Publications

2021

• On Riemannian Stochastic Approximation Schemes with Fixed Step-Size
  
  • Durmus Alain
  • Jiménez Pablo
  • Moulines Éric
  • Said Salem
  , 2021, 130. This paper studies fixed step-size stochastic approximation (SA) schemes, including stochastic gradient schemes, in a Riemannian framework. It is motivated by several applications, where geodesics can be computed explicitly, and their use accelerates crude Euclidean methods. A fixed step-size scheme defines a family of time-homogeneous Markov chains, parametrized by the step-size. Here, using this formulation, non-asymptotic performance bounds are derived, under Lyapunov conditions. Then, for any step-size, the corresponding Markov chain is proved to admit a unique stationary distribution, and to be geometrically ergodic. This result gives rise to a family of stationary distributions indexed by the step-size, which is further shown to converge to a Dirac measure, concentrated at the solution of the problem at hand, as the step-size goes to 0. Finally, the asymptotic rate of this convergence is established, through an asymptotic expansion of the bias, and a central limit theorem. (10.48550/arXiv.2102.07586)
  DOI : 10.48550/arXiv.2102.07586
• Phaseless inverse scattering with background information
  
  • Novikov Roman
  • Sivkin Vladimir
  Inverse Problems, IOP Publishing, 2021, 37 (5), pp.055011. We consider phaseless inverse scattering for the multidimensional Schr\"odinger equation with unknown potential $v$ using the method of known background scatterers. In particular, in dimension $d \geq 2$, we show that $|f_1|^2$ at high energies uniquely determines $v$ via explicit formulas, where $f_1$ is the scattering amplitude for $v+w_1$, $w_1$ is an a priori known nonzero background scatterer, under the condition that $supp\, v$ and $supp\, w_1$ are sufficiently disjoint. If this condition is relaxed, then we give similar formulas for finding $v$ from $|f|^2 , |f_1|^2$, where $f$ is the scattering amplitude for $v$. In particular, we continue studies of [Novikov, J. Geom. Anal. 26(1), 346–359, 2016], [Leshem et al, Nature Communications 7(1), 1–6, 2016]. (10.1088/1361-6420/abf36c)
  DOI : 10.1088/1361-6420/abf36c
• POT : Python Optimal Transport
  
  • Flamary Rémi
  • Courty Nicolas
  • Gramfort Alexandre
  • Alaya Mokhtar Zahdi
  • Boisbunon Aurélie
  • Chambon Stanislas
  • Chapel Laetitia
  • Corenflos Adrien
  • Fatras Kilian
  • Fournier Nemo
  • Gautheron Léo
  • Gayraud Nathalie T H
  • Janati Hicham
  • Rakotomamonjy Alain
  • Redko Ievgen
  • Rolet Antoine
  • Schutz Antony
  • Seguy Vivien
  • Sutherland Danica J
  • Tavenard Romain
  • Tong Alexander
  • Vayer Titouan
  Journal of Machine Learning Research, Microtome Publishing, 2021. Optimal transport has recently been reintroduced to the machine learning community thanks in part to novel efficient optimization procedures allowing for medium to large scale applications. We propose a Python toolbox that implements several key optimal transport ideas for the machine learning community. The toolbox contains implementations of a number of founding works of OT for machine learning such as Sinkhorn algorithm and Wasserstein barycenters, but also provides generic solvers that can be used for conducting novel fundamental research. This toolbox, named POT for Python Optimal Transport, is open source with an MIT license.
• Body-fitted topology optimization of 2D and 3D fluid-to-fluid heat exchangers
  
  • Feppon Florian
  • Allaire Grégoire
  • Dapogny Charles
  • Jolivet Pierre
  Computer Methods in Applied Mechanics and Engineering, Elsevier, 2021, 376, pp.113638. We present a topology optimization approach for the design of fluid-to-fluid heat exchangers which rests on an explicit meshed discretization of the phases at stake, at every iteration of the optimization process. The considered physical situations involve a weak coupling between the Navier-Stokes equations for the velocity and the pressure in the fluid, and the convection-diffusion equation for the temperature field. The proposed framework combines several recent techniques from the field of shape and topology optimization, and notably a level-set based mesh evolution algorithm for tracking shapes and their deformations , an efficient optimization algorithm for constrained shape optimization problems, and a numerical method to handle a wide variety of geometric constraints such as thickness constraints and non-penetration constraints. Our strategy is applied to the optimization of various types of heat exchangers. At first, we consider a simplified 2D cross-flow model where the optimized boundary is the section of the hot fluid phase flowing in the transverse direction, which is naturally composed of multiple holes. A minimum thickness constraint is imposed on the cross-section so as to account for manufacturing and maximum pressure drop constraints. In a second part, we optimize the design of 2D and 3D heat exchangers composed of two types of fluid channels (hot and cold), which are separated by a solid body. A non-mixing constraint between the fluid components containing the hot and cold phases is enforced by prescribing a minimum distance between them. Numerical results are presented on a variety of test cases, demonstrating the efficiency of our approach in generating new, realistic, and unconventional heat exchanger designs. (10.1016/j.cma.2020.113638)
  DOI : 10.1016/j.cma.2020.113638
• An entropy stable high-order discontinuous Galerkin spectral element method for the Baer-Nunziato two-phase flow model
  
