Program > Plenary talks

The role of fractional differentiation in the detection and characterization of chirps and oscillating singularities   Stéphane Jaffard Professeur at Université Paris Est Créteil (UPEC), France

Abstract:

Many types of signals display a very oscillatory behavior in the neighborhood of singularities. It is for example the case for gravitational waves, fully developed turbulence, or brain data. A major issue is to detect such behaviors (referred to as "oscillating singularities'' or "chirps'') which are the signature of important physical phenomena which can often be characterized mathematically using fractional differentiation or integration.  We will show how wavelet and related tools allow us to meet these challenges.

Biography:

Stéphane Jaffard is a professor of mathematics at LAMA (Laboratory of Analysis and Applied Mathematics) at Université Paris Est Créteil (UPEC).  He graduated at École Polytechnique, where  he obtained a PhD under the supervision of  Yves Meyer. He spent a postdoc at the Institute for Advanced Study. In 1995, he became a professor at Université Paris Est Créteil. From 2000 to 2005, he was a junior member of the Institut de France. He was President of the French Mathematical Society from 2007 to 2010. In 2022, he coordinated the project "Assises des Mathématiques", which aimed at promoting the importance of mathematics in science, industry, and society. His main works concern wavelet and multifractal analysis, with applications in signal and image processing. In 1991, together with Ingrid Daubechies and Jean-Lin Journé, he built Wilson bases, the existence of which had been conjectured by Nobel Prize-winning physicist Kenneth Wilson. Their exceptional properties for the temporal and frequency analysis of signals play a key role in the signal processing algorithm that led to the discovery of gravitational waves in 2015. He determined the wavelet characterization of pointwise Hölder regularity and applied it to the analysis of large classes of functions and stochastic processes. Recently, with Bruno Martin, he determined the pointwise regularity of Jean-Christophe Yoccoz's Brjuno function, which plays a central role in the theory of holomorphic dynamical systems. He introduced the wavelet leader method in multifractal analysis, whose purpose is to estimate the size of the sets of points where a function or a signal displays a given pointwise regularity exponent. Together with Patrice Abry and Herwig Wendt, he applied this technique to the analysis and classification of e.g. turbulence data, finance, physiological signals or the analysis of Vincent van Gogh paintings (attribution to different periods of creation, differentiation of forgeries). In 2021, he was awarded the Jacques-Louis-Lions prize by the French Academy of Sciences. In 2023, he held the Aisentstadt Chair at the CRM (Centre de Recherche Mathématique, Université de Montréal). In 2024 he holds the  Francqui Chair at VUB (Vrije Universiteit Brussel). 

Fractional integro-differentiation as case of FoxH transform   Oleg Igorevich Marichev Mathematician, Wolfram Alpha, USA and  Paco Jain, co-presenter Research Programmer, Wolfram Alpha

Abstract:

For many years, Oleg has worked with Meijer G and Fox H functions and has helped to build the largest collection of their particular cases, wherein one can find about 150 named functions and their combinations. Recently, in collaboration with Paco Jain, he has implemented his results in the Wolfram Function Repository with the four functions MeijerGForm, FoxHForm, GenericIntegralTransform, and FractionalOrderD, and made corresponding talks at the Wolfram's annual Technology Conference. In this talk, he will describe the structure of majority of the one-dimensional integral transforms (including Riemann-Liouville fractional integro-differentiation as case) in terms of Mellin-Barnes integrals containing Fox H functions in the kernel.

Biography:

Oleg Igorevich Marichev was born in 1945 in Velikiye Luki, Soviet Union) is a mathematician. In 1949 he moved to Minsk (Belarus) with his parents. He graduated from the University of Belarus, where he continued to study for the Ph.D. degree. His scientific supervisor was Fedor Gakhov.  He is the co-author of a comprehensive five volume series of  Integrals and Series (Gordon and Breach Science Publishers, 1986–1992) together with Yury Brychkov and A. P. Prudnikov. Around 1990 he received the D.Sc. degree (Habilitation) in mathematics from the University of Jena, Germany. In 1992, Marichev started working with Stephen Wolfram on Mathematica. His wife Anna helps him in his job. 

