADESAWP
+60.17
+33.81
+33.6
+57.07
++B
+63.67
+36.34
+36.15
+51.92
+LBGAT
+60.64
+35.13
+34.53
+30.65
++B
+66.04
+36.29
+36.01
+52.92
+LASAT
+61.8
+33.27
+32.83
+31.12
++B
+62.45
+36.60
+36.17
+49.60
+LASTRADES
+60.62
+32.82
+32.51
+29.51
++B
+62.58
+35.22
+34.96
+50.99
+LASAWP
+64.89
+37.11
+36.36
+33.92
++B
+65.50
+37.55
+37.27
+52.27
+Table 3: CIFAR-10 WRN-32-10.
+Method
+Clean
+FGSM
+PGD-20
+CW
+CAT
+77.43
+57.17
+46.06
+42.48
+DART
+85.03
+63.53
+48.70
+47.27
+FAT
+89.34
+65.52
+46.13
+46.82
++B
+84.59
+69.98
+57.02
+71.36
+3.7
+TSNE
+In Figure 3, we show the tsne plots for MNIST testset with
+classes 0 and 1. Here, we use a weak MNIST model which
+has only two dimensions before the classification layer. We
+deliberately chose a weak model so that we can easily show
+the effect in low dimensions. Though higher dmensions could
+be taken, the effect cannot be easily seen due to a highly non-
+linear transformation from high to low dimension of tsne. We
+can see that the two clusters yellow and purple are well sepa-
+rated for clean and barycenters of clean images. In case of ad-
+versarial samples, the points overlap on each other. However,
+when we take barycenter of adversarial samples, we again
+see that the clusters are well separated, similar to the case of
+clean images. Thus, it is evident that the barycenter nullifies
+the effect of adversarial noise.
+(a) Clean
+(b) Barycenter
+(c) Attacked
+(d) Adv.+Bary.
+Figure 3: Left to right: Plot of 2D features of Clean image, Barycen-
+ter of clean image, Attacked image, Barycenter of adversarial image.
+MNIST model obtains 51% accuracy and has only 2D feature vector
+before the classification layer.
+3.8
+Mutual Information
+In Table 6, we present the study of mutual information. We
+use LASAT and LASTRADES on CIFAR-10. The MI is com-
+puted using Eq 12 and Eq 14. Here we note that the MI for
+LASAT+B is more for training set compared to that of LASAT.
+
+Table 4: Tiny ImageNet PreActResNet18.
+Method
+Clean
+PGD-50
+CW
+AA
+LASAT
+44.86
+22.16
+18.54
+16.74
++B
+45.12
+24.54
+37.14
+27.78
+LASTRADES
+41.38
+18.36
+14.50
+14.08
++B
+43.07
+19.25
+35.13
+25.59
+LASAWP
+45.26
+23.42
+19.88
+18.42
++B
+46.91
+24.70
+37.93
+27.00
+Table 5: CIFAR-10 (C-10) WRN-70-16 and CIFAR-100 (C-100)
+WRN-34-20.
+Dataset
+Method
+Clean
+FGSM
+CW
+AA
+C-10
+LASAWP
+85.66
+70.25
+58.44
+57.61
++B
+88.87
+74.04
+75.40
+62.54
+C-100
+LBGAT
+62.55
+43.16
+31.72
+31.92
++B
+66.86
+50.92
+54.19
+40.26
+This indicates that the information available about the labels
+given the model parameters is high and in turn gives a better
+clean accuracy. In case of adversarial samples too, we see that
+the MI is higher for our case. This indicates that the model
+has better prediction for these samples. Further, the measure
+for test set is smaller compared to training set which is ex-
+pected as the model carries more information about train set
+compared to test set.
+Table 6: CIFAR-10 WRN-34-10.
+Method
+Train
+Test
+FGSM
+CW
+LASAT
+0.029
+0.026
+0.020
+0.019
++B
+0.034
+0.029
+0.023
+0.022
+LASTRADES
+0.048
+0.040
+0.033
+0.032
++B
+0.054
+0.045
+0.037
+0.036
+In Table 7, we present the results obtained using Eq 14. We
+observe that for the model trained with barycenter, the MI is
+higher between the barycenter of clean and adversarial sam-
+ples. Thus, the model does better on barycenter of adversarial
+samples compared to baseline LASAT and TRADESAWP. This
+is consistent across FGSM, PGD-10 and CW attacks.
+3.9
+Sensitivity to Barycenter Parameters
+In Figure 4, we demonstrate the sensitivity to different pa-
+rameters involved in the computation of barycenter. In the top
+row, we fix the number of iterations to 200 and τ1 = 1e − 1.
+Here we observe that increasing τ2 makes the barycenter
+brighter. In the second row, increasing τ1 makes the barycen-
+ter darker. Decreasing iterations has a similar effect in the last
+row. We see that unless there is a change of order of magni-
+tude, the appearance does not substantially change. Thus, our
+proposed Beckman barycenter is robust with respect to the
+parameter settings.
+4
+Conclusion
+In this work we introduce Beckman barycenter analogous to
+Wasserstein barycenter. We use the Beckman OT formula-
+Table 7: Mutial Information for CIFAR-10 WRN-34-10.
+Method
+FGSM
+PGD-10
+CW
+LASAT
+0.218
+0.198
+0.203
++B
+0.275
+0.241
+0.264
+LASTRADES
+0.576
+0.554
+0.563
++B
+0.629
+0.572
+0.618
+Figure 4: Top row: Blue boundary represent the given image.
+Barycenter for iterations = 200, τ1 = 1e − 1, τ2 = 1, 1e-1, 1e-2,
+1e-3. Second: barycenter for iterations = 200, τ1=1, 1e-1, 1e-2, 1e-
+3, τ2 = 1. Third: iterations = 200, 100, 50, 10, τ1=1e-1, τ2=1. The
+red boundary indicates the images obtained from default settings of
+the paramater which are used for all experiments.
+tion and analytically solve for the barycenter. Defining the
+baycenter using Beckman OT also has the advantage that the
+computational tools to obtain barycenter are well known and
+efficient. This overcomes the complexity in solving Wasser-
+stein barycenters. Further, we show that barycenter can be
+used for enhancing the performance of adversarially trained
+models. Our training is very efficient as we only need a sin-
+gle epoch. We theoretically show that our barycenters can
+help in defending against attacks. We perform rigorous qual-
+itative and quantitaive analysis to show the effectivenes of
+barycenter. Experimental analysis on CIFAR-10, CIFAR-100
+and Tiny ImageNet demonstrates state-of-art results against
+wide variety of attacks.
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+

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+Prepared for submission to JINST
+Improving primary-vertex reconstruction with a
+minimum-cost lifted multicut graph partitioning algorithm
+V. Kostyukhin,1 M. Keuper,2,3 I. Ibragimov,1 N. Owtscharenko,1 and M. Cristinziani1
+1Center for Particle Physics Siegen, Department Physik, Universität Siegen
+2Visual Computing, Department Elektrotechnik und Informatik, Universität Siegen
+3Max Planck Institute for Informatics, Saarland Informatics Campus
+Abstract: Particle physics experiments often require the simultaneous reconstruction of many
+interaction vertices. Usually, this problem is solved by ad hoc heuristic algorithms. We propose
+a universal approach to address the multiple vertex finding through a principled formulation as
+a minimum-cost lifted multicut problem.
+The suggested algorithm is tested in a typical LHC
+environment with multiple proton–proton interaction vertices. Reconstruction errors caused by the
+particle detectors complicate the solution and require the introduction of special metrics to assess
+the vertex-finding performance. We demonstrate that the minimum-cost lifted multicut approach
+outperforms heuristic algorithms and works well up to the highest vertex multiplicity expected at
+the LHC.
+Keywords: Vertexing algorithms; Pattern recognition, cluster finding, calibration and fitting
+methods
+arXiv:2301.05548v1  [hep-ex]  13 Jan 2023
+
+Contents
+1
+Introduction
+1
+2
+Minimum-cost multicuts and lifted multicut algorithm for cluster finding
+3
+3
+Data simulation
+5
+4
+Features of simulated data
+6
+5
+Edge weights and constraints
+6
+6
+Performance metrics
+10
+7
+Results
+12
+8
+Conclusion
+19
+A Non-clustered tracks and total reconstructed clusters
+21
+1
+Introduction
+In particle physics experiments, many problems require a precise reconstruction of vertices — points
+in 3D space where particle interactions occur. Knowledge of the positions and features of such
+vertices provides valuable information about the underlying physics of these interactions. There are
+numerous examples: 𝐵 physics, heavy-flavour jet identification, primary event vertex reconstruction,
+search for exotic particles as new physics manifestations, etc. Except for a few specifically designed
+detectors (emulsions, Wilson chamber, etc.), rarely used in modern experiments, the vertices are
+not directly detectable. The presence of vertices is usually inferred from 3D traces of charged stable
+particles produced in the interaction. Various tracking detectors measure the curved trajectories of
+these particles (tracks) in space. The trajectories can be extrapolated to a single 3D point, which
+represents the interaction vertex position [1].
+Despite the simplicity of the vertex reconstruction idea, its real-life exploitation encounters
+problems. For example, at the Large Hadron Collider (LHC) at the end of Run 2, a typical recorded
+event consisted of ∼80 primary proton–proton interactions, and numerous produced charged par-
+ticles underwent further interactions leading to additional vertices, distributed in significant 3D
+volumes. The expected number of proton–proton interactions in a single event at the LHC after
+the planned high-luminosity upgrade (HL-LHC) may reach 200–300, resulting in a few thousand
+reconstructed tracks. Therefore, prior to determining the vertex positions, one needs to determine
+how many vertices are present in a given event and assign the reconstructed tracks to these assumed
+vertices. The track measurement uncertainties, which may differ by a factor of 10 for different tracks
+– 1 –
+
+and often are comparable with the vertex–vertex distances, cause additional complications. These
+uncertainties make an exact crossing of track pairs in 3D space impossible: even if two charged
+particles are produced in the same interaction point, their reconstructed trajectories will only be
+close to the true vertex position and to each other, up to the corresponding uncertainties.
+The explicit reconstruction of multiple vertices in an event can be addressed in a graph-based
+approach. In fact, all space trajectories of the particles produced in a single vertex should be pairwise
+compatible, i.e. every pair of tracks should be close to each other in some volume around the true
+vertex position. Therefore, a compatibility graph can be constructed where every node represents
+a track. Two nodes are connected by an edge if and only if the distance of the corresponding
+trajectories is very small. In the ideal case, every vertex is represented by a fully connected, isolated
+subgraph in such a graph. In a realistic scenario, track measurement errors shuffle tracks among
+different vertices, resulting in a large number of fake edges in the compatibility graph. Yet, it can
+be tried to split the full graph into non-overlapping clusters by minimising the track–track distances
+for all track pairs in a cluster. The obtained set of isolated clusters should be a good approximation
+of the true vertices.
+The present paper focuses on finding primary proton–proton interaction vertices at the LHC.
+Subsequent decays of the particles produced in the detector volume and their interactions with
+the detector material will not be considered here. To illustrate the primary-vertex reconstruction
+problem, Figure 1 shows two zoomed-in regions of a typical LHC event with several pileup
+interactions. The upper plot presents a region where a pileup interaction vertex is identified, which
+has the largest sum of track transverse momenta.
+The bottom plot presents a region where a
+hard-scatter vertex, i.e. the point of interaction of interest, is identified. In both plots, the true
+positions of interaction vertices are shown, together with charged particle trajectories displaced
+due to reconstruction uncertainties. Several true interaction vertices in these plots do not have
+associated tracks because all emanated particles in this interaction are outside of the tracking
+detector’s sensitive volume, see Section 3 for the details. The overlap of the red (from hard-scatter
+vertex), blue and grey (from nearby pileup vertices) tracks in the centre of the bottom plot on
+Figure 1 is clearly visible.
+Experiments at the LHC use heuristic algorithms [3–5] to reconstruct multiple proton–proton
+interaction vertices. Several other approaches can be found in the literature, including medical
+imaging-inspired algorithms [6] and the RAVE package [7] implementing the deterministic anneal-
+ing algorithm [8].
+This article presents an implementation of the Lifted Multicut Graph Partitioning algorithm
+(LMC), which solves the inclusive vertex reconstruction problem described above.
+Section 2
+describes the LMC algorithm and details of its implementation for the vertex finding application.
+Section 3 describes the simulated samples which are used to test the algorithm performance. In
+Section 4, features of the simulated samples are discussed. Section 5 introduces edge cost functions
+used in the graph partitioning. In Section 6, the metrics are introduced to estimate the algorithm
+performance and to compare it with other existing approaches. Section 7 presents the performance
+of the LMC approach in simulation. In Section 8, conclusions are made.
+– 2 –
+
+61
+−
+60
+−
+59
+−
+58
+−
+57
+−
+56
+−
+55
+−
+54
+−
+53
+−
+52
+−
+z [mm]
+1
+−
+0.8
+−
+0.6
+−
+0.4
+−
+0.2
+−
+0
+0.2
+0.4
+0.6
+0.8
+1
+r [mm]
+Reco z = -56.71 mm
+Truth z = -2.59 mm
+2
+ = 470.3 GeV
+T
+2
+ p
+Σ
+w
+T
+ p
+Σ
+PU Vertex chosen by 
+Truth
+Reco
+HS tracks
+PU tracks
+HS jets
+PU jets
+truth PU vertex
+truth HS vertex
+ cut
+0
+ z
+T
+-p
+η
+Simulation Preliminary
+ATLAS 
+7
+−
+6
+−
+5
+−
+4
+−
+3
+−
+2
+−
+1
+−
+0
+1
+2
+z [mm]
+1
+−
+0.8
+−
+0.6
+−
+0.4
+−
+0.2
+−
+0
+0.2
+0.4
+0.6
+0.8
+1
+r [mm]
+Reco z = -2.60 mm
+Truth z = -2.59 mm
+2
+ = 121.2 GeV
+T
+2
+ p
+Σ
+Reconstructed Hard-Scatter Primary Vertex
+Truth
+Reco
+HS tracks
+PU tracks
+HS jets
+PU jets
+truth PU vertex
+truth HS vertex
+ cut
+0
+ z
+T
+-p
+η
+Simulation Preliminary
+ATLAS 
+Figure 1: Two regions of a typical LHC event in the ATLAS detector with many pileup interac-
+tions [2]. True positions of the proton–proton interactions are shown, as well as the reconstructed
+trajectories (tracks) of the produced particles scattered due to reconstruction uncertainties. Some
+truth interaction vertices do not have associated tracks because all emanated particles are outside
+of the sensitive detector phase space and not reconstructed. These pictures illustrate typical track
+densities and overlap of the tracks produced in nearby interaction vertices. Both, tracks associated
+with the hard-scattering (HS) and pileup (PU) are shown.
+2
+Minimum-cost multicuts and lifted multicut algorithm for cluster finding
+We formulate the primary-vertex reconstruction problem as a minimum-cost lifted multicut problem.
+This problem was originally proposed in Reference [9] in the context of image segmentation and
+mesh decomposition. It is a generalization of the better-known minimum cost multicut problem,
+also referred to as the weighted correlation clustering problem [10, 11]. The minimum cost multicut
+problem is a grouping problem defined for a graph 𝐺 = (𝑉, 𝐸) and a cost function 𝑐 : 𝐸 → R
+which assigns to all edges 𝑒 ∈ 𝐸 a real-valued cost or reward for being cut. Then, the minimum
+– 3 –
+
+cost multicut problem is to find a binary edge labelling 𝑦 according to
+min
+𝑦∈{0,1}𝐸
+∑︁
+𝑒∈𝐸
+𝑐𝑒𝑦𝑒
+(2.1)
+subject to
+∀𝐶 ∈ cycles(𝐺)
+∀𝑒 ∈ 𝐶 : 𝑦𝑒 ≤
+∑︁
+𝑒∈𝐶\{𝑒}
+𝑦𝑒 .
+(2.2)
+The constraints on the feasible set of labellings 𝑦 given in Equation (2.2) ensure that the solution
+of the multicut problem relates one-to-one to the decompositions of graph 𝐺, by ensuring for every
+cycle in 𝐺 that if an edge is cut within the cycle (𝑦𝑒 = 1), so needs to be at least one other. Trivial
+optimal solutions are avoided by assigning positive (attractive) costs 𝑐𝑒 to edges between nodes
+𝑣, 𝑤 ∈ 𝑉 that likely belong to the same component, while negative (repulsive) costs are assigned to
+edges that likely belong to different components.
+The minimum cost lifted multicut problem (LMC) generalizes over the problem defined in
+Equation (2.1)–Equation (2.2) by adding a second set of edges that defines additional, potentially
+long-range costs without altering the set of feasible solutions. It thus defines a second set of edges
+𝐹 between the nodes 𝑉 of 𝐺, resulting in a lifted graph 𝐺′ = (𝑉, 𝐸 ∪ 𝐹), on which we can define
+a cost function 𝑐′ : 𝐸 ∪ 𝐹 → R. Then, Equation (2.1) and Equation (2.2) are optimized over all
+edges in 𝐸 ∪ 𝐹 and two additional sets of constraints are defined according to [9]
+∀𝑣, 𝑤 ∈ 𝐹
+∀𝑃 ∈ 𝑣, 𝑤 − paths(𝐺) : 𝑦𝑣𝑤 ≤
+∑︁
+𝑒∈𝑃
+𝑦𝑒
+(2.3)
+∀𝑣, 𝑤 ∈ 𝐹
+∀𝐶 ∈ 𝑣, 𝑤 − cuts(𝐺) : 1 − 𝑦𝑣𝑤 ≤
+∑︁
+𝑒∈𝐶
+(1 − 𝑦𝑒)
+(2.4)
+to ensure that the feasible solutions to the LMC problem still relate one-to-one to the decompositions
+of the original graph 𝐺.
+For the vertex reconstruction problem, this formulation allows encoding Euclidean distance
+constraints in the structure of graph 𝐺 (e.g. point observations that are spatially distant can not
+originate from the same vertex), while the cost function can be naturally defined in the distance
+significance space to take into account the measurement errors. The Euclidean distance and its
+significance can be very different in case of significant reconstruction errors, the lifted multicut
+formulation encodes both metrics in the same graph.
+The minimum cost multicut problem is 𝑛𝑝-hard, and so is the minimum cost LMC problem
+[12]. Yet, efficient heuristic solvers provide practically good solutions [9, 13–16]. Here, we resolve
+to use the primal feasible heuristic KLj that has been proposed in Reference [9] and published in
+an open-source library1. KLj is an iterative approach that produces a sequence of feasible solutions
+whose cost decreases monotonically. It takes as input an initial edge labelling (for example, all
+edge labels are initially set to 0), a lifted graph and costs defined on all edges. In every step, it
+either moves nodes between two neighbouring components, moves nodes from one component into
+a new component or joins two components such as to decrease the cost of the multicut maximally
+according to Equation (2.1).
+1https://github.com/bjoern-andres/graph
+– 4 –
+
+3
+Data simulation
+To estimate the clustering performance, we simulated data using DELPHES [17]. The framework
+allows to perform a fast and realistic simulation of a general-purpose collider detector composed of
+an inner tracker, electromagnetic and hadron calorimeters, and a muon system. For this study, we
+added a detailed parameterisation of the ATLAS detector tracking resolution to the framework.
+To simulate the pileup vertices and hard-scattering events, a sufficiently large amount of
+minimum-bias interaction events was prepared, consisting of single, double, and non-diffractive
+processes. These events have been generated using the Pythia 8 [18] event generator. As the main
+source of hard-scattering interactions, 𝑡¯𝑡 events are used, also generated with Pythia 8. To simulate
+an LHC collision event with full pileup, a single 𝑡¯𝑡 event is mixed with a number of minimum-
+bias events, distributed according to a Poisson distribution with a mean corresponding to a chosen
+luminosity. The interaction vertices are then distributed along the LHC beam trajectory inside the
+detector, according to typical interaction region parameters for ATLAS, i.e. according to a Gaussian
+with 𝜎𝑧 = 42 mm.
+The acceptance of the ATLAS detector allows for reconstructing charged particle trajectories in
+a limited phase space of 𝑝⊥ > 500 MeV and |𝜂| < 2.5. Some minimum-bias proton–proton interac-
+tions produce only particles outside the sensitive phase space of the ATLAS detector, which makes
+them unreconstructable. Positions of interactions with a single track in the ATLAS acceptance can
+be reconstructed, but this vertex category is contaminated by tracks that are strongly displaced by
+measurement errors. In the following, a reconstructable truth vertex refers to the true position of a
+proton–proton interaction producing at least two tracks within the ATLAS detector acceptance.
+All tracks produced in an event and falling into the sensitive ATLAS detector phase space
+are smeared according to the parameterised ATLAS detector resolution.
+Tracks with smeared
+parameters are referred to as reconstructed tracks in the following. The set of reconstructed tracks
+corresponding to a full pileup event is used as input for the performance estimation of the clustering
+algorithms. DELPHES samples used in this paper have been prepared with different energies and
+different pileup conditions (Table 1).
+Energy
+�
+𝜇
+�
+Interaction region 𝜎��
+�
+𝑁event
+trk
+�
+�
+𝑁vrt
+trk =0
+�
+�
+𝑁vrt
+trk =1
+�
+�
+𝑁vrt
+trk >1
+�
+13 TeV
+63
+35 mm
+718
+9
+4
+50
+14 TeV 150
+42 mm
+1674
+22
+9
+119
+14 TeV 200
+42 mm
+2227
+28
+12
+160
+14 TeV 250
+42 mm
+2771
+35
+16
+199
+Table 1: The DELPHES samples used to estimate the LMC performance. Column 𝑁event
+trk
+reports
+the total number of reconstructed tracks in simulated events. The last three columns show the
+numbers of true vertices with 𝑁vrt
+trk = 0, 1, > 1, correspondingly.
+– 5 –
+
+4
+Features of simulated data
+The number of truth tracks in the detector acceptance in the simulated vertices and the position
+measurement errors of these tracks are shown in Figure 2. As can be seen in Figure 2a, in 14% of
+the cases, the simulated vertices do not have tracks in the detector acceptance, and in 6.5% of the
+cases, they have only one track. The number of tracks for all other vertices is widely spread up to
+80.
+0
+20
+40
+60
+80
+Number of tracks in a truth vertex
+0
+0.02
+0.04
+0.06
+0.08
+0.1
+0.12
+0.14
+Fraction
+a)
+0
+0.5
+1
+1.5
+2
+Track measurement error
+0
+0.02
+0.04
+0.06
+0.08
+0.1
+0.12
+0.14
+0.16
+0.18
+0.2
+Fraction
+b)
+Figure 2: a) Number of tracks per simulated vertex and b) Track measurement errors of the
+simulated tracks.
+Track measurement errors are shown in Figure 2b. From the sizes of the luminous regions and
+the number of vertices in Table 1 we can conclude that the track measurement errors are comparable
+or larger than a typical vertex–vertex distance in the simulated data. Smearing of the track positions
+due to measurement errors results in a significant overlap of the tracks from different truth vertices.
+The fraction of cases when a track from one vertex is entirely surrounded by tracks from other
+vertices for different pileup scenarios is shown in Table 2. An example of the track overlap can be
+seen in the bottom panel of Figure 1. Another example is shown in Figure 3.
+�
+𝜇
+�
+63
+150
+200
+250
+Track overlap fraction 20% 41% 53% 66%
+Table 2: Fraction of tracks, positioned in between tracks from other truth vertices due to measure-
+ment errors, as a function of the pileup.
+A priori, well-measured tracks with small errors should be easy to cluster according to the
+truth, while poorly measured tracks with large errors can easily migrate from one cluster to another,
+independently of their true origin. This random migration can be interpreted as noise, and thus, the
+overall problem may be considered as clustering in the presence of significant noise.
+5
+Edge weights and constraints
+To formulate the vertex finding problem in the presence of pileup as a minimum cost lifted multicut
+(LMC) problem, a track-pair compatibility graph needs to be constructed. A node in this graph
+– 6 –
+
+4
+6
+8
+10
+12
+Track and true vertex positions (mm)
+1.5
+−
+1
+−
+0.5
+−
+0
+0.5
+1
+1.5
+Track error (mm)
+Track positions linked to true vertices
+True vertices
+Figure 3: Example display of overlapping tracks from different vertices caused by measurement
+errors (zoom of a simulated DELPHES event with 𝜇 = 150). The crosses at the ordinate value
+of 0 represent the track positions, and the vertical error bars represent the corresponding position
+measurement errors. Squares at ordinate values of 1.3 represent the truth vertex positions. The
+connecting lines show the origin vertex for every track.
+represents a track, and two nodes are connected by an edge if and only if they are close in space and
+can be produced in the same vertex. The degree of track closeness, or equivalently the probability of
+originating in the same vertex, is estimated during the graph construction and is expressed as a weight
+assigned to the edge. The edge weights determine the efficiency of the clustering. Therefore, they
+should incorporate enough information, and the weight assignment procedure should be carefully
+designed. The following approaches are used in our study:
+1. Probability density function (PDF) ratio of the track–track geometrical distance significance
+based on measured uncertainties, 𝑆 =
+√︃
+(𝑧𝑖 − 𝑧 𝑗)2/(𝜎2
+𝑖 + 𝜎2
+𝑗 );
+2. Multivariate binary classification with Boosted Decision Trees (BDT);
+3. Logistic regression based on 𝑆.
+The LMC formulation assumes that the correct edges (two tracks from the same vertex) receive
+positive weights, while random (fake) edges receive negative weights. This can be achieved by
+using a logarithm of the ratio of the probability density functions for the correct and fake edges
+as the cost function of the problem log 𝑝true
+𝑝fake . According to the Neyman–Pearson lemma, this is
+the most efficient test statistic for the true/fake edge classification. An example of the track–track
+distance significance distributions and their ratio are shown in Figure 4. As the PDF of the fake
+edges is independent of the track–track distance significance, its overall normalisation depends on
+the significance range used for the parameterisation. Thus, the exact values of the PDF ratio can
+be scaled by the choice of the parametrisation range, which in principle, should not affect the LMC
+– 7 –
+
+clustering performance if the range is sufficiently large. Such a behaviour can be mimicked by a
+global multiplier of the PDF ratio function. The influence of this multiplier on the clustering will
+be studied in Section 7.3.