  • Coquel Frédéric
  • Marmignon Claude
  • Rai Pratik
  • Renac Florent
  Journal of Computational Physics, Elsevier, 2021, 431, pp.110135. In this work we propose a high-order discretization of the Baer-Nunziato two-phase flow model (Baer and Nunziato (1986) [5]) with closures for interface velocity and pressure adapted to the treatment of discontinuous solutions, and stiffened gas equations of states. We use the discontinuous Galerkin spectral element method (DGSEM), based on collocation of quadrature and interpolation points (Kopriva and Gassner (2010) [47]). The DGSEM uses summation-by-parts (SBP) operators in the numerical quadrature for approximating the integrals over discretization elements (Carpenter et al. (2014) [10]; Gassner et al. (2016) [32]). Here, we build upon the framework provided in (Renac (2019) [55]) for nonconservative hyperbolic systems to modify the integration over cell elements using the SBP operators and replace the physical fluxes with entropy conservative fluctuation fluxes from Castro et al. (2013) [12], while we derive entropy stable numerical fluxes applied at interfaces. This allows to prove a semi-discrete inequality for the cell-averaged physical entropy, while keeping high-order accuracy. The design of the numerical fluxes also formally preserves the kinetic energy at the discrete level. High-order integration in time is performed using strong stability-preserving Runge-Kutta schemes and we propose conditions on the numerical parameters for the positivity of the cell-averaged void fraction and partial densities. The positivity of the cell-averaged solution is extended to nodal values by the use of an a posteriori limiter. The high-order accuracy, nonlinear stability, and robustness of the present scheme are assessed through several numerical experiments in one and two space dimensions. (10.1016/j.jcp.2021.110135)
  DOI : 10.1016/j.jcp.2021.110135
• Chair in applied mathematics OQUAIDO Activity report
  
  • Roustant Olivier
  • Le Riche Rodolphe
  • Garnier Josselin
  • Ginsbourger David
  • Deville Yves
  • Helbert Céline
  • Pronzato Luc
  • Prieur Clémentine
  • Gamboa Fabrice
  • Bachoc François
  • Rohmer Jeremy
  • Perrin Guillaume
  • Marrel Amandine
  • Damblin Guillaume
  • Gliere Alain
  • Sinoquet Delphine
  • Richet Yann
  • da Veiga Sébastien
  • Huguet Frédéric
  , 2021. The research Chair OQUAIDO - for "Optimisation et QUAntification d'Incertitudes pour les Données Onéreuses" in French - gathers academic and technological research partners to work on statistical learning problems involving scarce and error-prone data. This Chair, was created in January 2016 for a period of 5 years. It has tackled problems where small data is described by statistical models that, in turn, serve to characterize uncertainties, calibrate computer codes and search for optimal configurations. Many of the investigated approaches rely on Gaussian processes and confront mathematical challenges such as high dimension (even functional inputs / outputs), mixed continuous and categorical inputs, specific constraints and medium data. This activity report highlights noticeable scientific contributions of OQUAIDO, provides bibliography indicators and summarizes the events that have marked the research Chair life. It is concluded by a few lessons on collaborative projects at the intersection between mathematics and engineering that were learnt during these 5 years.
• Understanding the Variability in Graph Data Sets through Statistical Modeling on the Stiefel Manifold
  