Oleg Marichev holds two doctorates in mathematics. His first dissertation, completed at the Belarusian State University, Minsk, was titled "Tricomi's Boundary Value Problem for Some Mixed Type Equations and Integral Equations with Special Functions in the Kernels." His second dissertation, for an advanced doctorate from Friedrich-Schiller-Universität, Jena, Germany, was titled "Functions of Hypergeometric Type and Some Applications to Integral and Differential Equations." Marichev is the author or coauthor of 70 papers and eight books, including his Handbook of Integral Transforms of Higher Transcendental Functions. He wrote the five-volume series Integrals and Series with A. P. Prudnikov and Yu. A. Brychkov. He is also a coauthor with S. G. Samko and A. A. Kilbas of Integrals and Derivatives of Fractional Order and Some of Their Applications, the first complete monograph on the topic. Marichev joined Wolfram Research in 1991 and has developed many of Mathematica's algorithms for the calculation of definite and indefinite integrals and hypergeometric functions including Meijer G.

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Diffusive Representations of Riemann-Liouville Integral Operators and Their Applications   Kai Diethelm Professor at Technical University of Applied Sciences Würzburg-Schweinfurt, Germany

Abstract:

The goal of this talk is to provide a theoretical foundation for the construction of very efficient numerical algorithms for computing fractional integrals and for solving fractional differential equations. In this context, efficiency relates to three aspects: (i) small run time, (ii) low memory requirements, and (iii) simple integrability into general computational frameworks like finite element software packages. While the first aspect has been discussed intensively for many years, the two others have attracted interest only much more recently. To achieve our goal, we recall the concept of diffusive representations (or infinite state representations) of fractional integral operators. We demonstrate how such representations naturally lead to numerical methods that automatically satisfy all three requirements. A few approaches of this type have been developed in the past, but often rather unsystematically. By looking at the issue from a slightly more abstract perspective, we attempt to provide some general structural information and to initiate a systematic analysis of such methods.

Biography:

Kai Diethelm is Professor for Mathematics and Applied Computer Science at the Technical University of Applied Sciences Würzburg-Schweinfurt, Germany, where he leads the Scientific Computing Laboratory. He holds a Diploma in Mathematics from TU Braunschweig and a PhD in Computer Science and a habilitation in Mathematics, both from the University of Hildesheim. His main research interests are approximation theory and its applications, analytical and numerical aspects of fractional calculus (especially, fractional ordinary differential equations), and high performance computing. His list of publications includes three textbooks and almost 100 peer-reviewed papers in journals and proceedings volumes. 

From a new approach of the epidemic dynamics to predictions of the CO2 concentration in the atmosphere*   Alain Oustaloup Professeur Emeritus at Bordeaux INP/enseirb-Matmeca, France

*At the origin of this plenary talk, two journal articles published in 2021 and 2023 in Annual Reviews in Control:

A. Oustaloup, F. Levron, S. Victor, and L. Dugard. Non-integer (or fractional) power model to represent the complexity of a viral spreading: Application to the COVID-19. Annual Reviews in Control, 52:523–542, 2021. doi: 10.1016/j.arcontrol.2021.09.003.

A. Oustaloup, F. Levron, S. Victor, and L. Dugard. Addendum: Predictive form of the FPM model. Annual Reviews in Control, 2023. doi: 10.1016/j.arcontrol.2023.02.001.

Abstract:

Recognized as a "CNRS 2021 highlight", the FPM (Fractional Power Model), which generalizes linear regression, is a three-parameter model whose power alone indicates the progression, stabilization or regression of an epidemic. Thanks to its non-integer power, this model contributes to the unification of the phenomena of diffusion in physics and viral spreading in epidemiology. Its representativity (of real data) is due to its ability to take into account an unlimited number of internal dynamics of different speeds. The model can thus represent all the internal dynamics of an epidemic, from the slowest (originating in the desolate countryside) to the fastest (originating in densely populated major cities). Its predictivity is due to its ability to take into account the whole of the past, weighting it as appropriate. In fact, the model is equipped with a long-memory predictive form that expresses the fact that any predicted value is a function of all past values, values that turn out to be favorably weighted according to a forgetting factor (which is not without evoking a subtle form of memory). The model thus has the advantage of making the best possible use of the past, all the more so as only the past can be used to predict the future - in fact, a predictive specificity likely to make this model a good predictor for decision-makers. The model's representativity has been validated with official data of French Ministry of Health on the spreading of COVID-19, including time series of contaminations and hospitalizations. Its predictivity has also been validated by verified predictions in containment and vaccination phases, and even for the vaccination itself. Finally, the model has also been validated by verified predictions in its favor, carried out in the field of air pollution.