+0
+1
+2
+3
+4
+5
+6
+Track-track distance significance
+0
+0.002
+0.004
+0.006
+0.008
+0.01
+0.012
+0.014
+0.016
+0.018
+0.02
+0.022
+0.024
+True edges
+0
+1
+2
+3
+4
+5
+6
+Track-track distance significance
+0
+0.001
+0.002
+0.003
+0.004
+0.005
+0.006
+Fake edges
+0
+1
+2
+3
+4
+5
+6
+Track-track distance significance
+0
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+4
+4.5
+5
+Ratio True/Fake pdf's
+Figure 4: Example track–track distance significance for true and fake edges and their ratio. The
+significance distributions are normalized to one.
+A better clustering performance could be achieved by encoding more information in the edge
+weight calculation. To test this approach, we use a BDT classifier combining seven features, listed
+in Table 3, to distinguish true edges from fake ones. The GradientBoost implementation (BDTG)
+from the TMVA [19] package is used to train the classifier. An example of the trained classifier
+response2 is shown in Figure 5. The output is negative for fake edges and positive for true ones,
+exactly as required by the KLj algorithm, and therefore can be used directly as the edge weight.
+n. Description
+1
+Squared significance 𝑆2 (or 𝜒2) of track–track distance along beamline
+2
+Average position of the track pair along beamline
+3
+Position uncertainty of track 1
+4
+Position uncertainty of track 2
+5
+Pseudorapidity 𝜂 of track 1
+6
+Pseudorapidity 𝜂 of track 2
+7
+Number of other tracks crossing the beamline between tracks 1 and 2
+Table 3: Input features for the edge classification BDT.
+Edge weights can also be assigned by using the logistic regression 𝑝 = 𝑒𝑧/(𝑒𝑧 + 1), where
+𝑧 = 𝛽0 + �𝑛
+𝑖=1 𝛽𝑖𝑥𝑖 and 𝑥𝑖 are explanatory variables. The negative inverse of the logistic function,
+logit(𝑝) = log[𝑝/(1 − 𝑝)], provides the necessary edge weight behaviour. Edges that need to be
+removed receive negative weights, and those that need to be preserved receive positive weights.
+The intercept value 𝛽0 is defined by the ratio between the amount of true and fake edges used
+for training, which can be linked to a prior probability of a given edge being true or fake. In the
+2TMVA GradientBoost uses the binomial log-likelihood loss 𝐿(𝐹, 𝑦) = ln[1 + exp(−2𝐹(𝑥)𝑦)] with Gini Index
+separation. We use the following training settings NTree=800, MaxDepth=10, MinNodeSize=1.5%, Shrinkage=0.07.
+– 8 –
+
+0.8
+−
+0.6
+−
+0.4
+−
+0.2
+−
+0
+0.2
+0.4
+0.6
+0.8
+BDTG response
+0
+1
+2
+3
+4
+5
+6
+dx
+ /
+ 
+(1/N) dN
+Signal
+Background
+U/O-flow (S,B): (0.0, 0.0)% / (0.0, 0.0)%
+TMVA response for classifier: BDTG
+Figure 5: Example BDTG classification weight distributions for true and fake edges.
+current problem, the prior probability depends on the true vertex density and cannot be defined
+unambiguously, e.g. it depends on the range of the track–track distance significance 𝑆, see above.
+Therefore, the value of the intercept 𝛽0 in this approach can be modified in some range to achieve
+an over- or undersegmentation in order to validate its optimality. This will be further discussed in
+Section 7.3. A one-dimensional regression is tested in this paper, using variable (1) from Table 3.
+The logistic regression for the edge weight calculation is illustrated in Figure 6.
+0
+5
+10
+15
+20
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+S2
+True
+Figure 6: Example one-variable logistic regression for true and fake edges using the squared
+track–track distance significance 𝑆2.
+The usage of the track–track distance significance for partitioning does not guarantee the com-
+pactness of the obtained cluster in Cartesian space, which may be beneficial when the vertex density
+is large. The compactness requirement can be imposed using the LMC constraint mechanism.
+Some edges in the connectivity graph can be additionally labelled as “have to be cut”, based on a
+priori information, different from the edge probability itself. To make clusters more compact, we
+– 9 –
+
+can constrain the edges to be cut if the corresponding Cartesian track–track distance is larger than
+some scale. In the following, a rather weak requirement of |𝑧𝑖 − 𝑧 𝑗| < 1 mm will be used, which
+removes tracks with very large errors, see Figure 2b. In addition to improving the quality of the
+solution, the constraint limits the phase space of possible solutions, and this leads to a significant
+algorithm speedup.
+6
+Performance metrics
+For a quantitative assessment of the performance of the vertex-finding algorithm, one or several
+metrics are to be established. To compare the performance of the clustering algorithms in, e.g.,
+image segmentation problems, metrics are usually employed, which are based on the assignment
+of graph nodes to clusters. One example of such a metric is the Variation of Information (VI)
+proposed in Reference [20]. The VI metric calculates the degree of compatibility of a clustering 𝐶
+with another clustering 𝐶′ as
+𝑉𝐼(𝐶, 𝐶′) = 𝐻(𝐶) + 𝐻(𝐶′) − 2 · 𝐼(𝐶, 𝐶′)
+(6.1)
+with
+𝐻(𝐶) = −
+𝐾
+∑︁
+𝑘=1
+𝑃(𝑘) · log(𝑃(𝑘)) and 𝐼(𝐶, 𝐶′) =
+𝐾
+∑︁
+𝑘=1
+𝐾 ′
+∑︁
+𝑘′=1
+𝑃(𝑘, 𝑘′) · log
+� 𝑃(𝑘, 𝑘′)
+𝑃(𝑘)𝑃(𝑘′)
+�
+.
+(6.2)
+Here 𝑃(𝑘) = 𝑛𝑘/𝑁, 𝑃(𝑘, 𝑘′) = |𝐶𝑘 ∩ 𝐶′
+𝑘′|/𝑁, 𝑛𝑘 is the number of nodes in the cluster 𝐶𝑘, 𝑁 is
+the total number of nodes in the graph, and 𝐾 and 𝐾′ are the number of elements in 𝐶 and 𝐶′,
+respectively. In our case, the VI metric can be used to compare the truth track-to-vertex assignment
+with the obtained clustering solution. When the obtained set of clusters and the track-to-cluster
+assignment reproduce the truth exactly, 𝑉𝐼 vanishes. Consequently, smaller VI values correspond
+to more truth-like (and therefore better) clustering solutions.
+Another track-to-cluster-based metric, which is investigated in the following, is the Silhou-
+ette [21] score
+𝑠(𝑖) =
+𝑏(𝑖) − 𝑎(𝑖)
+max{𝑎(𝑖), 𝑏(𝑖)}
+(6.3)
+with
+𝑎(𝑖) =
+1
+𝑛𝑘 − 1
+𝐶𝑘
+∑︁
+𝑗, 𝑖≠𝑗
+𝑑(𝑖, 𝑗) and 𝑏(𝑖) =
+min
+𝐶𝑘′≠𝐶𝑘
+1
+𝑛 𝑗
+𝐶𝑘′
+∑︁
+𝑗
+𝑑(𝑖, 𝑗)
+(6.4)
+for node 𝑖 in cluster 𝐶𝑘. Here 𝑑(𝑖, 𝑗) is a distance between nodes 𝑖 and 𝑗. In this study, we use the
+Cartesian distance between tracks and average over all tracks silhouette value
+�
+𝑠(𝑖)
+�
+as a quality
+estimator of the clustering solution. The silhouette value is limited −1 < 𝑠(𝑖) < 1, larger values
+corresponding to more compact clusters, better separated from each other.
+Several other track-to-cluster-based metrics can be found in Reference [20]. These metrics are
+expected to encounter problems in the present case due to the overlap of truth clusters, as explained
+in Section 4. Tracks are assigned most probably to the wrong cluster by any partitioning algorithm
+if placed in between tracks from other clusters by mismeasurement. This phenomenon inevitably
+reduces the accuracy of any track-to-cluster-based metrics. Nevertheless, at least the clustering of
+– 10 –
+
+the well-measured tracks should reproduce the truth closely, which the track-to-cluster metrics can
+still be sensitive to.
+As the metric accuracy is compromised by the presence of tracks with large measurement
+errors, it might be useful to downscale the contribution of such tracks to the metric. For the VI
+metric this can be achieved by weighting every track with 𝜎−2 in the metric calculations, namely
+𝑛𝑘 = �𝑘
+𝑖=1
+1
+𝜎2
+𝑖 , 𝑁 = �𝑁
+𝑖=1
+1
+𝜎2
+𝑖 , etc. For the Silhouette metric the Cartesian distance between two
+tracks can be replaced by its significance 𝑑(𝑖, 𝑗) = 𝑆𝑖 𝑗. The weighted versions of the VI and
+Silhouette metric will be used in the following, along with the original versions.
+The number of reconstructed clusters and the weighted average positions of these clusters,
+dominated by the well-measured tracks, are mostly decoupled from the details of the track-to-
+cluster assignment. The number of clusters can be directly used as a metric (up to the possible
+presence of fake clusters), but a Cartesian distance-based metric is not straightforward. One may
+try to introduce such a metric exploiting the cluster–cluster resolution 𝑅𝑐𝑐, i.e. the minimal distance
+between two reconstructed clusters, see Figure 7. The good, merged, bad cluster categories could
+be defined based on whether the cluster–truth vertex distance is smaller or larger than 𝑅𝑐𝑐. Such
+cluster categories could be used to compare various clustering solutions. But this categorisation
+explicitly depends on 𝑅𝑐𝑐, which itself depends on the clustering algorithm. To avoid such circular
+dependence, a scale-independent Cartesian distance-based metric is needed.
+4
+−
+3
+−
+2
+−
+1
+−
+0
+1
+2
+3
+4
+ z[mm] 
+∆
+ 
+0
+20
+40
+60
+80
+100
+120
+140
+160
+180
+Events / 0.1mm
+=0.35mm
+CC
+R
+Figure 7: Example of a fit to the cluster–cluster distance to determine the resolution. The used
+fitting function is 𝑎/{1 + exp[𝑏 · (𝑅𝑐𝑐 − |𝑥|)]} + 𝑐 where a, b, c are free fitting parameters and 𝑅𝑐𝑐
+is the cluster–cluster resolution, defined as the half-width at the half-depth of the dip in the centre
+of the cluster–cluster weighted centre distances, averaged over all clusters.
+To construct such a metric, we propose the following procedure. Every reconstructable truth
+vertex is linked to the closest reconstructed cluster in the Cartesian space that has 2 or more assigned
+tracks. Thus, a list of linked reconstructed clusters is obtained. Then, every reconstructed cluster is
+classified depending on how many times it enters into this list. If a cluster enters this list only once,
+there is just a single truth vertex referencing this cluster. Therefore it can be called unique, which
+means that a truth vertex is unambiguously reconstructed as a cluster. If a cluster enters several
+times into the list, it is referenced by several truth vertices, and therefore it combines tracks from
+– 11 –
+
+these vertices: this cluster can be called merged. Also, some clusters may not appear in this list
+at all: such clusters are not referenced by any truth vertex and are thus fake. The total number of
+obtained clusters and their classification as unique, merged, fake are scale-independent and can be
+used as a metric to compare various clustering options.
+7
+Results
+7.1
+LHC Run-2 13 TeV data
+First, the LMC clustering algorithm is tested with simulated DELPHES data at a collision energy of
+13 TeV, with pileup
+�
+𝜇
+�
+= 63 and 𝜎𝑧 = 35 mm. These parameters are chosen to provide simulated
+data close to the actual data collected by the ATLAS detector in Run 2. Edge-weight distributions
+for various edge-labelling approaches on these data are shown in Figure 8. The performance of
+the LMC algorithm on these data is shown in Table 4. The rows labelled “cnst” in these tables
+provide performance estimation with the applied constraints |𝑧𝑖 − 𝑧 𝑗| < 1 mm, while the “base”
+rows describe the baseline algorithm performance without constraints.
+15
+−
+10
+−
+5
+−
+0
+5
+Edge weight
+0
+0.02
+0.04
+0.06
+0.08
+0.1
+0.12
+0.14
+0.16
+0.18
+0.2
+0.22
+0.24
+Density (arbitrary units)
+PDF ratio
+20
+−
+15
+−
+10
+−
+5
+−
+0
+5
+Edge weight
+0
+0.02
+0.04
+0.06
+0.08
+0.1
+0.12
+0.14
+0.16
+0.18
+0.2
+Density (arbitrary units)
+Logistic regression 1var
+1
+−
+0
+1
+2
+Edge weight
+0
+0.02
+0.04
+0.06
+0.08
+0.1
+0.12
+0.14
+Density (arbitrary units)
+BDT
+Figure 8: Typical edge weight distributions for various edge labelling options.
+The column 𝑁wrong
+trk
+in Table 4 is the number of tracks assigned to one cluster but entirely
+surrounded by tracks from other clusters. This number is an estimator for the degree of cluster
+overlap in the obtained solution. The relevant truth data overlap for comparison can be found
+in Table 1.
+In addition, Table 9 in the Appendix gives the number of isolated nodes (tracks)
+reported by the LMC clustering algorithm. These non-assigned tracks do not represent the one-
+track truth vertices, considered non-reconstructable without a priori information, but rather reflect
+the clustering problems.
+The PDF ratio and the regression-based edge weight assignment result in approximately equal
+clustering performance. The BDT-based edge weight assignment leads to a significantly worse
+Silhouette metric value, a smaller value of the cluster overlap and a larger amount of fake clusters.
+As expected, the weighted versions of the VI and Silhouette metrics have significantly better values
+than the standard ones due to downscaling of the noise. Using constraints uniformly improves all
+quality estimators and provides ∼ 30% CPU reduction.
+In total, 70% of the reconstructable truth vertices are reconstructed as unique clusters, while
+the remaining 30% (i.e. 15) truth vertices are squeezed into 7.5 merged vertices. The amount of
+– 12 –
+
+Edge weight
+VI
+VI
+Silhouette
+Silhouette
+Unique
+Merged
+Fake 𝑁wrong
+trk
+CPU
+weighted
+weighted
+PDF ratio
+base 0.839
+0.407
+0.615
+0.646
+33.3
+8.2
+2.4
+15%
+0.25s
+cnst
+0.782
+0.362
+0.649
+0.660
+33.9
+7.9
+2.3
+8%
+0.18s
+Regression
+base 0.860
+0.416
+0.589
+0.623
+34.7
+7.6
+4.1
+14%
+0.27s
+cnst
+0.829
+0.387
+0.614
+0.633
+35.0
+7.5
+3.9
+8%
+0.18s
+BDT
+base 0.945
+0.399
+0.478
+0.230
+35.0
+7.5
+7.1
+5%
+0.23s
+cnst
+0.937
+0.377
+0.487
+0.234
+35.2
+7.4
+7.0
+4%
+0.14s
+Table 4: LMC performance for the collision energy 13 TeV, pileup 63 and interaction region width
+𝜎𝑧 = 35 mm. These simulation parameters are chosen to match the full ATLAS simulation for Run
+2 results used for comparison. The column 𝑁wrong
+trk
+shows the fraction of tracks wrongly associated
+by the clustering algorithm, which shall be compared to the truth fraction of 20% (Table 2).
+fake clusters is in the range of 5–15%. The number of tracks in the different cluster categories is
+presented in Figure 9. The number of tracks in the unique clusters is close to the track amount in
+the truth vertices, see Figure 2, while the merged clusters contain much more tracks. Finally, fake
+clusters have a very small number of tracks.
+0
+10 20 30 40 50 60 70 80 90 100
+Number of tracks in cluster
+0
+0.02
+0.04
+0.06
+0.08
+0.1
+Density (arbitrary units)
+Unique vertices
+0
+10 20 30 40 50 60 70 80 90 100
+Number of tracks in cluster
+0
+0.005
+0.01
+0.015
+0.02
+0.025
+0.03
+0.035
+Density (arbitrary units)
+Merged vertices
+0
+10 20 30 40 50 60 70 80 90 100
+Number of tracks in cluster
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+Density (arbitrary units)
+Fake vertices
+Figure 9: Number of tracks in a cluster for the unique, merged and fake cluster categories. The
+distributions are obtained for pileup
+�
+𝜇
+�
+= 63 data using a one-variable logistic regression for the
+edge weight assignment.
+7.2
+High-Luminosity LHC 14 TeV data
+The High Luminosity LHC (HL-LHC) project foresees a significant increase in interaction rates
+to collect significantly more data and thus increase the sensitivity for new physics. The exact
+parameters of the upgraded HL-LHC are not yet final; pileup values of 150, 200, and 250, and an
+interaction region width of 𝜎𝑧 = 42 mm are considered the most probable options. These options
+result in an increase in the density of pileup interaction vertices up to a factor of 4, as compared
+to the current LHC parameters. The degree of truth cluster overlap rises from 20% to 66%, see
+– 13 –
+
+Table 1. It is interesting to check the performance of the LMC problem formulation in such extreme
+conditions.
+For this test, the same PDF ratio and logistic regression function are used for the edge weight
+calculation, while the BDT classification is retrained using 𝜇 = 150, 200, 250 data. Results for
+nominal PDF ratio and logistic regression-based edge weight calculation functions are shown in
+Tables 5, 6, and 7.
+Edge weight
+VI
+VI
+Silhouette
+Silhouette
+Unique
+Merged
+Fake 𝑁wrong
+trk
+CPU
+weighted
+weighted
+PDF ratio
+base 1.318
+0.690
+0.535
+0.577
+57.7
+27.4
+4.8
+28%
+1.1s
+cnst
+1.211
+0.612
+0.581
+0.609
+59.4
+26.9
+4.1
+14%
+0.42s
+Regression
+base 1.316
+0.682
+0.514
+0.559
+63.0
+25.6
+8.8
+26%
+0.73s
+cnst
+1.259
+0.634
+0.546
+0.582
+63.6
+25.4
+8.2
+14%
+0.50s
+BDT
+base 1.303
+0.658
+0.394
+0.146
+61.8
+25.9
+13
+9%
+0.96s
+cnst
+1.275
+0.616
+0.409
+0.155
+62.8
+25.6
+12
+7%
+0.43s
+Table 5: LMC performance for pileup 𝜇 = 150 in an HL-LHC environment with collision energy
+14 TeV and interaction region size 𝜎𝑧 = 42 mm. The column 𝑁wrong
+trk
+shows the fraction of the
+tracks, wrongly associated by the clustering algorithm, which can be compared to the true fraction
+41% (Table 2).
+Edge weight
+VI
+VI
+Silhouette
+Silhouette
+Unique
+Merged
+Fake 𝑁wrong
+trk
+CPU
+weighted
+weighted
+PDF ratio
+base 1.574
+0.852
+0.500
+0.546
+64.3
+40.3
+5.7
+36%
+2.3s
+cnst
+1.441
+0.756
+0.552
+0.586
+66.6
+39.8
+4.8
+18%
+0.69s
+Regression
+base 1.546
+0.825
+0.492
+0.539
+70.3
+38.6
+9.0
+32%
+2.4s
+cnst
+1.470
+0.765
+0.529
+0.568
+71.0
+38.4
+8.1
+18%
+0.69s
+BDT
+base 1.512
+0.805
+0.312
+0.040
+69.9
+38.6
+15.6
+13%
+1.8s
+cnst
+1.479
+0.755
+0.332
+0.051
+71.3
+38.2
+15.0
+7%
+0.66s
+Table 6: LMC performance for pileup 𝜇 = 200 in an HL-LHC environment with collision energy
+14 TeV and interaction region size 𝜎𝑧 = 42 mm. The column 𝑁wrong
+trk
+shows the fraction of the
+tracks, wrongly associated by the clustering algorithm, which can be compared to the true fraction
+53% (Table 2).
+Similarly to the 𝜇 = 63 results, the BDT-based edge weight assignment leads to a significantly
+worse Silhouette metric value, a much smaller value of the cluster overlap and a larger number of
+fake clusters, while the PDF ratio and regression-based edge weight calculation approaches provide
+similar performances. The weighted versions of the VI and Silhouette metrics have significantly
+better values than the standard ones due to downscaling of the noise.
+The use of constraints
+– 14 –
+
+Edge weight
+VI
+VI
+Silhouette
+Silhouette
+Unique
+Merged
+Fake 𝑁wrong
+trk
+CPU
+weighted
+weighted
+PDF ratio
+base 1.782
+0.990
+0.477
+0.526
+68.7
+53.2
+6.4
+42%
+3.0s
+cnst
+1.638
+0.887
+0.531
+0.569
+71.0
+52.7
+5.3
+21%
+1.7s
+Regression
+base 1.753
+0.961
+0.467
+0.517
+77.1
+51.2
+11.
+38%
+3.2s
+cnst
+1.672
+0.895
+0.505
+0.547
+77.8
+51.1
+9.9
+21%
+1.7s
+BDT
+base 1.691
+0.941
+0.307
+0.040
+72.8
+52.4
+15.
+12%
+3.0s
+cnst
+1.651
+0.882
+0.330
+0.055
+74.5
+52.0
+14.
+9%
+1.2s
+Table 7: LMC performance for pileup 𝜇 = 250 in an HL-LHC environment with collision energy
+14 TeV and interaction region size 𝜎𝑧 = 42 mm. The column 𝑁wrong
+trk
+shows the fraction of the
+tracks, wrongly associated by the clustering algorithm, which can be compared to the true fraction
+66% (Table 2).
+significantly improves all quality estimators and provides ∼ 30% CPU reduction.
+The number of unambiguously reconstructed unique clusters is 53% (44%, 37%) out of the
+total amount of the reconstructable truth vertices for the pileup 𝜇 = 150 (200, 250). The remaining
+56 (90, 125) reconstructable truth vertices are clustered into 25 (40, 52) merged clusters. The
+correctness of representation of the initial truth vertices by merged clusters is not granted. Truth
+vertices with a large number of tracks might “absorb” vertices with a small number of tracks.
+7.3
+LMC performance adjustment
+As can be seen from Tables 4–7, different edge weight assignment approaches lead to non-coinciding
+clustering results. For a practical application of the LMC approach for primary vertex finding in
+the LHC experiments, it is important to verify whether a unique optimal clustering solution exists
+in this problem and, if so, whether the different LMC cost functions can be tuned to provide the
+same clustering. As explained in Section 5, parameters of the PDF ratio and regression function for
+the edge weights can be modified to enforce under- or over-segmentation.The PDF ratio function
+can be scaled up and down. In the logistic regression function, the intercept term can be shifted by
+a constant. The cost function modifications are tried on the 𝜇 = 150 data. The obtained clustering
+results are shown in Figure 10 and Figure 11.
+In the performed test, the exploited metrics change monotonically depending on the scale factor
+for the PDF ratio and the intercept shift for the linear regression function. It doesn’t seem possible
+to adjust the PDF ratio and logistic regression parameters so that both approaches provide exactly
+the same clustering performances in all used metrics. In addition, the BDTG-based Silhouette and
+Silhouette weighted metrics results (see Table 5) are not reproducible by any modification of the
+PDF ratio and logistic regression cost functions. However, the overall variations of the clustering
+results remain limited, which means that the LMC approach performance stays close to optimal in
+the full scanned parameter range.
+To conclude, the cost function modification test doesn’t demonstrate the presence of an evident
+unique globally optimal clustering solution for the problem in consideration. Three used edge
+– 15 –
+
+0.8
+0.9
+1
+1.1
+PDF ratio scale
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+1.1
+1.2
+1.3
+1.4
+Metrics
+VI
+VI weighted
+Silhouette
+Silhouette weighted
+0.8
+0.9
+1
+1.1
+PDF ratio scale
+0
+20
+40
+60
+80
+100
+Clusters
+All
+Unique
+Merged
+Fake
+Resolution
+0
+0.2
+0.4
+Resolution (mm)
+Figure 10: PDF ratio cost-based clustering results as a function of the applied scaling.
+0.2
+−
+0
+0.2
+0.4
+Regression intercept shift
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+1.1
+1.2
+1.3
+1.4
+Metrics
+VI
+VI weighted
+Silhouette
+Silhouette weighted
+0.2
+−
+0
+0.2
+0.4
+Regression intercept shift
+0
+20
+40
+60
+80
+100
+Clusters
+All
+Unique
+Merged
+Fake
+Resolution
+0
+0.2
+0.4
+Resolution (mm)
+Figure 11: Logistic regression cost-based clustering results as a function of the logistic regression
+intercept term shift.
+weight assignment strategies provide different clustering results, which can be additionally changed
+by simple modification of the cost functions. Therefore, for a practical application as a primary
+vertex finder, an exact LMC formulation should be chosen based on desired physics requirements,
+e.g. minimal amount of fake vertices or best vertex–vertex resolution, disregarding the clustering
+metrics.
+7.4
+Influence of tracks with large measurement errors
+As the truth cluster overlap is caused by the track position mismeasurement, the overlap degree
+can be reduced by removing the badly measured tracks by cutting on the track measurement error
+shown in Figure 2b. A moderate decrease in the total amount of tracks due to this rejection should
+not significantly affect the overall clustering efficiency as the total amount of tracks per truth vertex
+is big enough, see Figure 2a. Reduction of the amount of the selected tracks and the degree of the
+truth cluster overlap due to strongly mismeasured track removal is shown in Table 8. The results
+– 16 –
+
+Track error cut
+𝑁trk
+Truth overlap
+-
+1674
+41%
+0.8
+1540
+31%
+0.6
+1444
+27%
+0.4
+1283
+22%
+Table 8: Number of selected tracks and the truth degree of overlap as a function of the track error
+cut for 𝜇 = 150 data.
+of the clustering are shown in Figure 12 for the PDF ratio cost function and in Figure 13 for the
+nominal logistic regression cost function.
+0.5
+1
+1.5
+Track error cut(mm)
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+1.1
+1.2
+1.3
+1.4
+Metrics
+VI
+VI weighted
+Silhouette
+Silhouette weighted
+0.5
+1
+1.5
+Track error cut (mm)
+0
+20
+40
+60
+80
+100
+Clusters
+All
+Unique
+Merged
+Fake
+Figure 12: PDF ratio cost-based clustering results as a function of the applied track error cut for
+the 𝜇 = 150 data.