  • Mantoux Clément
  • Couvy-Duchesne Baptiste
  • Cacciamani Federica
  • Epelbaum Stéphane
  • Durrleman​ Stanley
  • Allassonnière Stéphanie
  Entropy, MDPI, 2021, 23 (4), pp.490. Network analysis provides a rich framework to model complex phenomena, such as human brain connectivity. It has proven efficient to understand their natural properties and design predictive models. In this paper, we study the variability within groups of networks, i.e., the structure of connection similarities and differences across a set of networks. We propose a statistical framework to model these variations based on manifold-valued latent factors. Each network adjacency matrix is decomposed as a weighted sum of matrix patterns with rank one. Each pattern is described as a random perturbation of a dictionary element. As a hierarchical statistical model, it enables the analysis of heterogeneous populations of adjacency matrices using mixtures. Our framework can also be used to infer the weight of missing edges. We estimate the parameters of the model using an Expectation-Maximization-based algorithm. Experimenting on synthetic data, we show that the algorithm is able to accurately estimate the latent structure in both low and high dimensions. We apply our model on a large data set of functional brain connectivity matrices from the UK Biobank. Our results suggest that the proposed model accurately describes the complex variability in the data set with a small number of degrees of freedom. (10.3390/e23040490)
  DOI : 10.3390/e23040490
• Optimal control of energy flexibilities in a stochastic environment
  
  • Grangereau Maxime
  , 2021. In this PhD dissertation, we use tools from stochastic optimal control, stochastic optimization and convex optimization to design mechanisms to control energy storage systems, to deal with the challenges created by the uncertain production of intermittent energy sources. First, we introduce a commitment mechanism where an individual consumer chooses a consumption profile, then controls its storage devices to track in real-time this profile. We formulate a Mean-Field Control problem to model this situation, for which we establish theoretic and numerical results. Second, we introduce a control problem for a large population of Thermostatically Controlled Loads (TCLs) subject to a common noise and providing ancillary services to the grid. We show that the centralized control problem can be replaced by a stochastic Stackelberg differential game with minimal information-sharing. This allows for a decentralized control scheme with performance guarantees, while preserving privacy of consumers and limiting telecommunication requirements. We then develop a Newton method for stochastic control problems. We show that the computation of the Newton step reduces to solving Backward Stochastic Differential Equations, then we design an appropriate line-search procedure and prove global convergence of the Newton method with line-search in an appropriate space. Its performance is illustrated on a problem of control of a large number of batteries providing services to the grid. Last, a multi-stage stochastic Alternating Current Optimal Power Flow problem is formulated in order to control a power network equipped with energy storage systems. A priori conditions ensuring a vanishing relaxation gap are derived and an easily computable a posteriori bound on the relaxation gap of the problem is given. Using Shapley-Folkman-type results, a priori bounds on the duality gap of non-convex multi-stage stochastic problems with a generic structure are derived.
• A Lagrangian approach for aggregative mean field games of controls with mixed and final constraints
  
  • Bonnans Joseph Frédéric
  • Gianatti Justina
  • Pfeiffer Laurent
  , 2021. The objective of this paper is to analyze the existence of equilibria for a class of deterministic mean field games of controls. The interaction between players is due to both a congestion term and a price function which depends on the distributions of the optimal strategies. Moreover, final state and mixed state-control constraints are considered, the dynamics being nonlinear and affine with respect to the control. The existence of equilibria is obtained by Kakutani's theorem, applied to a fixed point formulation of the problem. Finally, uniqueness results are shown under monotonicity assumptions.
• On rapid oscillations driving biological processes at disparate timescales
  
  • Lunz Davin
  Physical Biology, Institute of Physics: Hybrid Open Access, 2021, 18 (3), pp.036002. We consider a generic biological process described by a dynamical system, subject to an input signal with a high-frequency periodic component. The rapid oscillations of the input signal induce inherently multiscale dynamics, motivating order-reduction techniques. It is intuitive that the system behaviour is well approximated by its response to the averaged input signal. However, changes to the high-frequency component that preserve the average signal are beyond the reach of such intuitive reasoning. In this study, we explore system response under the influence of such an input signal by exploiting the timescale separation between high-frequency input variations and system response time. Employing the asymptotic method of multiple scales, we establish that, in some circumstances, the intuitive approach is simply the leading-order asymptotic contribution. We focus on higher-order corrections that capture the response to the details of the high-frequency component beyond its average. This approach achieves a reduction in system complexity while providing valuable insight into the structure of the response to the oscillations. We develop the general theory for nonlinear systems, while highlighting the important case of systems affine in the state and the input signal, presenting examples of both discrete and continuum state spaces. Importantly, this class of systems encompasses biochemical reaction networks described by the chemical master equation and its continuum approximations. Finally, we apply the framework to a nonlinear system describing mRNA translation and protein expression previously studied in the literature. The analysis shines new light on several aspects of the system quantification and both extends and simplifies results previously obtained. (10.1088/1478-3975/abd9db)
  DOI : 10.1088/1478-3975/abd9db
• Bayesian inference and uncertainty quantification of reduced chemical schemes
  