1 – The FPM model: a new model initially conceived for a viral spreading

The complexity of a viral spreading does not escape from diversity that naturally associates multiplicity and difference(Oustaloup et al. (2014)). Then why not explicit a form of diversity in conformity with a mode of spreading by contamination, and likely to give a systemic definition of such a spreading mode.

Founded on the state change of several elements, such a diversity form is precisely a multitude of elements liable to change state in different time and in different number: it is the case of numerous complex phenomena, among others spreading phenomena by contamination, not to mention phenomena whose state changes can come from vaccinations, births, deaths, ..., or even financial motions. In this form of diversity, the difference associated with the multiplicity thus turns, both, on the time and the number of state changes. If one considers that each state change is defined by the change from a state 0 to a state 1, and if one assumes that state 0 is a state of non-contamination and that state 1 is a state of contamination, the studied diversity form then corresponds to the spreading process of a viral contamination.

1.1. An unlimited number of internal dynamics

In terms of dynamics and from the state changes as defined, a phenomenological analysis of the effects of a joint dispersion of the times and numbers of state changes enables detecting a dispersion of internal dynamics. These dynamics are convex if the epidemic increases or concave if the epidemic decreases: in fact, in the sense of diversity, a multitude of different dynamics that well answers to multiplicity and difference. The analysis idea explaining this dispersion of internal dynamics is founded on the two following cases: if the state changes are far in time and low in number, they generate slow dynamics (as those stemming from very desertified countryside); if the state changes are close in time and high in number, they generate fast dynamics (as those stemming from very densified cities). Between these two cases, all the possible combinations in time and in number generate intermediate dynamics. 

1.2. A dynamics in power-law

Arranging from the slowest to the fastest dynamics, thus corresponding to a regular growth of rapidities, all these internal dynamics constitute a regular distribution of concave or convex dynamics. Actually, such a distribution evokes the distribution of aperiodic dynamics which is at the origin of a (global) dynamics in power-law, that the power-law, , defines, up to a factor, where  is a non-integer power (Oustaloup (1995)). It is true that the power-law, , is well composed of concave or convex internal dynamics according to the value of . The decomposition of  into its internal dynamics indeed reveals an indefinite (or unlimited) number of internal dynamics that are concave for   or convex for .

 1.3. On the power-law, , in epidemiology

Already apprehended by its essence, the power-law, , that via its non-integer power participates in the unification of the diffusion and viral spreading phenomena, is presented here more for its use interest in epidemiology. For positive , this power-law, then appropriate to model a viral spreading, appears as a modeling tool particularly fitted in the matter. Indeed, if a first reason is the simplicity of the law in comparison with the complexity to be represented, a second reason is that the power, ,  determines on its own the curvature of the phases of an epidemic, even a null curvature, namely: a convexity, for , corresponding to a progression of the epidemic; a  straight-line, for , corresponding to a stabilization of the epidemic; a concavity, for , corresponding to a  regression of the epidemic. The power, , is thus, according to its value, a simple indicator of an aggravation, a stabilization or an improving of the epidemic.

1.4. From the power-law, , to the FPM model

To change from the power-law, , to a complete model, this law must be completed by two constants: an additive constant, , to take into account the whole of the contaminations, including the former contaminations prior to the initial instant; a multiplicative constant, , to express that the highlighted power-law dynamics is defined by  up to a factor. Thus, the complete model, likely to correctly represent the evolution of the totality of the contaminations, is presented under the form of an affine function of , namely : this model with three parameters, ,  and , where  is a non-integer parameter ``of high level'', is called model with non-integer (or fractional) power, or FPM model (Fractional Power Model); its parameters are determined by identification with real data, including those on the evolution of an epidemic.  

1.5. Validation of the model by its application to the COVID-19

The results that are presented in this section result from the application of the FPM model to the official data relative to the contaminations by the COVID-19, in this case the data collected in France from March 1^st to October 1^st 2020, namely on 7 months or 215 days. Determined in the identification phase, the model, , then gives a (mathematical) representation of the evolution of the real data; then, in the prediction phase where it is used as a predictor, the model then gives a prediction (or estimation) of the evolution of the future values.