+0.5
+1
+1.5
+Track error cut(mm)
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+1.1
+1.2
+1.3
+1.4
+Metrics
+VI
+VI weighted
+Silhouette
+Silhouette weighted
+0.5
+1
+1.5
+Track error cut (mm)
+0
+20
+40
+60
+80
+100
+Clusters
+All
+Unique
+Merged
+Fake
+Figure 13: Logistic regression cost-based clustering results as a function of the applied track error
+cut for the 𝜇 = 150 data.
+– 17 –
+
+The distance-based metric demonstrates very small changes in the clustering results in a
+wide range of the badly measured track admixture and, correspondingly, the initial degree of the
+vertex overlap. One may conclude that the amount of clusters identified by the LMC algorithm is
+largely defined by the tracks with small measurement errors and, therefore, is stable with respect
+to significant track noise admixture. Redistribution of the tracks with big errors over the obtained
+clusters doesn’t change their amount but evidently strongly affects all track counting-based clustering
+metrics. The track weighting does mitigate this effect for the VI metric, its weighted version is
+practically independent of the track noise admixture. Surprisingly, the Silhouette metric is only
+weakly sensitive to this noise.
+7.5
+Comparison with the existing approaches
+6
+−
+4
+−
+2
+−
+0
+2
+4
+6
+z [mm]
+∆
+arbitrary units
+          
+t
+AMVF, t
+           
+t
+IVF, t
+Preliminary
+ Simulation
+ATLAS
+ = 60
+〉
+µ
+〈
+ = 13 TeV, 
+s
+0
+10
+20
+30
+40
+50
+60
+70
+80
+ interactions per bunch crossing
+pp
+Number of 
+0
+10
+20
+30
+40
+50
+60
+Average number of reconstructed vertices
+100% interaction reconstruction efficiency
+Reconstruction acceptance
+t
+AMVF, t
+t
+IVF, t
+AMVF - MATCHED
+AMVF - MERGED
+AMVF - SPLIT
+AMVF - FAKE
+ATLAS Simulation Preliminary
+ = 13 TeV
+s
+6
+−
+4
+−
+2
+−
+0
+2
+4
+6
+ z[mm] 
+∆
+ 
+0
+200
+400
+600
+800
+1000
+1200
+1400
+Events / 0.02mm
+DELPHES simulation LMC Cluster-Cluster distance 
+=0.37mm
+CC
+R
+0
+10
+20
+30
+40
+50
+60
+70
+80
+Number of pp interactions per bunch crossing
+0
+10
+20
+30
+40
+50
+60
+Average number of reconstructed clusters
+LMC All
+LMC Unique
+LMC Merged
+100% interaction reconstruction efficiency
+Reconstruction acceptance
+Figure 14: The vertex–vertex resolution and the number of reconstructed vertices as a function
+of the number of 𝑝𝑝 interactions for typical ATLAS data. The upper plots are obtained with the
+the ATLAS baseline AMVF [4] and IVF [3] algorithms. The bottom plots are obtained using the
+LMC algorithm with the PDF ratio-based edge weight assignment on DELPHES 𝜇 = 63 data.
+The DELPHES 𝜇 = 63 simulation is specially tuned to match the ATLAS data used in [4]. The
+cluster–cluster resolution for the LMC algorithm on the bottom left picture is obtained as described
+in Section 6.
+The ATLAS Collaboration used the IVF algorithm [3] to reconstruct the 𝑝𝑝 collision vertices
+in Run 1 and the AMVF algorithm [4] in Run 2 and Run 3. Essential characteristics of a primary-
+vertex reconstruction algorithm are the vertex–vertex resolution and the number of reconstructed
+– 18 –
+
+vertices as a function of the number of 𝑝𝑝 interactions. The upper plots in Figure 14 present the
+corresponding distributions for typical ATLAS data for the AMVF and IVF algorithms. The bottom
+plots show the same distributions provided by the LMC algorithm using DELPHES data tuned to
+the same pileup conditions.
+Figure 14 clearly demonstrates that the LMC algorithm outperforms the ATLAS heuristic
+algorithms. It provides significantly better vertex–vertex resolution. This naturally leads to a larger
+amount of Unique/Matched vertices reconstructed by LMC, while the amount of Merged vertices
+remains practically the same. Routine application of the LMC for the primary vertex reconstruction
+can provide a significant gain in performance for LHC and future collider experiments.
+8
+Conclusion
+In this work, we have addressed a typical particle physics problem of reconstructing multiple
+interaction positions in a dense environment, where each interaction is represented by a cluster
+of tracks. Significant track reconstruction errors lead to a large overlap of truth track clusters,
+which makes their identification challenging. Heuristic algorithms are usually used to address this
+problem. In contrast, we propose to address this problem through a principled formulation as a
+minimum-cost lifted multicut problem. We construct several cost functions for the LMC from
+track–track distances and their significance. We study the performance of the LMC algorithm
+for different vertex densities, cost functions, constraint usage and varying degree of overlap. To
+address potential performance problems of existing track counting clustering metrics for strongly
+overlapped clusters, dedicated metrics are introduced.
+We demonstrate that the LMC approach outperforms the heuristic algorithms in the problem
+of vertex reconstruction in dense environments in terms of vertex–vertex resolution and vertex
+reconstruction efficiency. It works up to the highest vertex density expected at the HL-HLC project
+in spite of the strong truth cluster overlap reaching ∼ 60%. Variations of the LMC algorithm
+parameters and cost functions studied in this work resulted in relatively small variations of the
+obtained clustering solutions.
+Acknowledgments
+This work is supported by the German Science Foundation (DFG) through a research grant and a
+Heisenberg professorship under contracts CR-312/4-1 and CR-312/5-1.
+References
+[1] R. Frühwirth and A. Strandlie, Pattern Recognition, Tracking and Vertex Reconstruction in Particle
+Detectors. Springer, 2021, 10.1007/978-3-030-65771-0.
+[2] ATLAS Collaboration, “Primary Vertex Selection in VBF Higgs to Invisibles at 𝜇 = 200 with the
+ATLAS Experiment.” IDTR-2019-004, 2019.
+[3] ATLAS Collaboration, Reconstruction of primary vertices at the ATLAS experiment in Run 1
+proton–proton collisions at the LHC, Eur. Phys. J. C 77 (2017) 332.
+– 19 –
+
+[4] ATLAS Collaboration, “Development of ATLAS primary vertex reconstruction for LHC Run 3.”
+ATL-PHYS-PUB-2019-015, 2019.
+[5] CMS Collaboration, Description and performance of track and primary-vertex reconstruction with
+the cms tracker, JINST 9 (2014) P10009.
+[6] S. Hageböck and E. von Toerne, Medical imaging inspired vertex reconstruction at LHC, Journal of
+Physics: Conference Series 396 (2012) 022021.
+[7] W. Waltenberger et al., Rave—a detector-independent vertex reconstruction toolkit, Nuclear
+Instruments and Methods in Physics Research A (2007) 549.
+[8] K. Rose, Deterministic annealing for clustering, compression, classification, regression and related
+optimisation problems, Proceedings of the IEEE 86 (1998) 2210.
+[9] M. Keuper, E. Levinkov, N. Bonneel, G. Lavoue, T. Brox and B. Andres, Efficient decomposition of
+image and mesh graphs by lifted multicuts, Proceedings of the IEEE International Conference on
+Computer Vision, ICCV (2015) .
+[10] E. D. Demaine, D. Emanuel, A. Fiat and N. Immorlica, Correlation clustering in general weighted
+graphs, Theoretical Computer Science 361 (2006) 172.
+[11] S. Chopra and M. R. Rao, The partition problem, Mathematical Programming 59 (1993) 87.
+[12] A. Horňáková, J.-H. Lange and B. Andres, Analysis and optimization of graph decompositions by
+lifted multicuts, Proceedings of the International Conference on Machine Learning, ICML (2017)
+[1503.03791].
+[13] T. Beier, T. Kroeger, J. Kappes, U. Köthe and F. Hamprecht, Cut, glue & cut: A fast, approximate
+solver for multicut partitioning, Proceedings of the IEEE Conference on Computer Vision and Pattern
+Recognition, CVPR (2014) .
+[14] J. H. Kappes, B. Andres, F. A. Hamprecht, C. Schnörr, S. Nowozin, D. Batra et al., A Comparative
+Study of Modern Inference Techniques for Structured Discrete Energy Minimization Problems,
+International Journal of Computer Vision 115 (2015) 155 [1404.0533].
+[15] T. Beier, B. Andres, U. Köthe and F. A. Hamprecht, An efficient fusion move algorithm for the
+minimum cost lifted multicut problem, Computer Vision – ECCV 2016. Lecture Notes in Computer
+Science (2016) .
+[16] A. Kardoost and M. Keuper, Solving minimum cost lifted multicut problems by node agglomeration,
+Computer Vision – ACCV 2018 (2019) .
+[17] DELPHES 3 collaboration, DELPHES 3: a modular framework for fast simulation of a generic
+collider experiment, JHEP 02 (2014) 057 [1307.6346].
+[18] C. Bierlich et al., A comprehensive guide to the physics and usage of PYTHIA 8.3, 2203.11601.
+[19] A. Hoecker et al., TMVA - Toolkit for Multivariate Data Analysis, physics/0703039.
+[20] M. Meilă, Comparing clusterings—an information based distance, Journal of Multivariate Analysis
+98 (2007) 873.
+[21] P. J. Rousseeuw, Silhouettes: A graphical aid to the interpretation and validation of cluster analysis,
+Journal of Computational and Applied Mathematics 20 (1987) 53.
+– 20 –
+
+A
+Non-clustered tracks and total reconstructed clusters
+In this study, we use four simulated event samples representing realistic proton–proton interactions at
+the LHC with different energies and luminosities. The total amounts of interaction vertices with one
+reconstructed track and two and more tracks are shown in Table 9. Due to the track measurement
+errors, the one-track vertices are difficult to reconstruct correctly without a priori information.
+Finding two and more track vertices becomes problematic if the vertex–vertex distance is less than
+the typical track measurement error. Both problems are illustrated in Table 9, where the amounts
+of the one-track and multi-track clusters are given for every cost function and event sample.
+13 TeV
+14 TeV
+�
+𝜇
+�
+= 63
+�
+𝜇
+�
+= 150
+�
+𝜇
+�
+= 200
+�
+𝜇
+�
+= 250
+𝑁vrt
+ntrk=1
+𝑁vrt
+ntrk>1
+𝑁vrt
+ntrk=1
+𝑁vrt
+ntrk>1
+𝑁vrt
+ntrk=1
+𝑁vrt
+ntrk>1
+𝑁vrt
+ntrk=1
+𝑁vrt
+ntrk>1
+Truth
+4
+50
+9
+119
+12
+160
+16
+199
+𝑁cl
+ntrk = 1
+𝑁rec
+clust
+𝑁cl
+ntrk = 1
+𝑁rec
+clust
+𝑁cl
+ntrk = 1
+𝑁rec
+clust
+𝑁cl
+ntrk = 1
+𝑁rec
+clust
+PDF ratio
+11
+44
+19
+90
+23
+110
+25
+128
+Regression
+13
+46
+25
+97
+27
+118
+31
+139
+BDTG
+43
+50
+77
+101
+102
+124
+104
+140
+Table 9: Average numbers of non-clustered tracks and reconstructed clusters obtained by the
+LMC algorithm with different cost functions as compared to the truth numbers of single-track and
+multi-track vertices. Results are shown for all collision energies and pileup densities.
+The number of one-track clusters in each case is significantly larger than the truth amount of
+one-track interaction vertices, especially in the BDTG case. They should be thought of as non-
+clustered tracks, not as reconstructed one-track vertices. The fraction of multi-track clusters found
+decreases with the interaction vertex density, as expected.
+– 21 –
+

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+Computer Anxiety: Supporting the Transition
+from Desktop to Mobile
+Thiago Donizetti dos Santos
+Federal University of ABC (UFABC)
+Santo André, SP, Brazil
+[email protected]
+Vagner Figueredo de Santana
+IBM Research
+São Paulo, SP, Brazil
+[email protected]
+ABSTRACT
+Computer Anxiety is a phenomenon studied in multiple contexts
+and, in the actual COVID-19 scenario, it is gaining more and more
+importance as it impacts technology adoption and autonomy. Peo-
+ple with Computer Anxiety (PwCA) might feel intimidated, afraid
+of feeling embarrassed or scared of damaging computers, even
+before the actual interaction. Thus, supporting the detection of
+Computer Anxiety at scale has the potential to support the tech-
+nology industry to cope with this challenge. This position paper
+presents a user study involving 39 elderly participants in an inves-
+tigation on the feasibility of using interaction events common to
+desktop and smartphones to predict different levels of Computer
+Anxiety. Moreover, it also proposes research directions about the
+role of smartphones in the context of Computer Anxiety for elderly
+people as a mean of supporting good first user experiences with
+technology, meaningful daily use, privacy, and feeling safe even
+when doing mistakes. We expect this position paper motivates prac-
+titioners, designers, and developers to consider Computer Anxiety
+as one of the existing barriers when creating mobile applications
+for elderly people.
+CCS CONCEPTS
+• Human-centered computing → Field studies.
+KEYWORDS
+Computer Anxiety; Aging; Older Adults; Accessibility; Usability;
+User Experience; smartphones; mobile
+ACM Reference Format:
+Thiago Donizetti dos Santos and Vagner Figueredo de Santana. 2021. Com-
+puter Anxiety: Supporting the Transition from Desktop to Mobile. , 7 pages.
+1
+INTRODUCTION
+The use of smartphones in daily activities is increasing in a fast pace.
+The ownership of smartphones by adults, in US, grew from 35%
+to 81% in the period between 2011 and 2019 [6]. Smartphones are
+generally used to make and receive calls, to access the internet, to
+text, to access social media and services (e.g., such as food delivery,
+transport, and mobility). Bearing in mind the ownership of devices,
+the rate of people owning smartphones drops from 96% (for the
+Permission to make digital or hard copies of part or all of this work for personal or
+classroom use is granted without fee provided that copies are not made or distributed
+for profit or commercial advantage and that copies bear this notice and the full citation
+on the first page. Copyrights for third-party components of this work must be honored.
+For all other uses, contact the owner/author(s).
+CHI ’21, Workshop on Designing Interactions for the Ageing Populations, May 08–13,
+2021, Yokohama, Japan
+© 2021 Copyright held by the owner/author(s).
+group aging between 18 and 29 years) to 53% (for the group aging
+65+ years) [6]. This difference shows that elderly people are increas-
+ingly using smartphones, but have not yet adopted this technology
+in the same way as young adults, which may indicate the existence
+of aspects impacting the adoption of smartphones by elderly users.
+Since a variety of services and content are currently available
+online, the difficulties faced by elderly people while using smart-
+phones may impact their quality of life. Through the use of mobile
+applications, one could order food, get a driver using a transport
+app, travel alone using maps, communicate with family members
+using instant messaging, learn new things on e-learning platforms
+or just browse photos and other contents on the social media appli-
+cations. Such services foment autonomy and help to avoid common
+stereotypes of dependency and limitations.
+One phenomenon that can help in the understanding of issues
+faced by the elderly people when using new technologies is Com-
+puter Anxiety (CA). CA can be defined in terms of affective factors
+such as intimidation, fear, apprehension, hostility, and worries that
+one will be embarrassed, will look stupid, or thinks she/he could
+damage the computer [15]. Although generally related to the use of
+computers, CA can also impact the use of other electronic devices
+and previously was also called “Technophobia” [3]. CA is related
+to technology acceptance [36] and it is generally related to biologi-
+cal changes such as blood pressure, heart rate and electrodynamic
+responses that occur while a person is using a device [30]. CA
+symptoms can occur during the interaction with the system and
+even before it, affecting the perceived ease of use and acting as a
+barrier, impacting the system accessibility as well [35]. Although
+CA can affect people of all ages, the literature shows that CA is
+more present in older groups [7, 12, 35]. Adding this to the rela-
+tionship between CA and technology acceptance, elderly people
+may face problems to use mobile devices and, when they use it, CA
+could make the system difficult to use, make them perform poorly
+on tasks or fail to achieve their goals while using the device.
+In this context, this position paper presents results from a user
+study involving 39 elderly participants aiming at exploring the
+feasibility of predicting CA from interaction events common to
+desktop and smartphones, mapped here as a regression problem.
+In addition, it also discusses the role of smartphones in supporting
+people with CA (PwCA) in the process of learning how to use
+technology, first experience, daily use, and autonomy. Thus, the
+following research questions were defined to guide the study: (1)
+Is it possible to predict different levels of CA using interaction events
+common to desktop and smartphones? and (2) How can smartphones
+be used to support the adoption of new technologies by PwCA? The
+work is organized as follows: the section 2 presents the related
+work, the section 3 details the user study, the section 4 shows the
+arXiv:2301.02201v1  [cs.HC]  5 Jan 2023
+
+CHI ’21, Workshop on Designing Interactions for the Ageing Populations, May 08–13, 2021, Yokohama, Japan
+Thiago Donizetti dos Santos and Vagner Figueredo de Santana
+results, section 5 discusses the role of smartphone for PwCA and
+section 6 concludes.
+2
+RELATED WORK
+CA is also called Computerphobia, Computer Apprehension, and
+Technophobia in the literature [3]. Rosen et al. [32] pointed out the
+following methods and questionnaires to measure CA:
+• Computer Anxiety Index (CAIN) examines avoidance of,
+caution with, negative attitudes toward, and disinterest in
+computers [23].
+• Computer Attitude Scale (CAS) assesses computer liking,
+confidence, and anxiety through a Likert attitude-measurement
+format [21].
+• Attitudes Toward Computers Questionnaire (ATCQ) as-
+sesses attitudes towards computer appreciation, usage, and
+societal impact [31].
+• Computer Anxiety Rating Scale (CARS) assesses behav-
+ioral, cognitive, and affective components related to technol-
+ogy use [15, 32].
+• Mobile Computer Anxiety Scale (MCAS) assesses anxiety
+regarding mobile computer using a 38-item Likert scale [39].
+The literature presents multiple factors associated with CA. In
+sum, PwCA usually have less experience in using computers, have
+low Computer Self-efficacy (CSE)1, take too long to accomplish
+tasks, perform worse when compared to other users, have negative
+beliefs about computer/skills, or negative bodily sensations previ-
+ous/during the interaction with a computer [35]. Earlier studies find
+a strong relationship between age and CA levels, showing evidence
+that CA is more present in groups with older people and that they
+have more CA than younger ones [7, 12, 27]. This can be related
+to the pace in which technology advances and to the fact that 48%
+of older adults report that they usually need someone else to set
+up a new electronic device or show them how to use it [5]. In this
+scenario, mobile accessibility has potential to support PwCA in
+increasing CSE, reducing negative beliefs and worries of using or
+damaging the device in front of other people.
+CA is also present in a few acceptance models. The Technology
+Acceptance Model (TAM) is one example. It uses the CA as a compo-
+nent that changes the perceived ease of use2. So, when considering
+the adoption of new technology by elderly people, CA should be
+taken into account, from design to personalization features.
+The influence of CA was investigated in contexts such as accep-
+tance of e-learning tools, e-gov, new technologies and health-care
+systems [8, 14, 17, 18, 22, 29, 36]. These studies stated that:
+• PwCA tend to prefer traditional classes instead of e-learning
+systems and computer-based tests;
+• PwCA usually perform worse on virtual classes when com-
+pared to people without CA;
+• PwCA have more difficulties in accepting new technologies;
+• Older PwCA face difficulties using home telehealth services
+and to learn how to use smartphones.
+1Computer Self-efficacy (CSE) is the belief one has in his/her own abilities to perform
+a task in the computer [9]
+2Perceived ease of use is defined as the degree to which a person believes that using a
+particular system would be free of effort [1]
+Considering support offered to PwCA, the literature presents
+that instructional/technical support reduce CA in the context of
+e-learning systems [13, 24, 26]. Finally, smartphones have potential
+to provide instructional support in a privacy respecting way, sup-
+porting the user-technology dialog, reducing worriers associated
+to trial and error inherent to learning.
+3
+METHOD
+This section details how the user study was run, including its materi-
+als, procedure, setup, experiment design, and data analysis planned.
+The goal of the study was to collect data related to questionnaires
+to identify different CA levels and collect detailed interaction data
+while participants performed tasks on a website in order to detect
+CA. Moreover, this study also aimed at exploring interaction data
+common to desktop and smartphone and at understanding the role
+of smartphone usage for PwCA. The types of data captured will be
+detailed in the Data Analysis section. Before the main experiment,
+a pilot was performed in order to assess the user study plan as
+whole. The pilot included 4 elderly participants, recruited the same
+way the participants of the main experiment (detailed in the next
+section). The pilot achieved its goals in assessing protocol adopted,
+duration, and data capture procedure. Next, we detail the method
+followed in the main experiment.
+3.1
+Participants
+Elderly people may face difficulties in staying up-to-date with tech-
+nology and, since they have not used computers since childhood,
+many of them face CA even when performing a simple task to
+others age groups [35]. Hence, the target-audience considered in
+this work is elderly people.
+The participants of the experiment were recruited from a list
+of registered people at the elderly center of the city of São Paulo,
+Brazil, called Reference Centre for Citizenship of Elderly (CRECI@).
+São Paulo is the biggest city in Latin America, with a population
+of approximately 11.2 million people in the last census (2010) and
+the current estimate is of 12.2 million people3; the population of
+elderly people is approximately 11.9%4. Before recruiting partici-
+pants, a partnership was signed and the proper process for ethics
+committee was followed at the Federal University of ABC (process #
+2.808.392 and CAEE: 94704418.8.0000.5594), detailing the materials,
+procedure, questionnaires, data to be collected, and analysis to be
+performed.
+Moreover, only those who had never taken computer classes
+offered by CRECI@ were invited as potential participants, since
+results from the literature point that computer classes might reduce
+CA [35], which could result in a bias.
+3.2
+Materials
+Questionnaires about CA, use of smartphones and computers skills
+were applied (Appendix A). In order to isolate CA from other co-
+morbidities, questionnaires to assess cognitive abilities and levels
+of depression were also applied. Only participants with low levels
+of depression and those who do not present signals of dementia or
+3https://cidades.ibge.gov.br/brasil/sp/sao-paulo/panorama
+4http://produtos.seade.gov.br/produtos/retratosdesp/view/index.php?
+temaId=1&indId=4&locId=3550308
+
+Computer Anxiety: Supporting the Transition
+from Desktop to Mobile
+CHI ’21, Workshop on Designing Interactions for the Ageing Populations, May 08–13, 2021, Yokohama, Japan
+cognitive deficits had their data considered in the analyses. These
+two metrics were considered in the exclusion criteria, detailed in
+the procedure. The questionnaires applied are listed below:
+• Technology use and profile: Has questions about the par-
+ticipant’s age, educational level, and frequency of use of
+computers and smartphones. This questionnaire was applied
+to give an overview of how participants use technology on
+a daily basis (Appendix A).
+• Mini Mental: A cognitive screening test used for adults and
+the elderly to evaluate orientation, memory and attention,
+naming ability, obedience to verbal and writing commands,
+free writing of a sentence, and copying a complex drawing
+(two intersecting polygons). It is currently the most used test
+for this type of assessment in the world [25]. The rationale
+for using Mini Mental was to identify comorbidity to CA.
+• Geriatric Depression Scale (GDS): GDS is a scale with 30
+yes/no questions used for screening depression in elderly
+people [2, 40]. The rationale for using GDS was also to iden-
+tify comorbidity to CA.
+• Computer Anxiety Rating Scale (CARS): CARS has nine-
+teen questions in a five points Likert scale ranging from
+strongly disagree to strongly agree. It assesses the behavioral,
+cognitive and affective components related to technology
+use [15, 32]. The rationale for using CARS is that it is the
+most referenced questionnaire for screening CA [35].
+• Computer Self-Efficacy (CSE): CSE is a ten items scale used
+to assess Computer Self-efficacy [9, 38]. The rationale for
+using CSE was to cross check the results from this study
+with results from the literature that show that CSE has a
+strong but inverse relationship with CA.
+• System Usability Scale (SUS): A five points Likert scale ques-
+tionnaire with ten items. It is often used to assess the per-
+ceived usability [4]. The rationale for using it was to compare
+perceived usability of the website and the CARS values.
+In order to capture the interaction data, the participants used a
+desktop computer including a interaction logger and internet access.
+The interaction logger used was the open source logger called User
+Test Logger5. The User Test Logger captures all JavaScript events
+such as mouse movements, clicks, keys pressed, etc., and generates
+a raw log file where each line represents an event and information
+about when, where, and what is related to the event triggered [34].
+3.3
+Procedure
+The experiment was structured into three steps: pre-test, test, and
+post-test. The steps are detailed next.
+3.3.1
+Pre-test. First, screening tests for cognitive deficit, depres-
+sion, and literacy were applied to identify participants whose scores
+fall outside the inclusion criteria. The Mini Mental presents a score
+indicating good cognitive capacity relating the answer points ob-
+tained and years of education of the participant as follows:
+• No formal education: ≤ 21 points;
+• 1 to 5 years of formal education: ≤ 24 points;
+• 6 to 11 years of formal education: ≤ 26 points;
+• 12+ years of formal education: ≤ 27 points
+5https://github.com/IBM/user-test-logger
+For GDS screening test, a score of four points or less on the scale
+indicates low levels of depression. Hence, the exclusion criterion
+was: 𝐺𝐷𝑆 ≥ 5 points. After the tests considered in the exclusion
+criteria, CARS and CSE were applied.