  • Armengol J M
  • Le Maitre Olivier
  • Vicquelin R
  , 2021. This work concerns the uncertainties arising from the derivation of global chemistry models and their impact on the predictions using modern combustion simulations. We perform the inference of the parameters of a two-step reaction mechanism for CH 4 , using synthetic observations of one-dimensional laminar flames generated using detailed mechanism simulations. Introduction of Principal Component Analysis (PCA) and the Polynomial Chaos (PC) expansion, to approximate the global model predictions, enables a full assessment of the inferred global model's posterior. In particular, we employ the Bayesian posteriors' extensive sampling to estimate mean, Maximum a Posteriori, and confidence intervals of the inferred global model's predictions. We contrast the posterior distributions of global quantities of the flame, namely the laminar flame speed, the thermal flame thickness, and the reaction zone thickness, depending on the inference's observations. Finally, we propagate the global chemistry model's posterior distribution through two-dimensional direct numerical simulations (DNS) of a flame-vortex interaction problem. This study highlights the importance of quantifying posterior uncertainties to fully appreciate the impact of using a global model in real-world reactive simulations.
• Shapes enhancing the propulsion of multiflagellated helical microswimmers
  
  • Berti Luca
  • Binois Mickael
  • Alouges François
  • Aussal Matthieu
  • Prud'Homme Christophe
  • Giraldi Laetitia
  , 2021. In this paper we are interested in optimizing the shape of multi-flagellated helical microswimmers. Mimicking the propagation of helical waves along the flagella, they self-propel by rotating their tails. The swimmer's dynamics is computed using the Boundary Element Method, implemented in the open source Matlab library $Gypsilab$. We exploit a Bayesian optimization algorithm to maximize the swimmer's speeds through their shape optimization. Our results show that the optimal tail shapes are helices with large wavelength, such that the shape periodicity is disregarded. Moreover, the best propulsion speed is achieved for elongated heads when the swimmer has one or two flagella. Surprisingly, a round head is obtained when more flagella are considered. Our results indicate that the position and number of flagella modify the propulsion pattern and play a significant role in the optimal design of the head. It appears that Bayesian optimization is a promising method for performance improvement in microswimming.
• A new McKean-Vlasov stochastic interpretation of the parabolic-parabolic Keller-Segel model: The two-dimensional case
  
  • Tomasevic Milica
  The Annals of Applied Probability, Institute of Mathematical Statistics (IMS), 2021, 31 (1), pp.28. In Talay and Tomasevic [20] we proposed a new stochastic interpretation of the parabolic-parabolic Keller-Segel system without cutoff. It involved an original type of McKean-Vlasov interaction kernel which involved all the past time marginals of its probability distribution in a singular way. In the present paper, we study this McKean-Vlasov representation in the two-dimensional case. In this setting there exists a possibility of a blow-up in finite time for the Keller-Segel system if some parameters of the model are large. Indeed, we prove the well-posedness of the McKean-Vlasov equation under some constraints involving a parameter of the model and the initial datum. Under these constraints, we also prove the global existence for the Keller-Segel model in the plane. To obtain this result, we combine PDE analysis and stochastic analysis techniques. (10.1214/20-AAP1594)
  DOI : 10.1214/20-AAP1594
• Some contributions to decision-making problems
  