When the FPM model is used as a predictor, the idea is then to use the first effects of a contamination or a sanitary measure (lockdown, vaccination, etc.) in order to capture (by the model) the inherent evolution to these first effects, and this, to predict the following by prolonging this evolution as far as the epidemic and the model present similar evolutions. To prove the conservation of the model validity in high vaccinal rise, the model has been calculated with the data of one week, from April 26 to May 2 2021, then used as a predictor on a prediction horizon of 6 weeks, from May 3 to June 13. The predictivity of the model has then been proved by a low prediction error, the maximum of the relative error modulus, obtained at the last day, being 1.18%, and the mean of these errors being only -0.23%, not to mention that, at the end of the 5^th week, the maximum and the mean were only 0.54% and 0.09%.

2 – Why the FPM model is a good predictor

2.1. Borrowing at best from the past to better predict the future

As only the past can be used to predict the future, then why not make the best use of the past. This formula which is worth a citation defines our strategy on the subject: indeed, this strategy consists of borrowing at best from the past to better predict the future; indeed, it is a way to seek a good prediction, as the one consisting of a global consideration of the past such as precised in what follows.

2.2. A predictive form with long memory

Having previously shown the predictive performances of the FPM model, and notably its capacity to predict on an unusually extensive prediction horizon, the aim is to explain why this model intrinsically has such a capacity of prediction. The idea is then to highlight its predictive specificity (because specificity there is), namely a predictive form with long memory which turns out to consider the whole past to express the future. This form, such as mathematically established in Oustaloup (2021), is the one of an equation that expresses that any future (or predicted) value is a function of all the past values. More precisely, any predicted value is a weighted sum of all the past values. The weighting coefficients of the past values turn out to weight the past as appropriate, notably according to an attenuation of the past in conformity with a forgetting factor, which recalls a subtle form of memory. 

3 – Application of the FPM model to the CO₂ concentration in the atmosphere

Through CO₂ emissions, state changes defined for an epidemic are here replaced by level changes (of CO₂) having the same effects, notably by internal dynamics whose rapidity is different according to the CO₂ emission intensity: in fact, a unified vision of the complex phenomena that can be measured by cumulative data, such as data coming from the Hawaii Mauna Loa observatory and turning on the CO₂ concentration evolution in the atmosphere during 64 years, from 1959 to 2022. 

Concerning the application of the FPM model to these data, the section that develops it strives to compare the results obtained in prediction with the FPM model, a polynomial model and an exponential model. Each model has three parameters, and the polynomial model (of degree 2) is a model in   and   to be coherent with the FPM model, in  , whose power, , is here between 1 and 2. If the prediction horizon if of 34 years in all cases, the identification period, for which each model is estimated, is different according to the case, indeed presenting different durations, 15 years, 20 years, 25 years and 30 years. Although the modeling errors obtained with the three models for the different identification periods are similar in first analysis, the prediction errors clearly prove to be in favor of the FPM model, as proven by the mean of the absolute values of the relative errors at the end of prediction, in 2022: 0.65% with the FPM model; 3.35% with the polynomial model; 7.87% with the exponential model. If the FPM model is the most predictive, the exponential model is the least one, which classifies the polynomial model between these two models in terms of predictivity. Finally, being the most predictive, only the FPM model is then used as a predictor in the case of a prediction horizon truly belonging to the future, namely a prediction horizon also of 34 years, but from 2023 to 2056; the identification periods then finish in 2022 and take again the previous durations, 15 years, 20 years, 25 years and 30 years. The predictions so obtained turn out to be little different despite relatively different identification periods, thus expressing a prediction robustness with respect to identification: for example, concerning the predictions for the year 2056, whose mean is 512.93, the maximal prediction, 515.70, and the minimal prediction, 509.36, correspond to relative gaps that are only 0.54% and -0.70%.

Biography:

A graduate of ENSEIRB in 1973, Alain Oustaloup is currently Professor Emeritus at ENSEIRB-MATMECA - Bordeaux INP. After synthesizing complex non-integer differentiation or integration, then overcoming the stability-accuracy dilemma in automatic control and the mass-damping dilemma in mechanics, he invented the CRONE control and the CRONE suspension. More recently, he has come back with a new non-integer power predictor, which is applied to epidemiology and air pollution. His work was rewarded with a CNRS Silver Medal in 1997 and a Grand Prize of the French Academy of Science in 2011.