+3.3.2
+Test. The tasks were performed individually on a computer
+with the User Test Logger installed. First, each participant was asked
+to access SESC homepage (Figure 1). The Social Service of Com-
+merce (SESC) is a private entity maintained by the entrepreneurs of
+the trade in goods, tourism, and services. SESC aims to provide the
+welfare and quality of life to workers in this sector and their fami-
+lies 6. SESC offers many activities for elderly people such as courses,
+sports, art exhibition and culture-related events in multiple units in
+the metropolitan area of São Paulo. The tasks were defined aiming
+to encourage participants to search online for activities offered by
+SESC and others services in the city, hoping to help them to see the
+internet as a tool which they can use as a means of improving their
+quality of life. The same way they do at CRECI@. In addition, the
+tasks were structured to be as familiar and as close to real tasks as
+possible. The tasks read out loud to participants were the following:
+(1) Search for an event, class, or activity he/she might be inter-
+ested;
+(2) Find the address of the unit where the chosen activity/event
+is offered;
+(3) Find the route to the unit.
+Figure 1: Homepage of the SESC’s website.
+There was no maximum time limit for the tasks. Thus, the task
+duration depended on participants saying whether they finished or
+gave up on the task. Finally, Thinking-Aloud Protocol [20] was used
+to understand the rationale of users while performing the tasks.
+3.3.3
+Post-test. In order to evaluate the perceived usability and
+any relationship with CA levels, they were asked to answer the
+SUS questionnaire.
+3.4
+Data analysis
+According to the exclusion criteria defined, the data from partic-
+ipants who did not score the points required by Mini Mental or
+scored five or more points on GDS were removed from the data set
+to be analyzed. The resulting data set combined data from the inter-
+action logger, questionnaires, and thinking-aloud protocol. Since
+the logged data were in raw format, all data captured were pro-
+cessed in order to extract usage metrics. The following metrics were
+considered having in mind they could also be applied in a mobile
+interaction setting: time to perform the task, number of clicks and
+double-clicks, click duration, typing velocity, and total time typing.
+6https://www.sescsp.org.br/
+
+3 Pagina Inicial - Sesc SP - Mozilla Firefox
+Debugging with Firefox Developer T X
+S Pagina Inicial - Sesc SP
++
+@ https://www.sescsp.org.br
+Q Pesquisar
+国三不
+Um Portal para cada um!
+Login
+Esqueci a senha I Cadastre-se
+Para ver os destaques da programagao de acordo com seus interesses
+fEntrar com Facebook
+ou
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+ senha
+ok
+online, cadastre-se aqui.
+OPORTUNIDADES
++ CREDENCIAL PLENA
+Meu perfil
+Sesc
+SAO PAULO
+O que voce procura?
+0000.
+Projeto Sawe
+Encontros no Sesc Ipiranga debatem a luta dos
+indios pela defesa de seus territorios e os desafios de
+construir o futuro mediante contextos impostos pela
+sociedade nao indigenaCHI ’21, Workshop on Designing Interactions for the Ageing Populations, May 08–13, 2021, Yokohama, Japan
+Thiago Donizetti dos Santos and Vagner Figueredo de Santana
+Finally, all metrics and the questionnaires scores were combined
+in a single comma-separated-values (CSV) file used to perform
+the data analysis. This CSV file is the main data source for the
+regression analysis performed to predict CARS values based only
+on interaction data, which could allow its use at scale.
+4
+RESULTS
+The experiment included 39 participants, but data from 8 partici-
+pants were not considered in the analysis due to exclusion crite-
+ria. Thus, considering the data from the remaining 31 participants
+(51.61% of males, 48.39% of females). The participants’ ages ranged
+between 62 and 87 years (𝑥 = 72.84). Regarding the computer usage,
+25.81% of the participants reported that they do not own a computer
+and do not have frequent access to computers, while 12.90% do not
+own a computer, but use it at lanhouses or those available in public
+places; 61.29% reported owning computers. Regarding frequency
+of use, 61.29% reported that rarely (less than once a month) use a
+computer, 16.13% reported that use it sometimes (more than once
+a month), 6.45% reported that usually (more than once a week)
+use it, and 16.13% reported that always (everyday) use computers.
+When considering ownership and use of smartphones, 87.10% of
+the participants reported that they own smartphones and 48.39%
+reported that they always use it. The education level of the par-
+ticipants varies from 0 to 15 years of formal education (𝑥 = 10.42)
+(Figure 2).
+Figure 2: Distribution of years of formal education.
+The obtained scores for CARS ranged from 20 to 59 (𝑥 = 42.19).
+Based on [10], the maximum and minimum scores were used to
+divide the data into 3 groups as follows: 𝐶𝐴𝑅𝑆_𝑟𝑎𝑛𝑔𝑒 : 59 − 20 = 39
+and 𝐺𝑟𝑜𝑢𝑝_𝑟𝑎𝑛𝑔𝑒 : 𝐶𝐴𝑅𝑆_𝑟𝑎𝑛𝑔𝑒/3 = 13.
+• No CA: CARS < 33 (6 participants);
+• Moderate CA: 33 ≤ CARS < 46 (14 participants);
+• High CA: CARS ≥ 46 (11 participants).
+Considering the three CA groups, Figure 3 shows the presence of
+CA considering the participants’ age. It can be seen that participants
+in the no CA group are among the youngest. The age of these
+participants ranged from 63 to 78 years old (𝑥 = 68.43, 𝜎 = 5.68).
+The age of the participants in the moderate CA group ranged from
+62 to 83 years old (𝑥 = 73.92, 𝜎 = 5.60). And, for the high CA group,
+the age of the participants ranged from 63 to 87 (𝑥 = 74.36, 𝜎 = 6.77),
+showing that high levels of CA are present over almost the entire
+age range covered in the study. Mann-Whitney non-parametric test
+shows that age was different between no CA and moderate CA
+groups (p-value = 0.03); no significant difference was found in other
+pairwise group comparisons.
+Figure 3: Age distributions for different CA groups.
+Bearing in mind the use of smartphones, Figure 4 shows smart-
+phone ownership and different uses by different CA groups. It can
+be seen that high CA group uses smartphones more for calls and less
+for leisure and other communication activities (e.g., games, music,
+video, instant messages, and social networks). On the other hand,
+people in the no CA group use smartphone heavily to access social
+network, instant messages, and internet. Analyzing the ownership
+rate by CA groups, it can be seen that 73% of the participants with
+high CA own smartphones, while the ownership is greater in the
+moderate CA group (92%) and reaches 100% for the no CA group.
+Similarly, considering the frequency of use of computers, people
+who rarely use computers are the ones with greater CA levels. 82%
+of the participants with high CA use it rarely, while it is 61.5% of
+the participants of the moderate CA and 28% of the no CA group.
+Figure 4: Smartphone ownership and use by CA groups.
+Bearing in mind task completion, 23 (74.19%) participants found
+an activity and completed the first task. Four participants found an
+activity that was not of interest to them or was offered by a unit
+far from their home, but they did not find another one after that.
+The remaining four gave up without finding an activity. For the
+Task 2, nine out of 23 (39.13%) participants found the address of
+the unit where the selected activity is offered and 14 participants
+found only the name of the unit. Regarding the Task 3, six out of
+nine (66.67%) participants found the map available on the site, but
+all of them failed to find the route to the unit. Only two out of
+nine (22.22%) participants figured out how to put the starting point
+address on the map-based UI. In sum, task completion dropped from
+74.19% (task 1), to 39.13% (task 2), and to 0% (task 3); 22.22% (6.45%,
+considering 31 participants) partially completed the last task. This
+can be related to task difficulty, fatigue effects, and task dependence.
+
+Education levels (in years)
+12
+ participants
+10
+Number of 
+2
+0
+4
+6
+8
+10
+11
+13
+14
+15
+Education yearsAge vs. CARS groups
+85
+80 
+e
+75
+70 
+65
+。
+No CARS
+Moderate CARS
+High CARSSmartphone use x Computer Anxiety Leve
+Music/Video
+Message
+Games
+Call
+Instant Message
+Social networks
+Internet
+Smartphone ownership
+0%
+10%
+20%
+30%
+40%
+50%
+60%
+70%
+80%
+%06
+100%
+ No CA
+ Moderate CA
+High CAComputer Anxiety: Supporting the Transition
+from Desktop to Mobile
+CHI ’21, Workshop on Designing Interactions for the Ageing Populations, May 08–13, 2021, Yokohama, Japan
+However, after triangulating these results with thinking-aloud data,
+it was possible to identify that participants faced difficulties with
+the map-based UI. For instance, participant 11 said: “It doesn’t say
+where it is. I didn’t like SESC.”, participant 35 said: “Why don’t you
+have the address on the about the unit page?” and participant 42
+said: “It will take me a long time to find it (address)”. While using
+the map, for instance, participant 22 said: “What should I do here?
+I have never used it (map) before”; participants are numbered from
+1-4 for the pilot and 5-43 for the experiment.
+Table 1 summarizes the time taken by each group to complete
+the tasks, showing the mean time and standard deviation by CA
+group. It can be seen that the group of participants with high CA
+had a mean shorter task time than the other groups in some tasks.
+This might be related to the fact that PwCA usually gave up more
+because they feel frustrated, lost, or think that they would not be
+able to finish the task. This also shows the relationship between
+high CA and low CSE, as identified in previous studies [35] and
+[11]. This can be exemplified by the participants quotes as: “I think
+I will have difficulty in this task”, “This is difficult”, “I’m lost, I don’t
+know what to do”, and “I don’t know how to find it”.
+Task 1 in sec.
+Task 2 in sec.
+Task 3 in sec.
+Group
+𝑥 (𝜎)
+𝑥 (𝜎)
+𝑥 (𝜎)
+High CA
+411.18 (230.72)
+599.67 (220.30)
+398.00 (262.19)
+Mod. CA
+647.90 (666.46)
+450.13 (336.52)
+560.50 (30.50)
+No CA
+525.14 (331.39)
+587.00 (390.73)
+399.67 (375.05)
+Table 1: Average task time by group and standard deviations.
+Although the interaction data was collected during the use of a
+desktop computer, in this study we explore metrics which can be
+captured in a mobile setting as well, namely: task time, numbers
+of clicks and double clicks, mean click duration (interval between
+pressing and releasing), typing velocity and total time typing. All
+metrics were normalized for the regression analysis. Prior to fit-
+ting the regression model, a random oversampling algorithm was
+applied7 addressing the minority values and a train-test split of
+80% / 20% was applied. Figure 5 shows Random Forest regression
+predictions for CA values (y-axis) vs. CARS values in the test set
+(x-axis). The obtained regressor has a mean squared error (MSE) of
+22.21 and 𝑅2 = 0.84. The high MSE value is due to errors related to
+predictions for lower CA scores. This pessimist prediction would
+show that users need more support than they would actually need,
+so such regressor might be useful for indicating when support for
+PwCA could be applied.
+5
+DISCUSSION
+Although there are few studies reporting no significant relationship
+between age and CA [16, 19, 28], there are also evidences that older
+people manifest more CA than younger ones [7, 12, 27, 33, 37].
+Results suggest that younger participants were in the no CA group,
+while the older were in the moderate CA group. For high CA group,
+results suggest that there were participants with high CA almost
+in the whole age range considered, but it still shows a greater
+concentration among the older ones (median = 74 years, 𝑥 = 74.2,
+7https://imbalanced-learn.org/stable/over_sampling.html
+Figure 5: Regression test results of CARS scores prediction.
+𝑠𝑖𝑔𝑚𝑎 = 7.12). These results are similar to other findings in the
+literature, indicating that the CA is more present in the older groups.
+Moreover, unlike [30], the results suggest that previous experience
+may impact CA levels, since people who rarely use computers and
+smartphones were the ones with greater CA levels.
+The difficulty in adopting new technologies is suggested by re-
+sults in Figure 4. Although there is a high rate of PwCA owning
+smartphones, the most frequent reported use is making calls. Thus,
+the participants use the smartphone, but they use the same way
+they used to do with the old phones: making calls. Moreover, the no
+CA group presented the same rate (57%) when using smartphones
+to make calls, using instant message and social networks apps. In
+contrast, moderate and high CA groups use more to make calls
+than to any of the other functions analyzed. Also they use more
+to make calls then the no CA group and presented a lower rate of
+ownership of smartphones than the no CA group. The increasing
+rate of smartphone ownership and the decreasing rate of use of
+different smartphone functions show possible impacts of CA on
+the behavior of the elderly regarding technology. The second most
+frequent use is instant message apps, even though there is also a
+difference between groups for this use. The use of this application
+could be related to the presence of functions such as the ability to
+make calls using the app or using voice messages. They reported to
+be used to make calls and that the use of voice messages is easier for
+them, since they generally have difficulties using the keyboard of
+the smartphones to write and may have difficulty in reading due to
+the size of the screen and letters. These findings suggest that high
+levels of CA may affect people of all ages, but it is more present
+among older ones due to factors such as lack of practice or lack of
+knowledge about recent features that could improve UX.
+The results regarding task completion and time taken to complete
+tasks show how CA levels might affect task performance. Besides
+the low task completion rate for the high CA group, this group
+took less time trying to complete the Task 1, showing that PwCA
+usually gave up more. The implications for HCI researchers in this
+aspect are related to the design of shorter and simpler tasks and to
+improve user experience, accessibility and usability, since high CA
+levels impact negatively the perceived easy of use and CSE, as they
+may feel frustrated or lost when facing problems to achieve their
+goals during the interaction.
+
+Random Forest Regression
+50 
+30 
+区
+21
+31
+50
+CARSCHI ’21, Workshop on Designing Interactions for the Ageing Populations, May 08–13, 2021, Yokohama, Japan
+Thiago Donizetti dos Santos and Vagner Figueredo de Santana
+CA is also related to specific situations since it tends to arise or
+be stronger during the first use of a device [17]. Thus, in addition
+to promoting the contact of the elderly with new technologies, it is
+important to promote a good first experience. It means an experi-
+ence free of effort, that helps the user to feel safe and unafraid of
+making mistakes. A bad experience, during which the user feels lost
+or makes mistakes, can reinforce the fears of PwCA. Consequently,
+this can increase their CA levels, making them believe they are
+unable to use it and prevent them from trying again.
+The results of the Random Forest regression show that CA in-
+fluences on how users interact with the system. Although the data
+belongs to desktop computer interaction domain, the result sug-
+gests that such approach should be explored in the mobile settings
+as well, given that the interaction events selected are common to
+desktop and smartphones. The prediction of higher CARS values
+could trigger personalization features and additional support, for
+instance. Smartphones have a myriad of sensors that could be used
+for such personalization features and here we advocate the use of
+the following metrics as a starting point: task time, numbers of
+clicks and double clicks, mean click duration, typing velocity and
+total time typing.
+This paper defends the idea that the use of smartphones by the
+elderly can bring benefits such as autonomy, access to content and
+services and communication. However, CA may create barriers
+which prevent elderly from enjoying these benefits. Therefore, we
+argue that further studies are needed regarding the influences of
+levels of CA in acceptance of smartphones and apps by the elderly
+people. On his work about the development of a mobile computer
+anxiety scale (MCAS), [39] argues that MCAS is related to three
+distinct components: (1) traditional CA construct; (2) Internet anxi-
+ety construct and (3) special factors making up the mobile anxiety
+construct (e.g., equipment limitation). The limitations of mobile
+equipment listed by [39] and that elderly people report as being
+problematic for them are: small screens and small multi-function
+key pads; lower display resolution; unfriendly user-interfaces; and
+graphical limitations. Thus, these limitations should be considered
+when developing new technologies for elderly people as well. Fur-
+thermore, the importance of the first experience is found in the
+literature and reported in the interviews conducted in this study.
+They reported having purchased or been presented with a new
+smartphone and feeling lost or afraid to use it. So, it is important
+that the device has a simple, accessible and usable interface. Another
+common factor reported, is the fear of making mistakes, looking
+stupid or breaking the device. In this sense, it is important that the
+system provides a safe environment, which asks for confirmation
+for important (dangerous) actions. And, in case the user makes a
+mistake, the system must provide ways to recover from the error,
+returning to the previous state without difficulty.
+In Brazil, families often have a computer to be shared by family
+members. It can make elderly people afraid of breaking what be-
+longs to the family or afraid of losing some important data if they
+do something wrong. The smartphone, on the other hand, is seen
+as an personal device. According to the participants’ reports, this
+brings greater freedom to learn how to use through trial and error.
+Besides that, as it is mobile, it has the advantage that elderly people
+can avoid to use it in front of other people. This can help dealing
+with the fear of not knowing how to use it or making mistakes in
+front of younger people.
+Finally, we believe a research agenda about the role of smart-
+phones for elderly people in the context of CA should address the
+following research questions: (1) How to detect CA during the use
+of mobile phones at scale? (2) How to provide (first) good user
+experiences for this population? (3) How to create a secure envi-
+ronment for PwCA to recover from mistakes? (4) How to combine
+technology use with learning in order to increase CSE? (5) How
+CA on the desktop relates to CA on smartphones?
+6
+CONCLUSION
+This position paper discussed how elderly people use smartphones
+in a specific region of São Paulo, Brazil, and shows how CA is
+present in this sample of the population. Our findings suggest
+that higher CA levels are prevalent on higher age and CA impacts
+how users interact with technologies. In addition, results indicate
+that the behavior of elderly users when performing tasks can be
+negatively impacted not only because of age-related factors, but
+also by the CA levels. The results indicating the preference of some
+applications over others by elderly people indicate the need for
+further studies on why some technologies still present barriers
+for PwCA. Moreover, the shorter task time obtained by the high
+CA group and the fact that they usually gave up when feel lost
+shows the importance of shorter and simpler tasks. The differences
+between groups regarding ownership of smartphones show that
+CA may impact on the technology adoption. In addition, the results
+showing the preference for known functions can be important to
+designers and developers to consider when developing new systems,
+since the inclusion of functions considered easy to use may increase
+the system adoption and improve user experience by reducing
+frustration.
+Finally, tackling technology adoption by elderly people may im-
+prove their quality of life, since the use of smartphones and the
+wide variety of applications and services may help they achieve
+independence, make their lives more comfortable, promoting their
+better participation in the community and allowing access to the
+most diverse online content. The presented user test shows that
+metrics such as task time, number of clicks, click duration, typing
+speed and total time typing can support the prediction of different
+CARS scores (𝑅2=0.84). In the current context of the COVID-19 pan-
+demics, promoting autonomy, communication and leisure activities
+became core goals for any technology and here we emphasize that
+by understanding CA and considering it in design and development
+phases mobile apps have the potential to change the live of PwCA.
+ACKNOWLEDGMENTS
+We thank the CRECI@ for all the support. This study was financed
+in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível
+Superior - Brasil (CAPES) - Finance Code 001.
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+A
+TECHNOLOGY USE AND PROFILE
+(1) Do you have a computer available at home? If not, do you use
+a computer somewhere else (e.g., at work or in the lanhouse)?
+(2) How often do you use a computer?
+A: [ ] Rarely
+[ ] Sometimes
+[ ] Usually
+[ ] Always
+(3) What do you usually do on computer?
+(4) Do you own a smartphone?
+(5) How often do you use smartphones?
+A: [ ] Rarely
+[ ] Sometimes
+[ ] Usually
+[ ] Always
+(6) What do you usually do on smartphone?
+Internet: [ ] Yes
+[ ] No
+Social networks: [ ] Yes
+[ ] No
+Instant Message: [ ] Yes
+[ ] No
+Call: [ ] Yes
+[ ] No
+Games: [ ] Yes
+[ ] No
+Message: [ ] Yes
+[ ] No
+Music / Video: [ ] Yes
+[ ] No
+

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+Positivity-preserving entropy filtering for the ideal
+magnetohydrodynamics equations
+T. Dzanica,∗, F. D. Witherdena
+aDepartment of Ocean Engineering, Texas A&M University, College Station, TX 77843
+A R T I C L E I N F O
+Keywords:
+Discontinuous spectral element
+Ideal magnetohydrodynamics
+Shock capturing
+Positivity-preserving
+Entropy filtering
+A B S T R A C T
+In this work, we present a positivity-preserving adaptive filtering approach for discontinuous
+spectral element approximations of the ideal magnetohydrodynamics equations. This approach
+combines the entropy filtering method (Dzanic and Witherden, J. Comput. Phys., 468, 2022) for
+shock capturing in gas dynamics along with the eight-wave method for enforcing a divergence-
+free magnetic field. Due to the inclusion of non-conservative source terms, an operator-splitting
+approach is introduced to guarantee that the positivity and entropy constraints remain satisfied
+by the discrete solution. Furthermore, a computationally efficient algorithm for solving the op-
+timization process for this nonlinear filtering approach is presented. The resulting scheme can
+robustly resolve strong discontinuities on general unstructured grids without tunable parameters
+while recovering high-order accuracy for smooth solutions. The efficacy of the scheme is shown
+in numerical experiments on various problems including extremely magnetized blast waves and
+three-dimensional magnetohydrodynamic instabilities.
+1. Introduction
+The transport and interaction of a non-resistive conducting fluid and its electromagnetic field remain extensively
+investigated phenomena as they are instrumental in various applications ranging from the study of astrophysical accre-
+tion disks [1] and supernova remnants [2] to magnetic confinement fusion [3] and plasma physics [4]. These strongly
+nonlinear effects are governed by the equations of ideal magnetohydrodynamics (MHD), which are composed of a
+combination of the Euler equations of gas dynamics and Maxwell’s equations of electromagnetism. From this formu-
+lation, a strong coupling between the magnetic field and the conducting fluid can be observed, where the magnetic
+field induces a current in the fluid which, in turn, gives rise to a second, induced magnetic field. This interaction
+can introduce multi-scale, multi-physics behavior in the system, such that magnetohydrodynamic flows can become
+exceedingly complex.
+As a result of this complexity, the robust and accurate numerical approximation of ideal MHD can present many
+challenges. Since hyperbolic systems are known to produce discontinuities even with smooth initial conditions [5],
+the numerical scheme must be able to robustly resolve these discontinuities which, in the case of MHD, come in the
+form of hydrodynamic and magnetic shocks and contact waves. Furthermore, the approximation of the ideal MHD
+equations also requires an intrinsic constraint on the solution in the form of a solenoidal magnetic field which may
+not be satisfied by the scheme even if the magnetic field is initially solenoidal. Without a mechanism to enforce
+this constraint, unphysical dynamics can arise in the solution which can lead to numerical instabilities. The standard
+numerical schemes for approximating MHD flows are finite difference and finite volume methods, whose properties
+and robustness are well-established in the literature [6–11]. However, they possess certain drawbacks in that they
+are either difficult to extend to complex domains with unstructured grids or cannot recover high-order accuracy in a
+computationally efficient manner.
+A particular class of schemes which have more recently grown in popularity are high-order discontinuous spec-
+tral element methods (DSEM) as they possess the geometric flexibility of finite volume methods while retaining the
+arbitrarily high-order accuracy and efficiency of spectral methods. As such, they provide a promising avenue for
+significantly decreasing the computational cost and expanding the viability of simulating complex MHD problems.
+However, due to the presence of discontinuities in MHD, DSEM approximations of these systems may introduce spu-
+rious oscillations in the solution in the form of Gibbs phenomena. Without proper treatment, these oscillations can
+∗Corresponding author
+[email protected] (T. Dzanic)
+ORCID(s): 0000-0003-3791-1134 (T. Dzanic); 0000-0003-2343-412X (F.D. Witherden)
+T. Dzanic et al.: Preprint submitted to Elsevier
+Page 1 of 22
+arXiv:2301.03129v1  [math.NA]  9 Jan 2023
+
+aPositivity-preserving entropy filtering for the ideal MHD equations
+result in unphysical predictions or the failure of the numerical scheme altogether. To extend to use of DSEM to MHD,
+various numerical stabilization techniques have been proposed, ranging from artificial viscosity methods [12, 13] to
+limiting-type approaches [14, 15]. While these various methods may be sufficient to stabilize the solution in many
+cases, they may not guarantee that the solution will abide by physical constraints, may require problem-dependent tun-
+able parameters, can be computationally inefficient for general unstructured grids, or may be excessively dissipative in
+smooth regions of the flow.
+There is significant interest in the design of numerical schemes that are “provably robust” in the sense that they
+can guarantee that the solution will abide by certain physical constraints even in the presence of features such as
+discontinuities, the quintessential examples being positivity-preserving schemes for gas dynamics which guarantee the
+positivity of the density and internal energy/pressure. For DSEM, this property is typically achieved through some
+form of nonlinear limiting or filtering [14, 16–18]. However, designing schemes that possess this property without
+sacrificing the computational efficiency of DSEM for general unstructured grids and their advantageous scale-resolving
+properties in smooth flow regions can be challenging. In the context of MHD, this becomes even more difficult due
+to the additional complexity of the governing equations as well as the incorporation of differential constraints, namely
+solenoidal magnetic fields. As such, there is a need for numerical stabilization techniques for DSEM approximations
+of the ideal MHD equations that retain as many of these desirable properties as possible, namely that they:
+1. Guarantee that physical constraints of the solution are satisfied.
+2. Are compatible with numerical techniques for enforcing intrinsic constraints such as a solenoidal magnetic field.
+3. Do not require problem-dependent tunable parameters.
+4. Do not appreciably degrade the ability of the underlying DSEM to resolve smooth portions of the flow.
+5. Can be easily and efficiently implemented on general unstructured grids.
+In this work, we propose a nonlinear adaptive filtering approach as a numerical stabilization technique for DSEM
+approximations of the ideal MHD equations to address these points. The proposed technique can be considered as an
+extension of the entropy filtering approach originally introduced by the authors for shock capturing in gas dynamics to
+the ideal MHD system [17]. This technique relies on using the solution’s ability to preserve convex invariants of the
+system, namely positivity of the density and pressure and a discrete local minimum entropy principle, to compute the
+necessary filter strength to ensure a well-behaved solution in the vicinity of discontinuities. Extending this approach to
+the ideal MHD system presents several challenges, primarily stemming from the treatment of the divergence-free con-
+straint on the magnetic field. We utilize the eight-wave method of Powell et al. [7] which introduces non-conservative
+source terms in the equation proportional to the divergence of the magnetic field. As these non-conservative terms
+can conflict with the necessary assumptions of the entropy filtering approach, we present a modified set of conditions
+and introduce an operator splitting approach to the system which allows the filtering method to retain its positivity-
+preserving properties. Furthermore, as the original approach for performing the optimization process necessary in the
+adaptive filtering framework as presented in Dzanic and Witherden [17] was found to be quite computationally expen-
+sive, we develop a highly-efficient numerical approach which drastically reduces the overall computational cost. The
+resulting approach can robustly resolve strong hydrodynamic and magnetic discontinuities in the flow without appre-
+ciably degrading the accuracy of the underlying DSEM for smooth flows, does not require problem-dependent tunable
+parameters, and can be easily extended to unstructured grids with relatively low computational cost. The efficacy of
+the proposed method is demonstrated in a variety of numerical experiments including smooth transport, extremely
+magnetized blast waves, and three-dimensional magnetohydrodynamic instabilities computing using high-order ap-
+proximations on both structured and unstructured grids.