  • Logé Frédéric
  , 2021. This thesis, motivated by applications in the industrial and health sectors, is a collection of studies on different decision problems.In the first part, we focus on single-step decision-making problems where a predictive model is used upstream of the decision-making, and where explicit feedback is received. We propose to leave the task of defining the loss function associated with the predictive model to the end-user, in order to encode the real cost of using a forecast to make a decision. As an algorithmic approach, we consider decision trees, optimized with the adjusted loss function, and function approximation methods, similar to Q-value function approximation in reinforcement learning, in case where only the immediate reward is of interest. Three applications are studied: the calibration of an alarm system to fight against medical wandering; the problem of nomination in an electricity market, from the point of view of a renewable energy supplier; the optimization of production in the uncertainty of customer demand.In the second part, we focus on two specific problems of sequential decision-making, which we address using a Markov decision-making framework and reinforcement learning algorithms. In the first application, we try to optimize meal timing and insulin management for people with type I diabetes who rely on self-injections. To do so, we rely on a patient simulator, which is based on medical knowledge of the interaction between glucose and insulin and on physiological parameters specific to the patients. In the second application, we try to build an adaptive predictive questionnaire for smooth interactions with users. For binary data, the questionnaire looks like a decision tree, optimized in a bottom-up way. For non-binary data, this new questionnaire only asks questions that have already been asked, remembers previously observed values, and exploits them fully once they arrive in a terminal node, where a specific prediction function is available.In our final section, we look at three decision processes that, by construction, do not require the agent to explore the environment. For example, we consider a system whose dynamics are sufficiently stochastic that, whatever our action, we explore the state space, while having some influence through our actions. We also consider a system where some actions are randomly unavailable depending on the epochs. In addition to the theoretical results found, this part emphasizes the importance of focusing the exploration where it is needed.
• Coherent interferometric imaging in a random flow
  
  • Gay Etienne
  • Bonnet Luc
  • Peyret Christophe
  • Savin Éric
  • Garnier Josselin
  Journal of Sound and Vibration, Elsevier, 2021, 494, pp.115852. This paper is concerned with the development of imaging methods to localize sources or reflectors in inhomogeneous moving media with acoustic waves that have travelled through them. A typical example is the localization of broadband acoustic sources in a turbulent jet flow for aeroacoustic applications. The proposed algorithms are extensions of Kirchhoff migration (KM) and coherent interferometry (CINT) which have been considered for smooth and randomly inhomogeneous quiescent media so far. They are constructed starting from the linearized Euler equations for the acoustic perturbations about a stationary ambient flow. A model problem for the propagation of acoustic waves generated by a fixed point source in an ambient flow with constant velocity is addressed. Based on this result imaging functions are proposed to modify the existing KM and CINT functions to account for the ambient flow velocity. They are subsequently tested and compared by numerical simulations in various configurations, including a synthetic turbulent jet representative of the main features encountered in actual jet flows. (10.1016/j.jsv.2020.115852)
  DOI : 10.1016/j.jsv.2020.115852
• Sample average approximation for risk-averse problems: A virtual power plant scheduling application
  
  • Lima Ricardo M
  • Conejo Antonio J
  • Giraldi Loïc
  • Le Maître Olivier
  • Hoteit Ibrahim
  • Knio Omar M
  EURO Journal on Computational Optimization, Springer, 2021, 9, pp.100005. In this paper, we address the decision-making problem of a virtual power plant (VPP) involving a self-scheduling and market involvement problem under uncertainty in the wind speed and electricity prices. The problem is modeled using a risk-neutral and two risk-averse two-stage stochastic programming formulations, where the conditional value at risk is used to represent risk. A sample average approximation methodology is integrated with an adapted L-Shaped solution method, which can solve risk-neutral and specific risk-averse problems. This methodology provides a framework to understand and quantify the impact of the sample size on the variability of the results. The numerical results include an analysis of the computational performance of the methodology for two case studies, estimators for the bounds of the true optimal solutions of the problems, and an assessment of the quality of the solutions obtained. In particular, numerical experiences indicate that when an adequate sample size is used, the solution obtained is close to the optimal one. (10.1016/j.ejco.2021.100005)
  DOI : 10.1016/j.ejco.2021.100005
• A continuation method for building invisible obstacles in waveguides
  
  • Bera Antoine
  • Bonnet-Ben Dhia Anne-Sophie
  • Chesnel Lucas
  Quarterly Journal of Mechanics and Applied Mathematics, Oxford University Press (OUP), 2021, 74 (1), pp.83-116. We consider the propagation of acoustic waves at a given wavenumber in a waveguide which is unbounded in one direction. We explain how to construct penetrable obstacles characterized by a physical coefficient ρ which are invisible in various ways. In particular, we focus our attention on invisibility in reflection (the reflection matrix is zero), invisibility in reflection and transmission (the scattering matrix is the same as if there were no obstacle) and relative invisibility (two different obstacles have the same scattering matrix). To study these problems, we use a continuation method which requires to compute the scattering matrix S(ρ) as well as its differential with respect to the material index dS(ρ). The justification of the method also needs for the proof of abstract results of ontoness of well-chosen functionals constructed from the terms of dS(ρ). We provide a complete proof of the results in monomode regime when the wavenumber is such that only one mode can propagate. And we give all the ingredients to implement the method in multimode regime. We end the article by presenting numerical results to illustrate the analysis. (10.1093/qjmam/hbaa020)
  DOI : 10.1093/qjmam/hbaa020
• Joint self-supervised blind denoising and noise estimation
  