+The organization of this work is as follows. We present some preliminaries regarding DSEM approximations and
+the ideal MHD equations in Section 2. The entropy filtering approach for ideal MHD is then introduced in Section 3,
+and its numerical implementation and computational optimizations are presented in Section 4. Results for various test
+cases are then shown in Section 5, and conclusions are drawn in Section 6.
+2. Preliminaries
+2.1. Ideal magnetohydrodynamics
+The governing equations for the evolution of an ideal magnetohydrodynamic fluid can be given in the form of a
+hyperbolic conservation law as
+휕푡퐮 + 훁⋅퐅 (퐮) = 퐒푩(퐮),
+(1)
+T. Dzanic et al.: Preprint submitted to Elsevier
+Page 2 of 22
+
+Positivity-preserving entropy filtering for the ideal MHD equations
+where 퐮 = 퐮(퐱, 푡) ∈ ℝ푚 is the solution of some number of field variables 푚 defined over a 푑-dimensional spatial
+domain 퐱 ∈ ℝ푑 and time 푡, 퐅(퐮) ∈ ℝ푚×푑, and 퐒퐁(퐮) is an additional source term to be defined in Section 2.3 whose
+purpose is to ensure a solenoidal magnetic field. The solution and flux are given as
+퐮 =
+⎡
+⎢
+⎢
+⎢⎣
+휌
+흆풗
+푩
+퐸
+⎤
+⎥
+⎥
+⎥⎦
+and
+퐅 =
+⎡
+⎢
+⎢
+⎢
+⎢
+⎢⎣
+흆풗
+흆풗 ⊗ 퐯 + 퐈
+(
+푃 + 1
+2퐁⋅퐁
+)
+− 퐁 ⊗ 퐁
+풗 ⊗ 퐁 − 퐁 ⊗ 풗
+(
+퐸 + 푃 + 1
+2퐁⋅퐁
+)
+퐯 − 퐁(퐯⋅퐁)
+⎤
+⎥
+⎥
+⎥
+⎥
+⎥⎦
+,
+(2)
+where 휌 is the density, 흆풗 is the momentum, 퐸 is the total energy, 푃 = (훾 − 1)
+(
+퐸 − 1
+2휌퐯⋅퐯 − 1
+2퐁⋅퐁
+)
+is the pressure,
+퐁 is the magnetic field, and 훾 is the specific heat ratio. Furthermore, the symbol 퐈 denotes the identity matrix in ℝ푑×푑
+and 퐯 = 흆풗∕휌 denotes the velocity. The solution can be more conveniently expressed in terms of a vector of primitive
+variables as 퐪 = [휌, 퐯, 퐁, 푃]푇 , and auxiliary quantities representing the magnetic pressure and plasma-beta can be
+defined as 푃푏 = 1
+2(훾 − 1)퐁⋅퐁 and 훽 = 2푃∕(퐁⋅퐁), respectively.
+Due to the lack of magnetic monopoles, the MHD equations have an intrinsic constraint on the solution in the form
+of a solenoidal magnetic field, i.e.,
+훁⋅퐁 = 0.
+(3)
+Although this constraint must be satisfied analytically by the MHD equations, numerical approximations do not nec-
+essarily satisfy it even if the magnetic field is initially solenoidal. If this constraint is not enforced by the scheme,
+numerical instabilities may arise in addition to the non-physical nature of the approximation. Many approaches exist
+to enforce this condition on the magnetic field, including the use of solenoidal basis functions [19], projection meth-
+ods [20], constrained-transport schemes [6], divergence cleaning methods [21], and the eight-wave method [7]. An
+overview of the salient techniques is presented in Wu and Shu [15].
+The entropy solution of Eq. (1) satisfies an entropy inequality of the form
+휕푡휎(퐮) + 훁⋅횺(퐮) ≥ 0,
+(4)
+where (휎, 횺) is any numerical entropy-flux pair [22] that satisfies the relation
+휕퐮횺 = 휕퐮휎휕퐮퐅.
+Note that this inequality may be negated depending on which notation is used for the numerical entropy. In Dao and
+Nazarov [23], it was shown that the entropy solution (in a vanishing viscosity sense) of the ideal MHD system satisfies
+a minimum entropy principle on the specific physical entropy 휎 = 푃휌−훾 in the form
+휎 (퐮(퐱, 푡 + Δ푡)) ≥ min
+퐱
+휎 (퐮(퐱, 푡)) ,
+(5)
+where Δ푡 > 0. This property is identical to the minimum entropy principle in gas dynamics [24], and it should be
+satisfied by the solution in both smooth regions and in the vicinity of discontinuities.
+2.2. Discontinuous spectral element methods
+For nodal discontinuous spectral element approximations of Eq. (1), including discontinuous Galerkin [25] and flux
+reconstruction [26] schemes, the domain Ω is partitioned into 푁푒 elements Ω푘 such that Ω = ⋃
+푁푒 Ω푘 and Ω푖 ∩Ω푗 = ∅
+for 푖 ≠ 푗. With a slight abuse of notation, the solution 퐮(퐱) within each element Ω푘 is approximated in a nodal manner
+as
+퐮(퐱) =
+∑
+푖∈푆
+퐮푖휙푖(퐱),
+(6)
+where 퐱푖 ∀ 푖 ∈ 푆 is a set of solution nodes, 휙푖(퐱) are their associated nodal basis functions that possess the property
+휙푖(퐱푗) = 훿푖푗, and 푆 is the set of nodal indices for the stencil. For brevity, we utilize the notation that 퐮푖 = 퐮(퐱푖). The
+order of the approximation of the solution is denoted as ℙ푝 for some order 푝, where 푝 is the maximal order of 퐮(퐱).
+This approximation formally yields a convergence rate of at least 푝 + 1 [25].
+T. Dzanic et al.: Preprint submitted to Elsevier
+Page 3 of 22
+
+Positivity-preserving entropy filtering for the ideal MHD equations
+The flux is approximated via the contribution of an interior term, denoted by the subscript Ω푘, and an interface
+term, denoted by the subscript 휕Ω푘, as
+퐅(퐮) ≈ 퐅Ω푘(퐮) + 퐅휕Ω푘(퐮).
+(7)
+For the interior component, the flux is computed through a collocation approach as
+퐅Ω푘(퐮) =
+∑
+푖∈푆
+퐅(퐮푖)휙푖(퐱),
+(8)
+such that the interior contribution to the divergence of the flux can be computed as
+훁⋅퐅Ω푘(퐮푖) =
+∑
+푗∈푆
+퐜푖푗퐅(퐮푗),
+where
+퐜푖푗 = ∇휙푖(퐱푗).
+(9)
+The interface component of the flux is formed over a set of interface nodes 퐱푖 ∈ 휕Ω푘 ∀ 푖 ∈ 퐼, where 퐼 is a set of nodal
+indices for the interface stencil. We assume that these interface nodes are a subset of the solution nodes (i.e., 퐼 ⊂ 푆)
+to avoid issues regarding interpolation for discontinuous solutions. At each interface node, there exist two values of
+the solution, 퐮−
+푖 and 퐮+
+푖 , representing the solution evaluated from the element of interest and the interface-adjacent
+element, respectively. The interface flux term can then be computed as
+퐅휕Ω푘(��푖) =
+∑
+푗∈퐼
+퐅(퐮−
+푗 , 퐮+
+푗 , 퐧푗)휙푗(퐱),
+(10)
+where 퐅(퐮−
+푖 , 퐮+
+푖 , 퐧푖) are the common interface flux values dependent on the interior and exterior values of the solution
+and their associated normal vectors 퐧푖 and 휙푖(퐱) are the interface bases. The common interface flux is generally
+computed using an approximate Riemann solver such as that of Rusanov [27]. The interface bases are dependent on
+the choice of spatial discretization, e.g., for flux reconstruction schemes, these terms can be given as
+휙푖(퐱) = 퐧푖⋅퐡푖(퐱) − 휙푖(퐱).
+(11)
+Here, 퐡푖 are a set of correction functions [28, 29] that posses the properties that
+퐧푖⋅퐡푗(퐱푖) = 훿푖푗
+and
+∑
+푖∈퐼
+퐡푖(퐱) ∈ RT푝,
+(12)
+where RT푝 is the Raviart–Thomas space [30] of order 푝. In this work, the flux reconstruction scheme with the equivalent
+discontinuous Galerkin correction functions [26] is used which recovers the nodal discontinuous Galerkin method [25].
+The interface contribution to the divergence of the flux can then be given as
+훁⋅퐅휕Ω푘(퐮푖) =
+∑
+푗∈퐼
+퐜푖푗퐅(퐮−
+푗 , 퐮+
+푗 , 퐧푗),
+where
+퐜푖푗 = ∇휙푖(퐱푗).
+(13)
+The semi-discrete form of Eq. (1) can then be given as
+휕푡퐮푖 = −
+(
+퐅휕Ω푘(퐮푖) + 훁⋅퐅휕Ω푘(퐮푖)
+)
++ 퐒퐁(퐮푖).
+(14)
+We assume that the spatial scheme satisfies the relation
+휕푡퐮 = − ∫휕Ω푘
+퐅 (퐱) ⋅ 퐧(퐱) d퐱 ≈ −
+∑
+푗∈퐼
+푚푗퐅(퐮−
+푗 , 퐮+
+푗 , 퐧푗)
+(15)
+where 푚푗 is the associated quadrature weight for 퐱푗 and 퐮 is the element-wise mean defined as
+퐮 = 1
+푉푘 ∫Ω푘
+퐮(퐱) d퐱
+and
+푉푘 = ∫Ω푘
+d퐱.
+(16)
+This assumption is appropriate for nodal discontinuous Galerkin schemes given appropriate quadrature and flux recon-
+struction schemes utilizing the equivalent discontinuous Galerkin correction functions.
+T. Dzanic et al.: Preprint submitted to Elsevier
+Page 4 of 22
+
+Positivity-preserving entropy filtering for the ideal MHD equations
+2.3. Eight-wave method
+A common method for enforcing a divergence-free magnetic field is to utilize the eight-wave method of Powell
+et al. [7]. This approach relies on an additional wave structure of the Riemann problem in MHD that arises when the
+magnetic field is not exactly solenoidal, and it can be utilized to force the magnetic field to a solenoidal state via a
+source term, given as
+퐒푩(퐮) = −
+⎡
+⎢
+⎢
+⎢⎣
+0
+푩
+풖
+풖⋅푩
+⎤
+⎥
+⎥
+⎥⎦
+훁⋅푩.
+(17)
+With the inclusion of this source term, the divergence of the magnetic field is typically suppressed to the order of mag-
+nitude of the approximation error [15]. As such, due to the simplicity of implementation and applicability to general
+unstructured grids, it remains a routine approach for robustly enforcing the divergence-free constraint on the solenoidal
+field. In addition, only this modified form of the ideal MHD equations is symmetrizable and Galilean invariant when
+the magnetic field is not exactly solenoidal [15]. However, as this form is non-conservative, it occasionally can cause
+inaccurate predictions around discontinuities in the flow (see Tóth [31]).
+The use of Powell’s method requires some clarification about the choice of the formulation for computing the
+divergence of the magnetic field. In the context of DSEM, there exist two formulations, a local divergence, consisting
+of just the interior component as
+훁⋅푩퐿(퐮푖) =
+∑
+푗∈푆
+퐜푖푗푩푗,
+(18)
+and a global divergence, consisting of both the interior component and the interface contribution as
+훁⋅푩퐺(퐮푖) =
+∑
+푗∈푆
+퐜푖푗푩푗 +
+∑
+푗∈퐼
+퐜푖푗푩푗,
+(19)
+where 푩푗 is a common interface value for the magnetic field, typically taken as the centered average of the interior and
+exterior values. Whereas the divergence-free constraint can be imposed on the local divergence through straightforward
+approaches such as projection to solenoidal bases, enforcing this constraint on the global divergence is typically more
+difficult as its domain of influence is not strictly contained within the element. It can be argued that the global approach
+is the “correct” choice as it is the one for which the space of the divergence is consistent with the space of the solution,
+but in practice, the local approach is typically sufficient. In this work, the global approach is used as the complexity of
+the two implementations is similar with Powell’s method.
+3. Methodology
+Due to the presence of discontinuities in MHD flows in the form of hydrodynamic and magnetic shocks, it is
+necessary to apply some sort of a numerical stabilization procedure to ensure robustness of the DSEM approximation.
+In Dzanic and Witherden [17], an adaptive filtering approach was introduced with goal of stabilizing the scheme by
+discretely enforcing convex constraints on the solution, given in the form of
+Γ(퐮푖) > 0 ∀ 푖 ∈ 푆,
+(20)
+where Γ(퐮) is some constraint functional. For a positivity-preserving scheme, these constraints are set as
+Γ1(퐮) = 휌
+and
+Γ2(퐮) = 푃,
+(21)
+corresponding to constraints on the positivity of density and pressure.
+While these constraints can ensure the positivity of these quantities, they are generally not restrictive enough to
+ensure that the solution remains well-behaved in the vicinity of discontinuities. It is necessary to attempt to form
+additional constraints on the solution that are restrictive enough to stabilize the solution in the vicinity of discontinuities
+without degrading the accuracy of the scheme in regions where the solution is smooth. By utilizing the fact that the
+minimum entropy principle presented in Section 2 should be satisfied by both smooth and discontinuous solutions, a
+third constraint on the solution is enforced corresponding to a discrete form of a local minimum entropy principle as
+Γ3(퐮) = 휎(퐮) − 휎min,
+(22)
+T. Dzanic et al.: Preprint submitted to Elsevier
+Page 5 of 22
+
+Positivity-preserving entropy filtering for the ideal MHD equations
+where 휎(퐮) = 푃 휌−훾 is the specific physical entropy and 휎min is some local minimum entropy bound. This minimum
+bound 휎min is computed in an element-wise manner as the discrete minima of the entropy functional across the element
+and its face neighbors prior to each stage of a temporal integration scheme, resulting in the enforcement of a discrete
+minimum entropy principle over the local domain of influence of the element (see Dzanic and Witherden [17], Section
+2 and 3). It was found in the context of gas dynamics that enforcing this constraint ensured well-behaved solutions in
+the vicinity of discontinuities while recovering high-order accuracy in smooth regions of the flow [17].
+3.1. Adaptive filtering
+The constraints are enforced by an adaptive filtering procedure, where the filtered solution ̃퐮 is given in terms of a
+filter kernel 퐻 applied to the solution, i.e.,
+̃퐮 = 퐻(퐮).
+(23)
+This filtering is performed in modal space given a modal decomposition of the solution in the form of
+퐮(퐱) =
+∑
+푖∈푆
+̂퐮푖휓푖(퐱),
+(24)
+where 휓푖(퐱) ∀ 푖 ∈ 푆 are a set of modal basis functions and ̂퐮푖 are their corresponding modes. We assume that this modal
+decomposition is chosen with respect to the unit measure (e.g., Legendre polynomials, Koornwinder polynomials, etc.).
+A discrete form of this change-of-basis operation can be given in terms of a Vandermonde matrix 퐕 as
+̂퐮 = 퐕−1퐮.
+(25)
+The filter kernel ̂
+퐻 is taken as a second-order exponential kernel in modal space, such that the filtered modal modes
+can be computed as
+̂
+퐻푖(̂퐮푖) = ̂퐮 exp(−휁푝2
+푖 ),
+(26)
+where 휁 is the filter strength and 푝푖 is the total order of the corresponding mode ̂퐮푖. It must be noted that the adaptive
+filtering approach is not restricted to this choice of filter and can be applied to any conservative filtering operation of
+one free variable that can recover both the unfiltered solution and the mean mode [17]. The filtering operation 퐻(퐮)
+can be cast in terms of a matrix-vector operation as
+̃퐮 = 퐻(퐮) = 퐕횲퐕−1퐮,
+(27)
+where 횲 is a diagonal matrix of 푝 + 1 unique values with its entries equal to 횲푖,푖 = exp(−휁푝2
+푖 ).
+The filter strength is computed via an element-wise nonlinear optimization process, taken as the minimum filter
+strength necessary such that the constraints are satisfied, i.e.,
+휁 = arg min
+휁 ≥ 0
+s.t. [Γ1
+(̃퐮(퐱푖)) > 0, Γ2
+(̃퐮(퐱푖)) > 0, Γ3
+(̃퐮(퐱푖)) > 0 ∀ 푖 ∈ 푆] .
+(28)
+Existence of a solution of 휁 is guaranteed if the element-wise mean of the solution satisfies the constraints, an assump-
+tion that will be explored in Section 3.2. As this optimization process is a function of a scalar free variable, its solution
+can be obtained using any root-bracketing approach. Furthermore, as it is nonlinear and non-convex, convergence to
+a local minima is sufficient in the case of multiple values of 휁 existing such that the constraints are satisfied exactly.
+While this optimization problem seems computationally demanding due to the element-wise matrix-vector operations
+necessary to compute the filtered solution each iteration of the solve, we present a numerical approach to solving
+this problem in Section 4.1 that is much more computationally efficient than the original methodology in Dzanic and
+Witherden [17].
+3.2. Extensions to MHD
+Extending the entropy filtering approach to the MHD system requires some modifications, with special care neces-
+sary in regards to the treatment of the source terms. The adaptive filtering operation naturally relies on that assumption
+that there exists a filter strength such that the constraints are satisfied, and it is trivial to show that a solution exists if the
+element-wise mean satisfies the constraints [17]. The ability of discontinuous Galerkin-type approaches to preserve
+convex invariants of hyperbolic systems on the element-wise mean is a well established in the literature, and the reader
+T. Dzanic et al.: Preprint submitted to Elsevier
+Page 6 of 22
+
+Positivity-preserving entropy filtering for the ideal MHD equations
+is referred to a variety of works which utilize this property [15–17, 32–35]. However, the inclusion of the source term
+and the presence of entropy constraints introduces some caveats on this property of the scheme.
+Let the set 퐺 represent the set of solutions which satisfy the constraints (i.e., Γ1(퐮) > 0, Γ2(퐮) > 0, Γ3(퐮) > 0), and
+let the shorthand notation 퐮 ∈ 퐺 represent 퐮푖 ∈ 퐺 ∀ 푖 ∈ 푆. To ensure that the filter can recover a constraint-satisfying
+solution, it is necessary for the temporal update of the element-wise mean to preserve these invariants, i.e., for some
+time step 푛, if 퐮푛 ∈ 퐺, then 퐮푛+1 ∈ 퐺. For brevity, we consider a temporal update in the form of a forward Euler
+approximation, given as
+퐮푛+1 = 퐮푛 + Δ푡 [퐿1(퐮푛) + 퐿2(퐮푛)] ,
+(29)
+where
+퐿1(퐮) = −훁⋅퐅(퐮)
+and
+퐿2(퐮) = 퐒퐁(퐮).
+(30)
+Without an exactly solenoidal magnetic field, the property 퐮푛+1 ∈ 퐺 is not necessarily satisfied in this form under the
+standard assumptions posed in works such as Zhang and Shu [32] and the original presentation of entropy filtering
+for gas dynamics in Dzanic and Witherden [17], e.g., appropriate Riemann solver, CFL condition, strong stability
+preserving temporal integration. If we consider the set of solutions 퐺푃 which satisfy just the positivity constraints
+(i.e., 퐮 ∈ 퐺푃 if Γ1(퐮) > 0, Γ2(퐮) > 0), then the work of Wu and Shu [15] showed that the property 퐮푛+1 ∈ 퐺푃 is
+satisfied under a potentially more restrictive condition on the time step dependent on the discrete divergence of the
+magnetic field (see Theorem 3.1 in Wu and Shu [15]). Furthermore, if we neglect the source term and consider an
+intermediate temporal update as
+퐮∗ = 퐮푛 + Δ푡퐿1(퐮푛),
+(31)
+then the work of Bouchut et al. [36] (paired with the equivalency of the element-wise mean and Godunov methods
+presented in Zhang and Shu [32] and subsequent works) shows that this intermediate state satisfies the property 퐮∗ ∈ 퐺
+under the standard assumptions.
+These two observations motivate an operator splitting approach for the filter. Two separate filtering operations are
+considered, a more restrictive filter which enforces both the positivity and entropy constraints, denoted by 퐻푒[퐮], and
+a more relaxed filter that enforces only positivity constraints, denoted by 퐻푝[퐮]. As the assumption on the positivity
+and entropy constraints on the element-wise mean are satisfied by the intermediate state, the more restrictive filter can
+be applied, i.e.,
+̃퐮∗ = 퐻푒
+[퐮푛 + Δ푡퐿1(퐮푛)] .
+(32)
+Since the entropy constraints are the most restrictive constraint and the contribution of the source term is typically
+minimal compared to the divergence of the flux (since it is proportional to 훁⋅퐁), this filtering operation can usually
+mitigate the majority of the spurious oscillations in the vicinity of discontinuities. The contribution of the source term
+is then added onto this filtered state, after which the positivity constraints are then enforced again on the temporal
+update as
+̃퐮푛+1 = 퐻푝
+[̃퐮∗ + Δ푡퐿2(퐮푛)] .
+(33)
+A solution to this filtering optimization problem is also guaranteed to exist as the positivity of the element-wise mean
+is guaranteed [15].
+Several properties of this splitting approach must be noted. First, it is very rarely the case that the secondary filtering
+operation is necessary – the entropy constraints on ̃퐮∗ are typically restrictive enough to where ̃퐮∗ + Δ푡퐿2(퐮푛) retains
+its positivity-preserving properties, such that in most cases, the positivity constraints are typically just checked and no
+filtering is needed. However, to ensure that the scheme remains provably positivity-preserving, this secondary filtering
+operation must be included. Second, the splitting for the source term is calculated explicitly as 퐿2(퐮푛), not through
+a Strang-type splitting approach [37] as 퐿2(퐮∗). While the latter may potentially better approximate the necessary
+corrections to the solution for preserving a solenoidal magnetic field, these forms of splitting can introduce a limit on
+the temporal accuracy of the scheme and therefore are avoided. Finally, unless the linear filtering kernel which recovers
+the squeeze limiter of Zhang and Shu [32] is chosen (see Dzanic and Witherden [17], Remark 1), the divergence of
+the filtered magnetic field is not guaranteed to be equal or lower than the unfiltered state. As this work pertains to a
+nonlinear filter, it may introduce minor divergence errors similarly to any nonlinear limiting operation, but these are
+mitigated via the source term at the next temporal update with the explicit splitting approach such that its effects were
+found to be negligible.
+T. Dzanic et al.: Preprint submitted to Elsevier
+Page 7 of 22
+
+Positivity-preserving entropy filtering for the ideal MHD equations
+Extensions to higher-order strong stability preserving (SSP) schemes follow readily from this formulation, e.g., the
+temporal update for a third-order, three-stage SSP Runge–Kutta scheme, neglecting the notatioñ⋅ for brevity, is given
+as
+퐮(1) = 퐻푝
+[퐻푒
+[퐮푛 + Δ푡퐿1(퐮푛)] + Δ푡퐿2(퐮푛)] ,
+(34)
+퐮(2) = 퐻푝
+[
+퐻푒
+[3
+4퐮푛 + 1
+4퐮(1) + 1
+4Δ푡퐿1(퐮(1))
+]
++ 1
+4Δ푡퐿2(퐮(1))
+]
+,
+퐮푛+1 = 퐻푝
+[
+퐻푒
+[1
+3퐮푛 + 2
+3퐮(2) + 2
+3Δ푡퐿1(퐮(2))
+]
++ 2
+3Δ푡퐿2(퐮(2))
+]
+,
+where the entropy constraints for 퐻푒 are computed from the previous temporal stage (see Dzanic and Witherden [17],
+Appendix A).
+4. Implementation
+Ω푘
+Figure 1:
+Schematic of a two-dimensional ℙ2 triangular element Ω푘 showing interior solution points (red circles), interior
+interface flux/solution points (red circles, blue outline), and exterior interface flux points (blue circles).
+The governing equations and the adaptive filtering approach were implemented in PyFR [38], a high-order GPU-
+accelerated unstructured flux reconstruction solver. The solution nodes were distributed along the Gauss–Legendre–
+Lobatto quadrature points and 훼-optimized points [25] for tensor-product and simplex elements, respectively. An
+example of the solution and flux point distributions for a two-dimensional ℙ2 triangular element is shown in Fig. 1.
+Temporal integration was performed using a three-stage, third-order SSP Runge–Kutta scheme as presented in Eq. (34).
+Unless otherwise stated, common interface fluxes were computed using the Harten-Lax-van Leer contact (HLLC)
+Riemann solver of Li [39] and Gurski [40] with the Davis wavespeed estimate [41], although for most test cases, we
+observed negligible differences in comparison to Rusanov-type [27] and Harten-Lax-van Leer (HLL) [42] Riemann
+solvers. To avoid a vacuum state for the Riemann solver and apply a numerical tolerance to the entropy condition, the
+constraints were instead implemented as
+Γ1(퐮) = 휌 − 휖,
+Γ2(퐮) = 푃 − 휖,
+and
+Γ3(퐮) = 휎 − 휎min − 휖,
+where 휖 = 10−8 is a small constant taken as some arbitrary factor of the machine precision.
+Boundary conditions were enforced in a weak sense through the imposition of an exterior ghost state to the inter-
+face Riemann solver [43]. Three types of boundary conditions were considered in this work: 1) Dirichlet boundary
+conditions, where the exterior state is explicitly defined; 2) Neumann boundary conditions, where the exterior state is
+identical to the interior state; and 3) reflecting boundary conditions, where the exterior state is identical to the interior
+state with the normal component of the velocity and magnetic field negated.
+T. Dzanic et al.: Preprint submitted to Elsevier
+Page 8 of 22
+
+Positivity-preserving entropy filtering for the ideal MHD equations
+4.1. Filter optimization
+Each time the filtering operation is called, the constraints are first checked on the solution. If the solution satisfies
+the constraints, no filtering is applied, otherwise the filter strength is computed using the Illinois root-bracketing ap-
+proach [44] with a stopping tolerance of 10−8 and a maximum of 20 iterations. While the filter strength can be simply
+iterated by repeatedly evaluating the element-wise filtered solution as per Eq. (27) and computing the minima of the
+constraints, several optimizations can be performed to drastically decrease the computational cost of performing this
+filtering operation.
+First, instead of solving for 휁, it beneficial to solve for 푓 = exp(−휁) and utilize the relation
+exp(−휁푝2
+푖 ) = 푓 푝2
+푖 .
+This bounds the search space of the root-bracketing approach to 푓 ∈ [0, 1], and the evaluation of the filter coefficients
+reduces to simple integer powers of the argument 푓. Then, to avoid the costly computation of the matrix-vector
+product in Eq. (27) each iteration of the root-bracketing process, certain properties of the matrix 횲 can be exploited.
+As previously mentioned, 횲 is a diagonal matrix of 푝 + 1 unique values with its entries equal to 횲푖,푖 = exp(−휁푝2
+푖 ). If
+we define a set of diagonal matrices 퐈(푘) for 0 ≤ 푘 ≤ 푝 as
+퐈(푘)
+푖,푖 =
+{
+1,
+if푝푖 = 푘,
+0,
+else,
+(35)
+then the filtering operation can be equivalently represented as
+̃퐮 =
+푝
+∑
+푖=0
+푓 푝2
+푖 퐮(푘),
+(36)
+where
+퐮(푘) = 퐕퐈(푘)퐕−1퐮.
+(37)
+Note that the values 퐮(푘) are independent of the value of 푓, such that these values can be pre-computed and the fil-
+tered solution can be efficiently evaluated each iteration of the root-bracketing approach without having to repeatedly
+compute the matrix-vector product 퐕횲퐕−1퐮.
+This approach can be even further optimized by utilizing the fact that the nodal values of the solution can now
+be decoupled, such that the root-bracketing process can be applied across each solution node sequentially which is
+particularly beneficial for computing architectures where memory bandwidth is the bottleneck. In this sequential
+approach, each solution node 퐱푗 for 푗 ∈ 푆 solves for a value of 푓푗 such that ̃퐮푗 satisfies the constraints. It is trivial to
+show that if
+푓 = min
+푗∈푆 푓푗,
+then ̃퐮 satisfies the constraints at all nodes. It is therefore advantageous to then use 푓푗 as the upper bound for the root-
+bracketing process for the node 퐱푗+1 as the constraints can be checked for ̃퐮푗+1 using this upper bound and the root-
+bracketing process for that node can be skipped if they are satisfied. As the proposed algorithm requires effectively only
+one full evaluation of Eq. (27) irrespective of the number of iterations of the root-bracketing approach, the memory
+bandwidth requirements are significantly decreased, such that the filtering process becomes only a relatively small
+portion of the total compute time that is typically less than the cost of the evaluation of the divergence of the flux. An
+example of the implementation of this approach is provided in the electronic supplementary material of this work, and
+an evaluation of the efficiency improvements of this proposed algorithm in comparison to the original methodology in
+Dzanic and Witherden [17] which utilizes repeated evaluations of Eq. (27) is presented in Section 5.
+5. Results
+5.1. Near-vacuum convecting vortex
+To verify that the proposed scheme retains the high-order accuracy of the underlying DSEM for smooth solutions,
+the rate of convergence was calculated for the smooth magnetized convecting vortex problem introduced by Christlieb
+T. Dzanic et al.: Preprint submitted to Elsevier
+Page 9 of 22
+
+Positivity-preserving entropy filtering for the ideal MHD equations
+et al. [45]. For this problem, the domain is taken as Ω = [−10, 10]2 with periodic boundary conditions discretized on
+a structured quadrilateral mesh, and the initial conditions are given as
+퐪(퐱, 0) =
+⎡
+⎢
+⎢
+⎢
+⎢
+⎢
+⎢⎣
+휌
+푢
+푣
+퐵푥
+퐵푦
+푃
+⎤
+⎥
+⎥
+⎥
+⎥
+⎥
+⎥⎦
+=
+⎡
+⎢
+⎢
+⎢
+⎢
+⎢
+⎢⎣
+1
+1 − 푦훿푢
+1 + 푥훿푢
+−푦훿퐵
+푥훿퐵
+1 + 훿푃
+⎤
+⎥
+⎥
+⎥
+⎥
+⎥
+⎥⎦
+(38)
+where
+훿푢 =
+휇
+√
+2휋
+휙(푟),
+훿퐵 = 휇
+2휋 휙(푟),
+훿푃 = −휇2(1 + 푟2)
+8휋2
+휙(푟)2,
+(39)
+and
+휙(푟) = exp(1 − 푟2),
+푟 =
+√
+푥2 + 푦2.
+(40)
+The specific heat ratio was set as 훾 = 5∕3. These conditions give a non-isentropic nature to the flow field, which
+allows for a proper assessment of the proposed entropy-based constraints for smooth flows where the filter should be
+primarily inactive. To make this problem more numerically challenging, the parameter 휇 is chosen such as to give
+a near-vacuum state for the pressure field [45]. This value was set as 휇 = 5.38948938512, which gives a minimum
+pressure value in the domain of approximately 2휖 = 2⋅10−8.
+The problem was solved until a non-dimensional time of 푡 = 0.05 using a fixed time step of Δ푡 = 1⋅10−4, after
+which the 퐿1 norm of the magnetic field error was computed as
+푒퐵 = 1
+퐴 ∫Ω
+||퐵푥 − 퐵exact
+푥
+|| + |||퐵푦 − 퐵exact
+푦
+||| d퐱,
+(41)
+where 퐴 = 202. The exact solution was computed through a translation of the initial conditions with a translation
+velocity of [1, 1], and the integration was computed using a 9th order Gauss–Legendre quadrature rule. The error
+with respect to the mesh resolution 푁푒 is shown for various approximation orders in Table 1 in addition to the rate of
+convergence. High-order convergence, on the order of 푝 to 푝 + 1, was observed for all approximation orders between
+ℙ2 and ℙ5. Furthermore, the ℙ2 results can be compared to the positivity-preserving third-order DG scheme in Wu and
+Shu [15] (Table 2), which can be considered as a subset of the adaptive filtering approach without entropy constraints
+(see Dzanic and Witherden [17], Remark 1). The proposed scheme gives marginally lower error even after removing
+the 1∕퐴 normalization factor.
+푁푒
+ℙ2
+ℙ3
+ℙ4
+ℙ5
+202
+2.29 × 10−4
+3.07 × 10−5
+3.30 × 10−6
+4.04 × 10−7
+252
+1.18 × 10−4
+1.17 × 10−5
+1.37 × 10−6
+9.36 × 10−8
+332
+5.39 × 10−5
+4.07 × 10−6
+3.39 × 10−7
+2.44 × 10−8
+402
+3.04 × 10−5
+2.02 × 10−6
+1.44 × 10−7
+9.55 × 10−9
+502
+1.59 × 10−5
+8.72 × 10−7
+5.02 × 10−8
+2.92 × 10−9
+672
+6.92 × 10−6
+2.99 × 10−7
+1.31 × 10−8
+6.51 × 10−10
+RoC
+2.89
+3.80
+4.62
+5.23
+Table 1:
+Convergence of the 퐿1 norm of the magnetic field error at 푡 = 0.05 with respect to mesh resolution 푁푒 for the
+near-vacuum convecting vortex problem with varying approximation order. Rate of convergence shown on bottom.
+5.2. Brio–Wu shock tube
+Extensions to flows with discontinuities was then performed through the shock tube problem of Brio and Wu [8]
+which includes features of the Riemann problem such as shock waves, contact discontinuities, rarefaction waves, and
+compound waves. For this problem, the domain is set as Ω = [0, 1] and the initial conditions are given by
+T. Dzanic et al.: Preprint submitted to Elsevier
+Page 10 of 22
+
+Positivity-preserving entropy filtering for the ideal MHD equations
+퐪(퐱, 0) =
+⎡
+⎢
+⎢
+⎢
+⎢
+⎢
+⎢⎣
+휌
+푢
+푣
+퐵푥
+퐵푦
+푃
+⎤
+⎥
+⎥
+⎥
+⎥
+⎥
+⎥⎦
+=
+{
+퐪푙,
+if 푥 ≤ 0.5,
+퐪푟,
+else,
+where
+퐪푙 =
+⎡
+⎢
+⎢
+⎢
+⎢
+⎢
+⎢⎣
+1
+0
+0
+0.75
+1
+1
+⎤
+⎥
+⎥
+⎥
+⎥
+⎥
+⎥⎦
+and
+퐪푟 =
+⎡
+⎢
+⎢
+⎢
+⎢
+⎢
+⎢⎣
+0.125
+0
+0
+0.75
+−1
+0.1
+⎤
+⎥
+⎥
+⎥
+⎥
+⎥
+⎥⎦
+.
+(42)
+The specific heat ratio is set as 훾 = 2. The hydrodynamic components of this problem are identical to the Sod shock
+tube [46]. Although this problem is one-dimensional, it was instead solved on a one element wide two-dimensional
+mesh to facilitate the use of the vertical magnetic field component within the solver. Dirichlet boundary conditions were
+applied on the left/right boundaries while periodic boundary conditions were applied along the top/bottom boundaries.
+The problem was computed with a ℙ3 scheme using a coarser mesh of 200 elements and a finer mesh of 400
+elements with time steps of Δ푡 = 2⋅10−4 and 1⋅10−4, respectively. A reference solution was also computed using a
+highly-resolved ℙ0 scheme with 5⋅104 elements. The predicted density, pressure, and vertical magnetic field profiles
+at 푡 = 0.1 are shown in Fig. 2 for both the coarse and fine mesh. For all fields, both rarefaction waves and the shock
+were well-resolved, showing sub-element resolution without any noticeable spurious oscillations. Furthermore, similar
+behavior was observed for the contact and compound wave in the pressure and magnetic fields. Some minor oscillations
+were observed in the density profile in the region between the compound wave and contact discontinuity, although this
+behavior is not uncommon for some numerical schemes. The predicted density profile in that region converged to the
+reference results with increasing resolution, but minor undershoots in front of the contact discontinuity were observed.
+0
+0.2
+0.4
+0.6
+0.8
+1
+0
+0.2
+0.4
+0.6
+0.8
+1
+푥
+휌
+Reference
+푁푒 = 200
+푁푒 = 400
+(a) Density
+0
+0.2
+0.4
+0.6
+0.8
+1
+0
+0.2
+0.4
+0.6
+0.8
+1
+푥
+푃
+(b) Pressure
+0
+0.2
+0.4
+0.6
+0.8
+1
+−1
+−0.5
+0
+0.5
+1
+푥
+퐵푦
+(c) Vertical magnetic field
+Figure 2:
+Density, pressure, and vertical magnetic field profiles for the Brio–Wu shock tube problem at 푡 = 0.1 computed
+using a ℙ3 FR scheme with 200 and 400 elements.
+5.3. Orszag–Tang vortex
+Two-dimensional flows with more complex features were then considered through the canonical Orszag–Tang
+vortex problem [47]. This case is a well-known model problem for evaluating a scheme’s ability to handle MHD
+shocks and shock interactions as well as predicting transition to supersonic MHD turbulence. The domain is set as
+Ω = [0, 1]2 with periodic boundary conditions, and the initial conditions are given as
+퐪(퐱, 0) =
+⎡
+⎢
+⎢
+⎢
+⎢
+⎢
+⎢⎣
+휌
+푢
+푣
+퐵푥
+퐵푦
+푃
+⎤
+⎥
+⎥
+⎥
+⎥
+⎥
+⎥⎦
+=
+⎡
+⎢
+⎢
+⎢
+⎢
+⎢
+⎢⎣
+25∕(36휋)
+− sin(2휋푦)
+sin(2휋푥)
+sin(2휋푦)∕
+√
+4휋
+− sin(4휋푥)∕
+√
+4휋
+5∕(12휋)
+⎤
+⎥
+⎥
+⎥
+⎥
+⎥
+⎥⎦
+.
+(43)
+The specific heat ratio is set as 훾 = 5∕3. Uniform meshes of various resolution were generated, and the problem
+was solved using a ℙ3 scheme. The contours of density at 푡 = 0.5 computed on meshes with 푁푒 = 642, 1282, and
+T. Dzanic et al.: Preprint submitted to Elsevier
+Page 11 of 22
+
+Positivity-preserving entropy filtering for the ideal MHD equations
+2562 elements are shown in Fig. 3, computed using time steps of Δ푡 = 4⋅10−4, 2⋅10−4, and 1⋅10−4, respectively. The
+results show good prediction of the canonical flow field of the Orszag–Tang vortex, with better approximation of shock
+structure and small-scale flow features with increasing resolution. Minor spurious oscillations were observed in the
+density field at low resolutions, but these oscillations diminished with increasing mesh resolution, such that the flow
+field at 푁푒 = 2562 was virtually oscillation-free.
+(a) 푁푒 = 642
+(b) 푁푒 = 1282
+(c) 푁푒 = 2562
+Figure 3:
+Contours of density for the Orszag-Tang vortex at 푡 = 0.5 computed using a ℙ3 FR scheme with 642 (left),
+1282 (middle), and 2562 (right) elements.
+For a more quantitative comparison, the predicted pressure profile on the cross-section 푦 = 0.3125 at 푡 = 0.48 is
+shown in Fig. 4 in comparison to the results of Jiang and Wu [48] obtained using a high-order weighted essentially
+non-oscillator (WENO) scheme. It can be seen that similar observations can be made for the pressure field as with the
+density field, with minor spurious oscillations at lower resolutions that diminish with increasing resolution. The overall
+prediction of the pressure profile was in good agreement with the reference results at moderate and high resolutions,
+and strong discontinuities in the pressure field were generally well resolved at the sub-element level even at lower
+resolutions. At the highest resolution, there was very good agreement with the reference data with minor differences
+in the prediction of the location of some small-scale flow features on the left-hand side of the cross-section. Overall,
+the results showed good predictions of the various flow features of the Orszag–Tang vortex.
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+0.1
+0.2
+0.3
+푥∕퐿
+푝
+Reference
+푁푒 = 642
+푁푒 = 1282
+푁푒 = 2562
+Figure 4:
+Pressure profile of the Orszag-Tang vortex on the cross-section 푦∕퐿 = 0.3125 at 푡 = 0.48 computed using a ℙ3
+FR scheme with 642, 1282, and 2562 elements. Numerical results of Jiang and Wu [48] shown for reference.
+5.4. Shock cloud interaction
+The proposed scheme was then evaluated on the shock cloud interaction problem of Dai and Woodward [9], con-
+sisting of a high-density cloud interacting with an impinging shock wave which results in strong discontinuities and
+the development of small-scale flow instabilities. The problem setup as described by Balbás and Tadmor [11] is solved
+T. Dzanic et al.: Preprint submitted to Elsevier
+Page 12 of 22
+
+0.4
+0.35
+0.3
+0.25
+0.2
+0.15
+0.1Positivity-preserving entropy filtering for the ideal MHD equations
+on the domain Ω = [0, 1]2 with the initial conditions given as
+퐪(퐱, 0) = [휌, 푢, 푣, 푤, 퐵푥, 퐵푦, 퐵푧, 푃]푇 =
+⎧
+⎪
+⎨
+⎪⎩
+퐪푐,
+if 푟′ ≤ 0.15,
+퐪푙,
+else if 푥 ≤ 0.6,
+퐪푟,
+else,
+(44)
+where the cloud state, left state, and right state are given as
+퐪푐 =
+⎡
+⎢
+⎢
+⎢
+⎢
+⎢
+⎢
+⎢
+⎢⎣
+10
+0
+0
+0
+2.1826182
+−2.1826182
+1
+167.345
+⎤
+⎥
+⎥
+⎥
+⎥
+⎥
+⎥
+⎥
+⎥⎦
+,
+퐪푙 =
+⎡
+⎢
+⎢
+⎢
+⎢
+⎢
+⎢
+⎢
+⎢⎣
+3.86859
+0
+0
+0
+2.1826182
+−2.1826182
+1
+167.345
+⎤
+⎥
+⎥
+⎥
+⎥
+⎥
+⎥
+⎥
+⎥⎦
+,
+and
+퐪푟 =
+⎡
+⎢
+⎢
+⎢
+⎢
+⎢
+⎢
+⎢
+⎢⎣
+1
+−11.2536
+0
+0
+0
+0.56418958
+0.56418958
+1
+⎤
+⎥
+⎥
+⎥
+⎥
+⎥
+⎥
+⎥
+⎥⎦
+,
+(45)
+respectively. The specific heat ratio is set as 훾 = 5∕3. The cloud is centered at [0.8, 0.5] with a radius of 0.15, such
+that
+푟′ =
+√
+(푥 − 0.8)2 + (푦 − 0.5)2.
+To facilitate the use of the three-dimensional magnetic field within the solver, the problem is solved on a one
+element deep three-dimensional mesh. Additionally, while the original problem setup uses a [0, 1]2 domain with
+Neumann boundary conditions on the top/bottom boundaries, we instead extend the domain to [0, 1] × [−0.5, 1.5] and
+apply periodic boundary conditions on the top/bottom boundaries. As these boundaries on the extended domain are
+outside of the domain of influence of the shock cloud interaction over the time range of the simulation, the effect of this
+modified setup on the flow field is negligible but it helps alleviate any issues arising from numerical errors compounding
+at free boundaries. The remaining left and right boundary conditions were set as Neumann and Dirichlet, respectively,
+while periodicity was enforced along the 푧 direction.
+(a) Density
+(b) Pressure
+(c) Magnetic pressure
+Figure 5:
+Contours of density (left), pressure (middle), and magnetic pressure (right) on the subregion [0, 1]2 for the
+shock cloud interaction problem at 푡 = 0.06 computed using a ℙ2 FR scheme with 4002 elements.
+To perform a comparison of the proposed approach to a third-order DG scheme augmented with a WENO limiter
+presented in Wu and Shu [15], an identical problem setup is used with a ℙ2 scheme on 4002 mesh (with respect to
+the original domain size Ω = [0, 1]2). The predicted contours of density, pressure, and magnetic pressure at 푡 = 0.06
+computed using a time step of Δ푡 = 4⋅10−6 are shown in Fig. 5. The results show good resolution of the strong
+discontinuities in the various fields without any observable spurious oscillations. Furthermore, small-scale features in
+T. Dzanic et al.: Preprint submitted to Elsevier
+Page 13 of 22
+
+p
+2
+5
+10
+20
+40P
+100
+200
+300
+4000.2
+0.5
+2
+5
+10
+20
+50
+100Positivity-preserving entropy filtering for the ideal MHD equations
+the cloud region of the density and magnetic fields were not excessively dissipated, and the symmetry of the problem
+was well-preserved. A comparison to the method of Wu and Shu [15] (Fig. 1) is shown in Fig. 6. Note that some
+discrepancy in the color schemes between the two images may be present. The proposed scheme was roughly equally
+performative in terms of the resolution of discontinuities and marginally better at resolving small-scale flow features on
+the trailing side of the cloud. Without the positivity-preserving filtering approach, the scheme diverged due to negative
+pressure in the solution.
+(a) Entropy filter
+(b) DG with WENO limiter
+Figure 6:
+Comparison of the contours of density computed by the proposed entropy filtering approach (left) and the
+positivity-preserving DG scheme augmented with a WENO limiter of Wu and Shu [15] (right).
+5.5. Magnetized blast
+As a stress test for the positivity-preserving property of the proposed scheme for extreme flow conditions, a modified
+form of the magnetized blast wave problem of Zachary et al. [49] and Balsara and Spicer [10] was considered. In this
+problem, a blast wave is driven by a spherical overpressure region in the center of the domain surrounded by a low
+plasma-beta ambient state, resulting in strong magnetosonic shocks. The problem is solved on the periodic domain
+Ω = [−0.5, 0.5]2, and the initial conditions are given as
+퐪(퐱, 0) =
+⎡
+⎢
+⎢
+⎢
+⎢
+⎢
+⎢⎣
+휌
+푢
+푣
+퐵푥
+퐵푦
+푃
+⎤
+⎥
+⎥
+⎥
+⎥
+⎥
+⎥⎦
+=
+{
+퐪푒,
+if
+√
+푥2 + 푦2 ≤ 0.1,
+퐪푎,
+else,
+where
+퐪푒 =
+⎡
+⎢
+⎢
+⎢
+⎢
+⎢
+⎢⎣
+1
+0
+0
+퐵0
+0
+푃푒
+⎤
+⎥
+⎥
+⎥
+⎥
+⎥
+⎥⎦
+and
+퐪푎 =
+⎡
+⎢
+⎢
+⎢
+⎢
+⎢
+⎢⎣
+1
+0
+0
+퐵0
+0
+푃푎
+⎤
+⎥
+⎥
+⎥
+⎥
+⎥
+⎥⎦
+.
+(46)
+The specific heat ratio is set as 5∕3. While the original problem setup uses 푃푎 = 0.1, 푃푒 = 103, and 퐵0 = 100∕
+√
+4휋, we
+consider the much more extreme case presented in Wu and Shu [15] with 푃푎 = 0.1, 푃푒 = 104, and 퐵0 = 1000∕
+√
+4휋,
+resulting in a very large pressure ratio of 105 and a very small plasma-beta of 훽 ≈ 2.5⋅10−4. As these conditions are
+quite extreme, the scheme would diverge almost instantly in the absence of any positivity-preserving modifications.
+To verify that the proposed scheme can be easily extended to unstructured grids, the problem was solved on trian-
+gular meshes using a ℙ4 scheme. A coarse mesh and a fine mesh were generated, consisting of approximately 1.2⋅105
+elements with an average edge length of ℎ = 1∕200 and approximately 5⋅105 elements with an average edge length of
+ℎ = 1∕400, respectively. For this case, the HLL Riemann solver was used as it was found to be much better behaved
+in these extreme conditions than the HLLC Riemann solver, although both approaches were properly stabilized with
+the proposed entropy filtering method. The contours of density, velocity magnitude, pressure, and magnetic pressure
+at 푡 = 0.001 are shown in Fig. 7 and Fig. 8 for the coarse and fine meshes, respectively, computed using time steps of
+Δ푡 = 2⋅10−7 and Δ푡 = 1⋅10−7. Even with these extreme conditions on unstructured grids, the predicted solutions were
+T. Dzanic et al.: Preprint submitted to Elsevier
+Page 14 of 22
+
+d
+0
+10
+20
+30
+40Positivity-preserving entropy filtering for the ideal MHD equations
+(a) Density
+(b) Velocity Magnitude
+(c) Pressure
+(d) Magnetic Pressure
+Figure 7:
+Contours of density (top left), velocity magnitude (top right), pressure (bottom left), and magnetic pressure
+(bottom right) for the magnetized blast problem at 푡 = 0.001 computed using a ℙ4 scheme on an unstructured mesh with
+an average edge length of ℎ = 1∕200.
+well-behaved, and both the coarse and fine meshes showed excellent resolution of the various discontinuities in the
+velocity, pressure, and magnetic fields with sub-element resolution and no observable spurious oscillations. Further-
+more, the numerical width of the discontinuities decreased appropriately with increasing resolution. For the density
+field, minor spurious oscillations were observed, primarily at lower resolution and somewhat indicative of mesh im-
+printing, but the strength and distribution of these oscillations decreased with the finer mesh. These observations are
+consistent with the case of the Brio–Wu shock tube where the density field was marginally less well-behaved than the
+other fields. However, the predicted fields were still very good given such extreme conditions and an unstructured
+mesh, indicating that the proposed approach remains robust and accurate for such flows.
+To demonstrate the sub-element shock-resolving ability of the entropy filtering approach on unstructured grids, an
+enlarged view of the contours of pressure with the mesh overlaid is shown for the two meshes in Fig. 9. It can be seen
+that the shock was resolved well within the element, with the majority of the feature resolved across 1-2 solution nodes.
+Furthermore, this behavior persisted when the mesh resolution was increased, such that the discrete shock thickness
+T. Dzanic et al.: Preprint submitted to Elsevier
+Page 15 of 22
+
+3
+4
+5/vl
+O
+10
+20
+30
+40
+50P
+1000
+2000
+3000
+4000
+5000P
+23000
+24000
+25000
+26000
+27000
+28000Positivity-preserving entropy filtering for the ideal MHD equations
+(a) Density
+(b) Velocity Magnitude
+(c) Pressure
+(d) Magnetic Pressure
+Figure 8:
+Contours of density (top left), velocity magnitude (top right), pressure (bottom left), and magnetic pressure
+(bottom right) for the magnetized blast problem at 푡 = 0.001 computed using a ℙ4 scheme on an unstructured mesh with
+an average edge length of ℎ = 1∕400.
+decreased proportionally. Given the unstructured nature of the mesh and the resulting solution point distribution, the
+circular shape of the shock front was still well-represented, with even better approximation at higher mesh resolutions.
+These results indicate that the proposed approach can be extended to unstructured grids in a straightforward manner
+without appreciably sacrificing its efficiency or performance at resolving discontinuities.
+5.6. Three-dimensional Rayleigh–Taylor instability
+A final evaluation of the proposed approach and the extension to three-dimensional flows was performed through
+the simulation of a magnetized three-dimensional Rayleigh–Taylor instability. The problem consists of a denser gas
+resting on top of a lighter gas under the effect of a gravitational field initially in equilibrium with the pressure gradient.
+Instabilities arise in the form of “bubbles” of the lighter gas rising and “fingers” of the heavier gas descending, after
+which nonlinear momentum transport drives the flow to a turbulent mixing state. The problem is solved on the domain
+T. Dzanic et al.: Preprint submitted to Elsevier
+Page 16 of 22
+
+3
+4
+5I
+/vl
+10
+20
+30
+40
+50P
+1000
+2000
+3000
+4000
+5000P
+23000
+24000
+25000
+26000
+27000
+28000Positivity-preserving entropy filtering for the ideal MHD equations
+(a) Coarse mesh
+(b) Fine mesh
+Figure 9:
+Enlarged view of contours of pressure with mesh overlay for the magnetized blast problem at 푡 = 0.001 computed
+using a ℙ4 scheme on the coarse mesh (left) and fine mesh (right). Contour scale identical to Fig. 7 and Fig. 8.
+[−퐿∕2, 퐿∕2]2 × [−퐿, 퐿], where 퐿 = 1, and the initial conditions are given as
+퐪(퐱, 0) = [휌, 푢, 푣, 푤, 퐵푥, 퐵푦, 퐵푧, 푃]푇 =
+{
+퐪푙,
+if 푧 ≤ 0,
+퐪ℎ,
+else,
+(47)
+where
+퐪푙 =
+⎡
+⎢
+⎢
+⎢
+⎢
+⎢
+⎢
+⎢
+⎢⎣
+휌푙
+0
+0
+푊 (푥, 푦, 푧)
+퐵0
+0
+0
+푃(푧)
+⎤
+⎥
+⎥
+⎥
+⎥
+⎥
+⎥
+⎥
+⎥⎦
+and
+퐪ℎ =
+⎡
+⎢
+⎢
+⎢
+⎢
+⎢
+⎢
+⎢
+⎢⎣
+휌ℎ
+0
+0
+푊 (푥, 푦, 푧)
+퐵0
+0
+0
+푃(푧)
+⎤
+⎥
+⎥
+⎥
+⎥
+⎥
+⎥
+⎥
+⎥⎦
+,
+(48)
+for some vertical velocity perturbation 푊 (푥, 푦, 푧) and initial pressure distribution 푃 (푧). Periodic boundary conditions
+are enforced along the transverse (푥, 푦) directions while reflecting boundary conditions are enforced along the top and
+bottom boundaries. A gravitational field is added to the problem, given in the form of a source term as
+퐒퐆 = −[0, 0, 0, 휌푔, 0, 0, 0, 휌푤푔],
+(49)
+where 푔 = 1. The treatment of this gravitational field in the context of the entropy filter is taken simply as an additional
+term in the source term of Powell’s method, and as it not stiff for this problem, it does not appreciably affect the time
+step restrictions of the scheme.
+The parameters for the problem are taken similarly to a scaled form of the setup in Stone and Gardiner [50]. The
+densities of the light and heavy gases are taken as 휌푙 = 1 and 휌ℎ = 3, respectively, yielding an Atwood number of 1∕2.
+To enforce equilibrium in the flow, the initial pressure field is taken as
+푃(푧) = 푃0 − 휌푔푧,
+(50)
+where 푃0 = 10∕훾 for a specific heat ratio 훾 = 5∕3. To seed instabilities in the flow, perturbations were added in the
+form of a vertical velocity field component as
+푊 (푥, 푦, 푧) = 퐴 cos
+( 휋푧
+2퐿
+)
+sin
+(4휋푥
+퐿
+)
+sin
+(4휋푦
+퐿
+)
+,
+(51)
+T. Dzanic et al.: Preprint submitted to Elsevier
+Page 17 of 22
+
+Positivity-preserving entropy filtering for the ideal MHD equations
+where 퐴 = 0.05. This differs from the work of Stone and Gardiner [50] in that the transverse distribution of the
+perturbations is taken as a single deterministic mode instead of randomly-generated noise. As such, the predicted flow
+fields are expected to differ during the linear growth regime of the instability.
+The addition of a magnetic field significantly impacts the behavior of the Rayleigh–Taylor instability as it induces
+a stabilizing effect on the flow. In fact, linear stability analysis presents a cutoff magnetic field value,
+퐵푐 =
+√
+(휌ℎ − 휌푙)푔퐿 =
+√
+2,
+(52)
+above which the instability is completely damped by the magnetic field [50]. We consider three variations of this flow,
+a hydrodynamic case where 퐵0 = 0, a weakly magnetized case where 퐵0 = 0.1퐵푐, and a strongly magnetized case
+where 퐵0 = 0.5퐵푐. The term strongly magnetized is relative in the sense that the plasma-beta is still quite high, but
+the magnetic field can suppress almost all potential instability modes along its orientation.
+These three flow conditions were computed using a ℙ3 scheme on a 푁푒 = 64 × 64 �� 128 mesh with a time step of
+Δ푡 = 2⋅10−5. The flow, visualized in the form of a volume rendering of the density field, at various times is shown in
+Fig. 10, Fig. 11, and Fig. 12 for the hydrodynamic, weakly magnetized, and strongly magnetized cases, respectively.
+For the hydrodynamic case, the canonical flow pattern of the Rayleigh–Taylor instability was observed, with rising
+bubbles and descending figures. At later times, these features transitioned to a turbulent mixing state. When a weak
+magnetic field was applied, a significant degree of anisotropy was imparted on the flow, seen in the form of distortions
+in the bubbles and fingers aligned with the orientation of the magnetic field. Furthermore, the slowed growth rate of
+the instabilities due to the stabilizing effect of this weak magnetic field could be observed. For the strongly magnetized
+case, the magnetic field effectively damped all instabilities along the orientation of the field, such that the resulting
+flow only varied perpendicular to the orientation of the field. Given a long enough simulation time, this flow would
+be expected to transition to a three-dimensional turbulent mixing state as nonlinear transport effects overcome the
+stabilizing nature of the magnetic field. For all cases, the flow was numerically well-behaved, indicating that the
+proposed approach can be effectively applied to three-dimensional flows in both the magnetized and hydrodynamic
+regimes.
+(a) 푡 = 2
+(b) 푡 = 3
+(c) 푡 = 4
+(d) 푡 = 5
+Figure 10:
+Volume rendering of the density field for the hydrodynamic (퐵0 = 0) Rayleigh–Taylor instability problem at
+varying times computed using a ℙ3 scheme on a 푁푒 = 64 × 64 × 128 mesh.
+To quantify the efficiency improvements of the proposed algorithm in comparison to the original approach in
+Dzanic and Witherden [17] which utilizes repeated evaluations of the matrix-vector product in Eq. (27), a runtime
+comparison between the two methods was performed for the case of 퐵0 = 0.1퐵푐. The cost was evaluated on 16
+NVIDIA V100 GPUs with respect to the wall-clock time elapsed until the simulation reached 푡 = 1, and the results
+are shown in Fig. 13. While the original approach required 61.4 GPU hours, the proposed approach required only
+25.3 GPU hours, a speedup factor of approximately 2.4 across the entire simulation time. Furthermore, this speedup
+is expected to increase with higher approximation orders due to the increased number of solution points per element.
+T. Dzanic et al.: Preprint submitted to Elsevier
+Page 18 of 22
+
+2
+3
+dPositivity-preserving entropy filtering for the ideal MHD equations
+(a) 푡 = 2
+(b) 푡 = 3
+(c) 푡 = 4
+(d) 푡 = 5
+Figure 11:
+Volume rendering of the density field for the weakly magnetized (퐵0 = 0.1퐵푐) Rayleigh–Taylor instability
+problem at varying times computed using a ℙ3 scheme on a 푁푒 = 64 × 64 × 128 mesh.
+(a) 푡 = 2
+(b) 푡 = 3
+(c) 푡 = 4
+(d) 푡 = 5
+Figure 12:
+Volume rendering of the density field for the strongly magnetized (퐵0 = 0.5퐵푐) Rayleigh–Taylor instability
+problem at varying times computed using a ℙ3 scheme on a 푁푒 = 64 × 64 × 128 mesh.
+To confirm this, an identical comparison was performed using a ℙ4 approximation on a 푁푒 = 51 × 51 × 102 mesh,
+which results in approximately the same number of degrees of freedom. At this approximation order, the original
+approach required 257 GPU hours whereas the proposed approach required only 39.8 GPU hours, a speedup factor
+of approximately 6.5. These results indicate that the proposed algorithmic improvements both substantially decrease
+the overall computational cost of the entropy filtering approach and show much better scaling with respect to the
+approximation order.
+T. Dzanic et al.: Preprint submitted to Elsevier
+Page 19 of 22
+
+2
+3
+d2
+3
+dPositivity-preserving entropy filtering for the ideal MHD equations
+ℙ3
+ℙ4
+0
+100
+200
+300
+25.3
+61.4
+39.8
+257
+GPU hours per characteristic time
+Proposed algorithm
+Original algorithm
+Figure 13:
+Comparison of the wall-clock time to reach 푡 = 1 for the weakly magnetized (퐵0 = 0.1퐵푐) Rayleigh–Taylor
+instability problem with a ℙ3 (left) and ℙ4 (right) scheme with the same number of degrees of freedom using the original
+algorithm of Dzanic and Witherden [17] and the proposed algorithm.
+6. Conclusions
+In this work, a positivity-preserving adaptive filtering approach was proposed for shock capturing in discontinuous
+spectral element approximations of the ideal magnetohydrodynamics equations. The proposed scheme can be consid-
+ered as an extension of the entropy filtering approach [17] introduced by the authors for the gas dynamics equations
+to the ideal magnetohydrodynamics system. By formulating convex invariants such as positivity of density and pres-
+sure and a local discrete minimum entropy principle as discrete constraints on the solution, the amount of filtering
+necessary to satisfy the constraints was computed as an element-wise scalar optimization problem. This approach was
+combined with the eight-wave method of Powell et al. [7] for enforcing a solenoidal magnetic field. As this method
+introduced non-conservative source terms to the system, an operator splitting approach was proposed and its effects
+on the assumptions necessitated by the adaptive filtering approach to guarantee the satisfaction of the constraints were
+analyzed. An improved algorithm for solving the optimization problem for the filter strength was also introduced which
+significantly improved the computational efficiency of the proposed method.
+The proposed scheme could robustly resolve strong discontinuities while recovering high-order accuracy in smooth
+regions of the flow and could be easily and efficiently implemented on general unstructured grids. The efficacy of the
+approach was shown in a variety of numerical experiments, ranging from simple transport and shock tubes to extremely
+magnetized blast waves and three-dimensional magnetohydrodynamic instabilities. Furthermore, the proposed algo-
+rithmic enhancements yielded significant improvements in the computational cost and showed much better scaling with
+respect to approximation order, reducing the total runtime of the simulations by a factor of 2.4 for ℙ3 approximations
+and 6.5 for ℙ4 approximations. Future improvements to the proposed scheme could focus on applying different filter
+kernels to various components on the solution, alternate methods for enforcing a divergence-free magnetic field, and
+anisotropic filtering approaches.
+Acknowledgements
+This work was supported in part by the U.S. Air Force Office of Scientific Research via grant FA9550-21-1-0190
+("Enabling next-generation heterogeneous computing for massively parallel high-order compressible CFD") of the
+Defense University Research Instrumentation Program (DURIP) under the direction of Dr. Fariba Fahroo.
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+Page 22 of 22
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+EgoDistill: Egocentric Head Motion Distillation
+for Efficient Video Understanding
+Shuhan Tan1
+Tushar Nagarajan1
+Kristen Grauman1,2
+1The University of Texas at Austin
+2FAIR, Meta AI
+{shuhan,tushar.nagarajan,grauman}@cs.utexas.edu
+Abstract
+Recent advances in egocentric video understanding
+models are promising, but their heavy computational ex-
+pense is a barrier for many real-world applications. To ad-
+dress this challenge, we propose EgoDistill, a distillation-
+based approach that learns to reconstruct heavy egocen-
+tric video clip features by combining the semantics from
+a sparse set of video frames with the head motion from
+lightweight IMU readings. We further devise a novel self-
+supervised training strategy for IMU feature learning. Our
+method leads to significant improvements in efficiency, re-
+quiring 200× fewer GFLOPs than equivalent video models.
+We demonstrate its effectiveness on the Ego4D and EPIC-
+Kitchens datasets, where our method outperforms state-of-
+the-art efficient video understanding methods.
+1. Introduction
+Recent advances in augmented and virtual reality
+(AR/VR) technology have the potential to change the way
+people interact with the digital world, much like the smart-
+phone did in the previous decade.
+A fundamental re-
+quirement for AR/VR systems is the ability to recognize
+user behavior from egocentric video captured from a head-
+mounted camera.
+Towards this goal, several egocentric
+video datasets have been proposed in recent years, spurring
+increasing attention of the research community [11,26,56].
+Recent advances in egocentric action recognition, antic-
+ipation, and retrieval focus on building powerful clip-based
+video models that operate on video clips of a few seconds
+at a time [12, 16, 18, 25, 43, 44, 54, 55]. Despite encourag-
+ing performance, these models typically process densely-
+sampled frames with temporally-aware operations, making
+them computationally heavy. This makes them impractical
+for AR/VR devices with constrained resources, or for real-
+time video applications that require low latency. How to
+efficiently perform egocentric video understanding is there-
+fore an important, yet unsolved problem.
++
+Camera motion (IMU)
+Video Frame
+EgoDistill
+Figure 1. Illustration of EgoDistill. Given a single video frame
+and camera motion from IMU, EgoDistill learns to reconstruct the
+more expensive dense video clip feature. With its lightweight in-
+put, EgoDistill significantly improves efficiency.
+To address this issue, we take inspiration from how an-
+imals perceive the world with ego-motion. Neuroscience
+research has found that during active movement, the animal
+visual cortex receives and utilizes head motion signals from
+the motor cortex for visual processing [27,52,53]. This indi-
+cates that head motion signals support an embodied agent’s
+efficient understanding of the egocentric visual stream. In-
+spired by this phenomenon, we explore the relationship be-
+tween human head motion and egocentric video for efficient
+video understanding. In practice, we consider head motion
+signals captured by the inertial measurement unit (IMU) of
+a head-mounted camera. IMU measures motion from an ac-
+celerometer and gyroscope and is widely available on pop-
+ular wearable devices. Prior work leverages IMU as an ex-
+tra modality for human action recognition [13,68,69], (e.g.,
+jumping, walking, standing) or as geometric cues for visual-
+inertial odometry [7,20,71].
+In contrast, we propose to achieve efficient video under-
+standing by drawing on IMU as a substitute for dense video
+frame observations. The intuition is as follows. A video
+clip contains two things: semantic content (appearance of
+objects, places, people) and dynamics (how the scene and
+the camera move).
+While densely sampled frames are
+sure to capture both of the above—as done by current clip
+arXiv:2301.02217v1  [cs.CV]  5 Jan 2023
+
+models [16, 17, 54]—we hypothesize they are sometimes
+overkill. For a short video clip, much of the semantic con-
+tent is intelligible from even a single frame; meanwhile, the
+head motion provides a good portion of the dynamics, im-
+plicitly revealing how the visual appearance changes across
+neighboring frames.
+Building on this insight, we introduce EgoDistill, an ap-
+proach that learns to reconstruct dense egocentric video
+clip features using temporally sparse visual observations
+(as few as one RGB frame) together with the head motion
+from IMU. Specifically, EgoDistill employs a new form of
+knowledge distillation from video models. During training,
+we train a lightweight model that takes sparsely sampled
+image(s) and IMU to approximate the video representation
+extracted by a powerful but expensive video model. We fur-
+ther improve the model with a novel self-supervised train-
+ing stage for IMU feature learning. During inference, we
+directly utilize the lightweight model for egocentric video
+recognition, leading to much higher efficiency. Our model
+is flexible to the target heavy video feature, as we demon-
+strate with multiple current leading egocentric video mod-
+els [16–18,54]. See Figure 1.
+Importantly, EgoDistill offers a major efficiency gain:
+processing low-dimensional IMU and a few frames is much
+more efficient compared to processing a dense stack of
+frames. In practice, EgoDistill uses 200× fewer GFLOPs
+than the original video model.
+We experiment on the largest available egocentric ac-
+tion recognition datasets: Ego4D [26] and EPIC-Kitchens-
+100 [11]. We show that IMU coupled with an image offers
+better cross-modality knowledge distillation performance
+than images alone or images with audio. For a typical 50-
+minute egocentric video, EgoDistill reduces inference time
+of the source video model from 25 minutes to 36 seconds.
+Moreover, with only 1-4 frames, our lightweight distillation
+model achieves a better accuracy-efficiency trade-off than
+state-of-the-art models for adaptively sampling video con-
+tent [50,65]. Notably, we surpass the accuracy of these fast
+approaches by a large margin while requiring 4× to 8× less
+computation.
+2. Related Work
+IMU for activity recognition. Recent work explores us-
+ing the IMU sensor on mobile devices for human activ-
+ity recognition of actions like walking, jumping, or sit-
+ting [3,47,59–61]. Normally, these models take input from
+IMU sensors mounted on human body joints [9, 42, 60],
+waist-mounted [41] or in-pocket smartphones [33]. See [64]
+for a survey. Abundant work in video recognition explores
+ways to learn from RGB coupled with other modalities—
+audio [1, 22, 38], optical flow [19, 57, 58] or both [32, 36,
+51]—but comparatively fewer use IMU [48], and unlike our
+work, they focus on third-person video [13, 68, 69] and do
+not target at model efficiency. Our idea is for IMU to help
+reconstruct more expensive video features, rather than sim-
+ply fuse IMU with RGB for multi-modal recognition.
+IMU for odometry. Inertial odometry aims to estimate the
+position and orientation of the camera-wearer with readings
+from the IMU sensor. Traditionally, methods rely on IMU
+double integration [4] or enhancements thereof [5, 35, 40].
+Recent data-driven methods automatically learn to per-
+form inertial odometry with supervised [30, 70] or self-
+supervised learning [7], or combine IMU and visual in-
+put for more robust estimates with visual-inertial odome-
+try [20, 71]. While IMU can convey geometric ego-motion
+to our learned model, our goal is to produce efficient ego-
+centric video features rather than to output odometry.
+Visual feature learning with IMU. IMU is also used to
+learn better vision features [14,15,34,63], e.g., to encourage
+image features that are equivariant with ego-motion [34],
+to predict an IMU-captured body part (leg, hand) [14, 15],
+or to predict video-IMU correspondence [63], for applica-
+tions like action recognition [15,63] and scene understand-
+ing [14, 34]. While these results reinforce that IMU can
+inject embodied motion into visual features, our idea to use
+head motion to infer pretrained video features for speedy
+video understanding is distinct.
+Efficient video recognition. Being crucial for mobile ap-
+plications, efficient video recognition has received increas-
+ing attention in recent years. Several studies focus on de-
+signing lightweight architectures [17, 30, 37, 62, 72] by re-
+ducing 3D CNN operations across densely sampled frames.
+Our idea is orthogonal to them as we focus on inputs with
+sparsely-sampled frames. As we show in experiments, our
+method is compatible with different video architectures.
+Another line of research achieves efficiency by adap-
+tively selecting video content to process.
+Some reduce
+temporal redundancy by adaptively selecting which video
+clip [39], frames [24, 49], and/or feature channel [50] to
+process and which to skip, while others reduce spatial re-
+dundancy, efficient recognition by dynamically selecting
+selecting for each frame a smaller but important region to
+process [66,67]. Other work dynamically selects tokens in
+video transformers among both the spatial and temporal di-
+mensions [65]. Our idea is complementary: rather than dy-
+namically subsample the available video content, we show
+how to infer “full” video features for every clip using static
+image(s) and motion data. Our results outperform state-of-
+the-art sampling models (cf. Sec. 4). In addition, we fo-
+cus on egocentric video, where head motion is particularly
+meaningful for inferring unobserved visual content. To our
+knowledge, ours is the first technique specifically aimed at
+accelerating egocentric video processing.
+Multimodal distillation. Knowledge distillation aims to
+transfer knowledge learned by an expensive model to a
+lightweight model [31]. Recent work explores multimodal
+
+distillation, e.g., transferring from a RGB model to a flow or
+depth model [23,28], from a 3D model to a 2D model [45],
+or from a visual model to audio model [2,21]. The Listen-
+ToLook model [22] incorporates both clip subsampling and
+video-to-audio distillation for fast activity recognition in
+third-person video. In contrast, we explore the relationship
+between the camera-wearer’s head motion and RGB sig-
+nals for egocentric video. Our experiments show EgoDis-
+till’s advantage over ListenToLook in terms of the speed-
+accuracy tradeoff on egocentric video datasets.
+3. Approach
+We introduce EgoDistill, which uses sparsely-sampled
+frames and head motion from IMU to approximate the fea-
+tures of heavy video models for efficient egocentric video
+understanding.
+We first introduce the egocentric action
+recognition task (Sec. 3.1). Then, we introduce our pipeline
+(Sec. 3.2), our distillation model and training objective
+(Sec. 3.3), and our self-supervised IMU feature learning
+(Sec. 3.4). Figure 2 overviews our approach.
+3.1. Egocentric action recognition
+Given a fixed-length video clip V ∈ RT ×H×W ×3 con-
+sisting of T RGB frames of size H×W and a set of C action
+classes, the task of action recognition is to output a score for
+each action class, representing its likelihood. Typically, this
+is done with a powerful but expensive video model Ω, that
+directly operates on all the available frames to output the C
+class logits Ω(V) ∈ RC. Ω is trained with standard classifi-
+cation loss:
+LACT =
+�
+Vi
+LCE(ci, σ(Ω(Vi))),
+(1)
+where Vi is the i-th video clip in the dataset, ci is the
+corresponding ground-truth action label, σ is the softmax
+function, and LCE is cross-entropy loss.
+Popular video
+recognition models use clips that are typically �2 seconds
+long [16, 18, 54]. For longer videos, scores are averaged
+across all clips it contains to infer the video action label.
+3.2. Efficient video inference with head motion
+Processing the video clip V for action recognition is
+computationally intensive; however, the computation cost
+can be modulated depending on how frames from the clip
+are used. On the one hand, clip-based models [16–18, 54]
+process most (or all) frames in a video clip V to achieve
+strong recognition performance, but come at a high com-
+putational cost.
+On the other hand, frame-level mod-
+els [24, 49, 51, 67] only process one (or a small number)
+of frames from V and are more efficient, but suffer a drop
+in performance as a result. Our goal is to train a frame-
+based model that can approximate heavy clip-based model
+performance while maintaining high efficiency.
+For this, we turn to head motion captured by IMU. Along
+with RGB frames, each video clip is paired with IMU mea-
+surements M that record the camera (head) motion during
+the video. Specifically, the IMU readings are composed of
+6-dimensional accelerometer and gyroscope measurements
+in the xyz axes, which encode strong temporal motion in-
+formation about camera pose changes (both translation and
+rotation) across frames.
+For short video clips, a set of sparsely sampled frames I
+often already captures most appearance information. Com-
+plementary to this, the IMU readings capture camera mo-
+tion information (see below for discussion on scene mo-
+tion). Moreover, IMU is very efficient to process due to its
+low dimensionality. By processing inputs from these two
+sources with a lightweight frame-based model, we can infer
+the semantic and dynamic features of a heavier clip-based
+video model.
+Given I and M, we train an efficient lightweight model
+Φ to approximate the output of video model Ω. Specifically,
+we train our EgoDistill model Φ that achieves
+Φ(I, M) ≈ Ω(V).
+(2)
+Such a lightweight model will be able to approximate the
+result of the heavy video model, while being much more ef-
+ficient. Our approach is agnostic to the specific video model
+Ω; in experiments, we demonstrate its versatility for Mo-
+tionFormer [54], MViT [16], SlowFast [18] and X3D [17].
+In practice, we uniformly sample N frames1 from V to
+obtain I. We can achieve a trade-off between efficiency and
+performance by changing the number of frames N. In our
+experiments we use very low values of N (1 to 4 frames).
+In the next section, we discuss how we train Φ.
+3.3. Video feature distillation with IMU
+We achieve the objective in Equation 2 via knowledge
+distillation [31], where we transfer knowledge learned by
+the expensive teacher model Ω to a lightweight student
+model Φ. Next we present the design of Φ and the train-
+ing objectives, followed by our self-supervised IMU feature
+pretraining stage in Sec. 3.4.
+We design Φ to be a two-stream model. For a video clip
+and associated IMU signal (I, M), we extract image fea-
+tures zI = fI(I) and IMU features zM = fM(M) using
+lightweight feature encoders fI, fM respectively. Then,
+we fuse zI and zM with a fusion network Π to obtain the
+fused VisIMU feature zφ = Π(zI, zM). Finally, a fully-
+connected layer uses the fused feature to predict class logits
+Φ(I, M) ∈ RC.
+The fused feature zφ contains semantic information from
+the image frame coupled with complementary motion infor-
+1Other frame sampling heuristics (e.g., selecting from the start or center
+of the video) performed equivalently or worse than uniform sampling.
+
+Video clip
+   Video model
+Frame
+IMU
+Image Encoder
+IMU Encoder
+Fusion Layer
+FC-Softmax
+FC-Softmax
+Frame
+IMU
+Image Encoder
+IMU Encoder
+Frame
+Image Encoder
+IMU  Predictor
+Figure 2. EgoDistill architecture. Left: Self-supervised IMU feature learning. Given start and end frames of a clip, we train the IMU
+encoder to anticipate visual changes. Right: Video feature distillation with IMU. Given image frame(s) and IMU, along with our pre-
+trained IMU encoder, our method trains a lightweight model with knowledge distillation to reconstruct the features from a heavier video
+model. When the input includes more than one image frame, the image encoder aggregates frame features temporally with a GRU.
+mation from IMU, allowing us to accurately reconstruct the
+video clip feature. See Figure 2.
+We train Φ with a combination of three losses, as fol-
+lows. First, we train Φ to approximate the original video
+feature zV from the video model Ω:
+L1 =
+�
+(zVi,zφi)
+∥zVi − zφi∥1 .
+(3)
+This cross-modal loss encourages the fused feature zφ to
+match the video feature, i.e., the combined features from
+the different modalities should match in the feature space.
+Training with L1 alone does not fully capture the clas-
+sification output of Ω. Therefore, we also train Φ with a
+knowledge distillation loss:
+LKD =
+�
+(Vi,Ii,Mi)
+DKL(σ(Ω(Vi)/τ), σ(Φ(Ii, Mi)/τ)),
+(4)
+where (Vi, Ii, Mi) represents the i-th clip in the dataset,
+DKL measures KL-divergence between the class logits from
+the teacher model Ω and student model Φ, and τ is a tem-
+perature parameter. Intuitively, LKD casts the output of the
+video teacher model as a soft target for training the student
+model. In this way, the student model learns to better gen-
+eralize by mimicking the output distribution of the heavy
+video model.
+Finally, to further encourage the features to preserve el-
+ements useful for activity understanding, we also compute
+an action classification loss:
+LGT =
+�
+(Ii,Mi)
+LCE(ci, σ(Φ(Ii, Mi))),
+(5)
+where ci is the ground-truth action label, following Equa-
+tion 1. The final training loss is a combination of these three
+loss functions:
+L = αLKD + (1 − α)LGT + βL1,
+(6)
+where α controls the balance between knowledge distilla-
+tion and activity training [31], and β controls the weight for
+feature space matching.
+Critically, processing a few image frame(s) and the low-
+dimensional IMU readings is substantially faster than pro-
+cessing the entire video. Once trained, our model can ap-
+proximate the behavior of the source video model for recog-
+nition tasks, with the key benefit of efficient egocentric
+video recognition.
+What kind of motion does our model preserve? Video
+motion decomposes into scene motion (e.g., how the objects
+and the camera wearer’s hands are moving on their own),
+and camera motion (i.e., how the camera wearer is moving
+their head). By itself, IMU would directly account only for
+camera motion, not scene motion. However, by learning to
+map from the RGB frame and IMU to the full video fea-
+ture, we are able to encode predictable scene motions tied
+to scene content, e.g., how does hand and object movement
+in subsequent frames relate to the camera wearer’s head mo-
+tion (see Figure 7). Moreover, our model is applied to rel-
+atively short clips (1-2 seconds) in sequence, which means
+the appearance content is regularly refreshed as we slide
+down to process the longer video.
+3.4. Self-supervised IMU feature learning
+The success of EgoDistill depends on how well the IMU
+feature encoder fM extracts useful camera motion informa-
+tion and associates it with the visual appearance change in
+the video clip. In this way EgoDistill can learn to antic-
+ipate unseen visual changes in the video with I and M.
+We design a self-supervised pretraining task to initialize the
+weights of fM to achieve this.
+
+Specifically, for each clip V, we obtain its first and last
+frames (I0, IT ) as well as the IMU M. We first extract
+visual features z0
+I, zT
+I and IMU feature zM with feature
+extractors fI and fM mentioned above. Then, we train
+a feature predictor h to predict the IMU feature ˆzM =
+h(z0
+I, zT
+I ). By connecting ˆzM—which is a function of im-
+age features only—with zM, we encourage fM to extract
+useful camera motion features specifically associated with
+the visual appearance changes. Note that those appearance
+changes may include scene motion. Therefore, we include
+an L1 loss to train fM, which encourages fM to extract mo-
+tion features accounting for scene motion in the full video.
+In sum, we train fM, h, and the fusion network Π using
+L1 and NCE loss [29]: Lpretrain = LNCE + L1, where
+LNCE =
+�
+i
+− log
+sim(ˆzMi, zMi)
+�
+j sim(ˆzMi, zMj).
+(7)
+We sample negative examples zMj from other instances
+in the same mini-batch for j
+̸=
+i, and sim(q, k)
+=
+exp( q·k
+|q||k|
+1
+τ ′ ) with temperature τ ′ = 0.12.
+To summarize, prior to the main training stage of Equa-
+tion 6, we pretrain the IMU feature extractor fM and fu-
+sion network Π. As we will show below, both pretraining
+losses result in IMU features that are consistent with visual
+changes and lead to better finetuning performance.
+4. Experiments
+We evaluate our approach for resource-efficient action
+recognition.
+4.1. Experimental setup
+Datasets.
+We experiment on two large-scale egocen-
+tric action recognition datasets.
+Ego4D [26] contains
+3,670 hours of egocentric videos of people performing di-
+verse tasks (from cooking to farming) across the globe.
+As action recognition is not part of the original Ego4D
+benchmark, we construct this task with annotations from
+the Hands+Objects temporal localization benchmark [26]
+(see Supp. for details).
+We include clips with paired
+IMU and audio3, and consider classes with at least 2 la-
+beled instances.
+This results in a 94-class action recog-
+nition dataset with 8.5k training videos and 3.6k evalua-
+tion videos. EPIC-Kitchens [11] contains 100 hours of
+egocentric videos capturing daily activities in kitchen en-
+vironments. We use annotations from the action recogni-
+tion benchmark. Similar to Ego4D, we select videos that
+have paired IMU and audio data, and split the resulting data
+by camera-wearer. This results in a 62-class action dataset
+2We keep the ImageNet-pretrained fI model frozen, as finetuning it
+leads to mode collapse.
+3We require audio to compare with the audio-based baseline [22].
+with 29k training videos and 6.2k evaluation videos. For
+both datasets, we use “verb” labels as the target for action
+recognition as they are well aligned to activity motions.
+Evaluation metrics. To measure action recognition per-
+formance, we report the per-video top-1 accuracy on the
+validation set. We densely sample clips from each video
+and average their predictions to compute accuracy.
+To
+benchmark efficiency, we measure computational cost with
+FLOPs (floating-point operations) during inference.
+Implementation details. In our main experiments, we
+use MotionFormer [54] as the video teacher model Ω due
+to its strong performance for egocentric video. For EPIC-
+Kitchens, we use the authors’ provided checkpoint.
+For
+Ego4D, we finetune the above model for 50 epochs with
+1e−4 learning rate and 64 batch size on the training set.
+We use 16-frame input with sample rate 4. For the stu-
+dent model Φ, we use a ResNet-18 as the image backbone
+fI and a 1D Dilated CNN [6] for the IMU backbone fM.
+The feature fusion module Π uses a concatenation operation
+following a two-layer fully-connected layer with hidden di-
+mension 1024. For each video clip, the input image(s) is
+resized to 224 × 224, and the IMU is a 422 × 6 matrix
+(around 2 seconds with 198Hz frequency), representing the
+accelerometer and gyroscope readings along the xyz axes.
+For the image input, we uniformly sample N frames from
+the video clip. If N > 1, we use fI to sequentially gen-
+erate features for each frame and aggregate them with a
+GRU module [10]. For both datasets, we first pretrain the
+model with the self-supervised objective (Section 3.4) for
+50 epochs with AdamW [46] using batch size 64 and learn-
+ing rate 1e−4. Then, we finetune all the models with the
+same setting (Equation 6). We set α = 0.95 and β = 1.0
+based on validation data. For Ego4D, we set τ = 10.0 and
+train the model for 150 epochs. For EPIC-Kitchens, we set
+τ = 1.0 and train for 50 epochs.
+4.2. Baselines
+We compare to the following methods:
+• AdaFuse [50] trains a lightweight policy network to
+adaptively compute (or skip) feature map channels for
+each frame during inference. We use the AdaFuseTSN
+R50
+model with the provided hyper-parameters.
+• STTS [65] trains a module to rank spatio-temporal
+tokens derived from videos in a transformer-based
+model, and selects only the top-K tokens to speed up
+inference.
+• ListenToLook [22]: uses the audio-based feature dis-
+tillation module from [22] following the same audio
+processing and model architecture.
+These methods represent recent advances in efficient
+video recognition models. AdaFuse represents state-of-the-
+
+2
+4
+6
+8
+10
+12
+14
+16
+Inference cost per video clip (GFLOPs)
+33
+34
+35
+36
+37
+38
+Ego4D accuracy (%)
+Ego4D
+2
+4
+6
+8
+10
+12
+14
+16
+Inference cost per video clip (GFLOPs)
+30
+35
+40
+45
+50
+EPIC-Kitchens accuracy (%)
+EPIC-Kitchens
+EgoDistill (ours)
+AdaFuse [50]
+STTS [65]
+ListenToLook [22]
+VisOnly-Distill
+VisIMU
+VisOnly
+Figure 3. Accuracy vs. efficiency for action recognition on Ego4D (left) and EPIC-Kitchens (right). EgoDistill outperforms state-of-
+the-art efficient video recognition methods that adaptively sample video content, while using 4× to 8× fewer GFLOPs.
+LKD
+L1
+LGT
+L1-pretrain
+LNCE-pretrain
+Ego4D
+EPIC-Kitchens
+✓
+34.15
+35.04
+✓
+✓
+✓
+✓
+35.51
+39.33
+✓
+✓
+✓
+✓
+37.71
+42.20
+✓
+✓
+✓
+✓
+37.46
+43.17
+✓
+✓
+✓
+36.99
+41.21
+✓
+✓
+✓
+✓
+37.26
+42.30
+✓
+✓
+✓
+✓
+37.49
+43.51
+✓
+✓
+✓
+✓
+✓
+37.95
+44.95
+Table 1. Ablation study of model components. We compare the
+accuracy of EgoDistill with different components under N = 1.
+art approaches that achieve efficiency by reducing tempo-
+ral redundancy in CNN models. STTS is one of the most
+recent approaches that efficiently reduces both spatial and
+temporal redundancy in ViT models, which achieves the
+state-of-the-art on Kinectics-400 [8]. ListenToLook also re-
+lies on distillation, but using audio rather than head motion.
+For each model we generate multiple versions with differ-
+ent computation budgets to plot accuracy vs. GFLOPs. We
+train all AdaFuse and STTS models with 4 input frames to
+align with the maximum frames used by our model.
+For
+AdaFuse, we use the only provided hyper-parameter in the
+paper.4 For STTS, we use three provided variants: T0
+0.5-
+S4
+0.7, T0
+0.8-S4
+0.9 and the full model without token selection.
+For ListenToLook we adopt the same efficiency-accuracy
+trade-off as our method, i.e., varying the number of input
+frames.
+In addition, we test variants of our method:
+• VisOnly-Distill is our model without the IMU branch
+and fusion layer but trained with the same loss func-
+tion. Performance of this model reveals the role of
+IMU in the process of distillation.
+• VisIMU is our model trained with only LGT in Equa-
+4Modifying hyper-parameters to control the accuracy-efficiency trade-
+off results in unstable training and unreliable performance.
+Source Model
+Ego4D
+EPIC-Kitchens
+Video
+EgoDistill
+VisOnly-D
+Video
+EgoDistill
+VisOnly-D
+MFormer [54]
+46.38
+37.95
+34.32
+77.28
+44.95
+37.20
+MViT [16]
+40.32
+36.46
+33.40
+53.38
+36.90
+31.22
+SlowFast [18]
+40.52
+33.29
+33.04
+58.34
+39.42
+33.47
+X3D [17]
+37.56
+33.57
+32.90
+52.28
+36.34
+31.71
+Table 2. Versatility to model architectures. EgoDistill outper-
+forms the baseline for multiple common architectures, showing
+the generality of our idea. “Video” refers to the more expensive
+source model. We show the model accuracy under N = 1.
+tion 5. It shows the effectiveness of distillation from
+the video model compared with directly training the
+features with action labels.
+• VisOnly is an image-only model trained with LGT,
+which serves as the baseline.
+4.3. Main Results
+Importance of IMU-guided distillation.
+Figure 3
+shows the accuracy vs. efficiency curves. Methods towards
+the top-left of the plot represent those with both high ac-
+curacy and efficiency.
+Our method achieves good accu-
+racy with low computational cost. Specifically, on EPIC-
+Kitchens, when N = 1, EgoDistill improves over VisOnly-
+Distill by 8.4% with only a small increase in computa-
+tion. This result shows the effectiveness of IMU for recon-
+structing egocentric video features. Compared to VisIMU,
+EgoDistill improves by 9.9%, showing the effectiveness of
+knowledge distillation from the video model. Importantly,
+this reveals that EgoDistill does not simply benefit from the
+extra IMU context; our idea to approximate video features is
+necessary for best results. We see similar results on Ego4D.
+Comparison with the state of the art. Figure 3 also
+shows that EgoDistill achieves better accuracy with less
+computation than existing efficient video recognition mod-
+els AdaFuse [50], STTS [65], and ListenToLook [22].
+
+pack
+sew
+iron
+spray
+unscrew
+paint
+dip
+water
+play
+file
+hit
+throw
+inspect
+pull
+insert
+clean
+close
+detach
+smooth
+cut
+press
+hang
+shuffle
+tighten
+20
+0
+20
+40
+60
+Accuracy improvement(%)
+Ego4D
+break
+mix
+drink
+shake
+dry
+pour
+close
+wash
+put
+cut
+turn-off
+apply
+open
+peel
+turn-on
+throw
+flip
+scrub
+insert
+move
+take
+empty
+fill
+scoop
+10
+0
+10
+20
+30
+40
+Accuracy improvement(%)
+EPIC-Kitchens
+Figure 4. Per-class accuracy improvement over VisOnly-Distill.
+Best and worst performing classes are shown.
+GFLOPs
+Runtime (ms)
+Parameters (M)
+Video [54]
+369.51
+10.70
+108.91
+AdaFuse [50]
+15.20
+2.04
+38.85
+STTS [65]
+7.19
+1.63
+36.63
+ListenToLook [22]
+3.10
+0.43
+25.53
+EgoDistill
+1.91
+0.25
+20.56
+Table 3. Efficiency analysis. Our approach is the most efficient.
+“Video” refers to the original (full-clip) feature. Lower is better.
+With N = 4 frames, EgoDistill surpasses STTS by 7.4%
+and AdaFuse by 4.2% on EPIC-Kitchens, with 2× fewer
+GFLOPs, and surpasses both methods by 2.1% on Ego4D.
+In addition, EgoDistill surpasses ListenToLook by 7.4%
+and 2.9% on EPIC-Kitchens and Ego4D respectively, which
+suggests that head motion is more informative than audio
+for feature reconstruction in egocentric video.
+4.4. Analysis
+Model component ablations. Table 1 ablates different
+design choices in our model, setting N = 1 for all exper-
+iments. We observe that training EgoDistill without L1,
+LKD or LGT deteriorates performance. Specifically, training
+without LKD leads to the largest performance drop, which
+indicates that knowledge distillation is an essential compo-
+nent in our approach. Training without L1 also leads to a
+significant performance drop, which shows the importance
+of our idea to align features from the different modalities.
+Further, our self-supervised pretraining stage is very effec-
+tive at training the IMU extractor to encode useful motion
+information that is consistent with visual feature change.
+Finally, we compare with a model that simply does multi-
+modal recognition with IMU (top row). The strong contrast
+here indicates the importance of our idea to use IMU to pre-
+dict video model features, as opposed to simply adding IMU
+as an additional input modality.
+Impact of teacher video model architecture. In our
+main experiments we use MotionFormer [54] as the teacher
+video model due to its strong performance on egocentric
+video tasks.
+To emphasize the generality of our idea,
+we show the performance of EgoDistill with other video
+teacher architectures in Table 2.
+Similar to the Motion-
+Former model, we train these models on each of the labeled
+Figure 5. Best (top) and worst (bottom) reconstructed videos.
+datasets, and then train our model using the resulting video
+models as the teacher. As expected, better video teacher
+models lead to better student model performance. More im-
+portantly, we observe consistent improvement by EgoDis-
+till over the VisOnly-Distill baseline on both datasets and
+with different video teacher models, highlighting our idea’s
+generality and versatility.
+Where does our model work best/worst? In Figure 3
+we saw that using IMU leads to an overall performance
+improvement on action recognition, indicating better video
+feature prediction capability. Next, we explore what kinds
+of clips are better reconstructed using EgoDistill. Figure 4
+shows the improvement of EgoDistill over the VisOnly-
+Distill model on Ego4D and EPIC-Kitchens split by action
+class. We observe that IMU is more useful for actions with
+predictable head motion (e.g., break, cut, close), and is less
+helpful for actions where head motion may be small or un-
+related (e.g., empty, fill, press).
+Figure 5 shows clip examples whose video features are
+best and worst reconstructed. We observe that the best re-
+constructed clips (top) contain moderate head motion that
+is predictive of scene motion and action semantics. For ex-
+ample, the camera wearer’s head moves slightly backwards
+while opening the cabinet. On the other hand, more poorly
+reconstructed clips tend to contain little head motion (third
+row)—in which case IMU is redundant to the RGB frame—
+or drastic head motion that is weakly correlated with the
+camera wearer’s activity and introduces blur to the frame
+(last row).
+Efficiency analysis. To compare the efficiency of dif-
+ferent models, aside from GFLOPs, we also compare their
+inference run-time and number of parameters. For run-time,
+we record the time spent to infer a single video clip’s label
+with a single A40 GPU, and take the average time over the
+full validation datasets of Ego4D and EPIC-Kitchens with
+batch-size of 32. Table 3 shows the results. EgoDistill runs
+much faster than the other methods. Notably, it reduces the
+GFLOPs of MotionFormer by nearly 200×. Furthermore,
+
+EgoDistill
+EgoDistill
+VisOnly-D
+EgoDistill
+EgoDistill
+EgoDistill
+VisOnly-D
+EgoDistill
+Figure 6. Retrieving video clips with EgoDistill. Given a query frame (bottom left) and a paired IMU segment (red camera frustums) ,
+we retrieve the nearest clip in the video dataset according to EgoDistill and visualize its (unobserved) frames (strip to the right). Compared
+to VisOnly-Distill, which outputs a single feature for a given input frame (bottom row), EgoDistill outputs a distinct feature by condi-
+tioning on IMU, showing its ability to preserve both semantic and motion during reconstruction. For instance, in the top-right example,
+EgoDistill retains the cabinet interaction semantics in the frame as well as the upward camera-motion in the IMU. Zoom in to view best.
+close: 0.88
+open: 0.01
+close: 0.18
+open: 0.34
+GT: close
+EgoDistill
+VisOnly-D
+put: 0.40
+take: 0.10
+put: 0.14
+take: 0.44
+GT: put
+EgoDistill
+VisOnly-D
+Figure 7. Anticipating scene motion with EgoDistill. For each clip, we show the head motion and video frames. Note, only the center
+frame (red border) is observed by the model. Action classification scores are shown on the right. EgoDistill can successfully anticipate
+scene motion and disambiguate the action semantics in the input frame. For example, in the top center frame, the image alone cannot reveal
+if the door is being opened or closed, whereas our feature, learned with head motion, recovers correlations with the scene motion (i.e., hand
+motion and door motion) to disambiguate “close” from “open”. A similar effect for “put” vs. “take” is seen in the second example.
+it runs 6.5× faster than STTS [65] while achieving 4.4%
+higher accuracy on EPIC-Kitchens.
+4.5. Qualitative Results
+What do EgoDistill features capture? To explore this,
+we pair a single input frame with different IMU clips as in-
+puts to EgoDistill, then retrieve the nearest video clip for
+each resulting anticipated video feature. Figure 6 illustrates
+this. We see that EgoDistill outputs video features that all
+involve interaction with the cabinet (right panel), and is able
+to use different IMU inputs to retrieve different video clips
+that show consistent camera motion. In contrast, VisOnly-
+Distill only retains the semantic context to retrieve a single
+clip. These results indicate that EgoDistill is able to approx-
+imate video features that capture both semantic and motion
+information. See Supp. for more (and animated) results.
+Is there evidence EgoDistill captures scene motion?
+Figure 7 shows how our features learned with head motion
+can nonetheless expose certain scene motion cues. EgoDis-
+till improves the accuracy over VisOnly-Distill on ambigu-
+ous categories (like close and put) by a large margin (20.3%
+and 10.4% on EPIC-Kitchens, 8.5% and 3.9% on Ego4D).
+See caption for details.
+5. Conclusion
+We present EgoDistill, the first model to explore ego-
+centric video feature approximation for fast recognition.
+Experiments on action recognition on Ego4D and EPIC-
+Kitchens demonstrate that our model achieves a good bal-
+ance between accuracy and efficiency, outperforming state-
+of-the-art efficient video understanding methods. Our ap-
+proach has great potential to accelerate video understand-
+ing for egocentric videos using a data stream that is already
+ubiquitous in egocentric cameras. In the future, we plan to
+investigate how to use head motion for long-term human
+activity understanding with room context and visual corre-
+spondence learning for multi-view videos.
+
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+The supplementary materials of this work consist of:
+A. Supplementary video.
+B. Dataset details.
+C. Implementation details.
+D. Additional analysis of our model.
+A. Supplementary Video
+In our supplementary video, we have a brief introduction
+of our work. More importantly, we show animated videos
+of Best and Worse reconstructed clips (Figure 5), Retriev-
+ing video clips with EgoDistill (Figure 6), and Anticipating
+scene motion with EgoDistill (Figure 7).
+Animated version of these figures better show head mo-
+tion and video dynamics. We recommend viewing the sup-
+plementary video for better understanding of our method
+and results.
+B. Dataset Details.
+We use two datasets in our experiments: Ego4D [26] and
+EPIC-Kitchens-100 [11]. In this section we describe more
+details about how we create our training and evaluation data.
+1. Ego4D [26] contains 3,670 hours of egocentric videos
+of people performing diverse tasks (from cooking to
+farming) across the globe. As action recognition is not
+part of the original Ego4D benchmark, we construct
+this task with annotations from the Hands+Objects
+temporal localization benchmark [26].
+Specifically,
+for each hand-objects interaction temporal annotation,
+we take the video clip between the pre-frame and post-
+frame of the annotation as input, and use the annotated
+verb for this interaction as label.
+We include clips with paired IMU and audio, and con-
+sider classes with at least 2 labeled instances, resulting
+in 94 action categories with 12.1k videos in total. In
+average, each clip has 2.2 second duration. Then, we
+randomly split data from each category into training
+and evaluation sets with 70%:30% ratio. Finally, we
+obtain a 94-class action recognition dataset with 8.5k
+training videos and 3.6k evaluation videos.
+2. EPIC-Kitchens [11] contains 100 hours of egocen-
+tric videos capturing daily activities in kitchen environ-
+ments. We use annotations from the action recognition
+benchmark in our experiment.
+We select videos that have paired IMU and audio data,
+and split the resulting data by camera-wearer, ensuring
+non-overlapping splits following the original bench-
+mark setting. Specifically, we take videos captured by
+camera-wearer id starting with P30, P35, P37 as evalu-
+ation videos and use all the remaining videos as train-
+ing videos. This results in a 62-class action dataset
+with 29k training videos and 6.2k evaluation videos.
+C. Implementation Details.
+IMU input processing. For each input clip, IMU in-
+put is a 422 × 6 matrix (around 2 seconds with 198Hz
+frequency), representing the accelerometer and gyroscope
+readings along the xyz axes. We observe that the raw IMU
+input has significant drifting and bias issues. This induces
+inconsistent correspondence between camera motion and
+IMU reading across different clips and videos. Therefore,
+for IMU reading of each clip, on each dimension we sep-
+arately subtract raw readings by the mean values on the
+corresponding dimension. This operation normalizes IMU
+readings in each dimension to have zero average value. In
+this way, our model can only focus on the temporal motion
+patterns in each clip.
+Audio input processing. For ListenToLook [22], we
+process the audio input in the same way mentioned in the
+paper. Specifically, we subsample the audio at 16kHZ, and
+compute STFT using Hann window size of 400 and hop
+length of 160. Please refer to [22] for more details.
+Model architecture. For the image backbone, we use
+the ImageNet-pretrained ResNet-18 model. For the IMU
+backbone, we use a 5-layer 1D Dilated CNN, as found ef-
+fective for IMU data processing [6]. We use the same net-
+work setting (kernel dimension, dilation gap and channel
+dimension) as in prior work [6]. The feature fusion model
+consists of a concatenation operation following two fully-
+connected layers with hidden dimension of 1024.
+Each
+layer except for the output layer is followed by a ReLU ac-
+tivation. The output dimension is the same as the teacher
+video model’s feature dimension (768 in the case of Mo-
+tionFormer). When N > 1, we use a one-layer GRU mod-
+ule to aggregate extracted features for each frame. We use
+a single-directioal GRU with hidden dimension of 512.
+Model training.
+We train our models in two stages.
+In the self-supervised IMU feature learning stage, we train
+random initialized IMU encoder fM, IMU predictor h and
+the fusion network Π with LNCE. Here the image encoder
+fI is a fixed ImageNet pretrained model. On both datasets,
+we train the model for 50 epochs with AdamW and batch
+size 64. The initial training rate is 1e−4. We decay the
+training rate by 0.1 at epoch 30 and epoch 40. In the sec-
+ond video feature distillation stage, we initialize the model
+with parameters obtained in the last stage and finetune. On
+both datasets, we use AdamW with batch size 64 and ini-
+tial learning rate 1e−4. On Ego4D, we train for 150 epochs.
+We decay the training rate by 0.1 at epoch 90 and epoch
+120. On EPIC-Kitchens, we train for 50 epochs. We decay
+the training rate by 0.1 at epoch 30 and epoch 40.
+
+Ego4D
+EPIC-Kitchens
+uniform
+38.46
+52.43
+random
+36.85
+48.48
+first
+38.68
+46.40
+last
+35.46
+41.72
+center
+37.04
+44.85
+Table 4. Effect of frame selection. We compare the accuracy of
+using different frame selection heuristics for EgoDistill when N =
+4. We observe that Uniform on average achieves better results.
+D. Analysis.
+Effect of frame selection.
+In Section 3.2, we men-
+tioned that we use uniform sampling to obtain the N frames
+from each video clip. In this section, we compare the per-
+formance of our work under uniform sampling with other
+heuristics. Specifically, we compare with random sampling,
+the first N frames, the last N frames and the center N
+frames. We show the results in Table 4 under N = 4. These
+results indicate that uniform sampling leads to the best per-
+formance on average. Intuitively, uniform sampling on av-
+erage leads to a broader coverage of both semantic contexts
+as well as scene motion.
+Why we set N to be small. In our experiments, we set
+N to be 1 to 4. Using larger N (e.g., 8 or 16) with densely
+sampled frames could lead to better results of all the meth-
+ods with more computational cost. Efficient video under-
+standing methods could benefit more as they have better
+temporal aggregation mechanisms given densely-sampled
+frames.
+However, the core purpose of our model is to
+deal with cases where we only use a few number of sam-
+ples. Therefore, our model is not comparable to video clip
+models under dense-frame setting. Furthermore, setting N
+to be a small number is very important in many applica-
+tions. As loading more image frames takes additional time
+and memory, applications with streaming videos or low-
+resource AR/VR devices will benefit from loading only a
+few frames.
+

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