  • Ollion Jean
  • Ollion Charles
  • Gassiat Elisabeth
  • Lehéricy Luc
  • Le Corff Sylvain
  , 2021. We propose a novel self-supervised image blind denoising approach in which two neural networks jointly predict the clean signal and infer the noise distribution. Assuming that the noisy observations are independent conditionally to the signal, the networks can be jointly trained without clean training data. Therefore, our approach is particularly relevant for biomedical image denoising where the noise is difficult to model precisely and clean training data are usually unavailable. Our method significantly outperforms current state-of-the-art self-supervised blind denoising algorithms, on six publicly available biomedical image datasets. We also show empirically with synthetic noisy data that our model captures the noise distribution efficiently. Finally, the described framework is simple, lightweight and computationally efficient, making it useful in practical cases.
• Randomized Derivative Free Optimization via CMA-ES and Sparse Techniques - Applications to Radars
  
  • Varelas Konstantinos
  , 2021. In this thesis, we investigate aspects of adaptive randomized methods for black-box continuous optimization. The algorithms that we study are based on the Covariance Matrix Adaptation Evolution Strategy (CMA-ES) algorithm and focus on large scale optimization problems. We start with a description of CMA-ES and its relation to the Information Geometric Optimization (IGO) framework, succeeded by a comparative study of large scale variants of CMA-ES. We furthermore propose novel methods which integrate tools of high dimensional estimation within CMA-ES, to obtain more efficient algorithms for large scale partially separable problems. Additionally, we describe the methodology for algorithm performance evaluation adopted by the Comparing Continuous Optimizers (COCO) platform, and finalize the bbob-largescale test suite, a novel benchmarking suite with problems of increased dimensions and with a low computational cost. Finally, we present the formulation, methodology and obtained results for two applications related to Radar problems, the Phase Code optimization problem and the Phased-Array Pattern design problem.
• Randomized Derivative Free Optimization via CMA-ES and Sparse Techniques : Applications to Radars
  
  • Varelas Konstantinos
  , 2021. In this thesis, we investigate aspects of adaptive randomized methods for black-box continuous optimization. The algorithms that we study are based on the Covariance Matrix Adaptation Evolution Strategy (CMA-ES) algorithm and focus on large scale optimization problems.We start with a description of CMA-ES and its relation to the Information Geometric Optimization (IGO) framework, succeeded by a comparative study of large scale variants of CMA-ES. We furthermore propose novel methods which integrate tools of high dimensional analysis within CMA-ES, to obtain more efficient algorithms for large scale partially separable problems.Additionally, we describe the methodology for algorithm performance evaluation adopted by the Comparing Continuous Optimizers (COCO) platform, and finalize the bbob-largescale test suite, a novel benchmarking suite with problems of increased dimensions and with a low computational cost.Finally, we present the formulation, methodology and obtained results for two applications related to Radar problems, the Phase Code optimization problem and the Phased-Array Pattern design problem.
• On continuum approximations of discrete-state Markov processes of large system size
  
  • Lunz Davin
  Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal, Society for Industrial and Applied Mathematics, 2021, 19 (1), pp.294-319. Discrete-state continuous-time Markov processes are an important class of models employed broadly across the sciences. When the system size becomes large, standard approaches can become intractable to exact solution and numerical simulation. Approximations posed on a continuous state space are often more tractable and are presumed to converge in the limit as the system size tends to infinity. For example, an expansion of the master equation truncated at second order yields the Fokker--Planck equation, a widely used continuum approximation equipped with an underlying process of continuous state. Surprisingly, in [Doering \textit{et. al.} Multiscale Model. Sim. 2005 3:2, p.283--299] it is shown that the Fokker--Planck approximation may exhibit exponentially large errors, even in the infinite system-size limit. Crucially, the source of this inaccuracy has not been addressed. In this paper, we focus on the family of continuous-state approximations obtained by arbitrary-order truncations. We uncover how the exponentially large error stems from the truncation by quantifying the rapid error decay with increasing truncation order. Furthermore, we explain why this discrepancy only comes to light in a subset of problems. The approximations produced by finite truncation beyond second order lack underlying stochastic processes. Nevertheless, they retain valuable information that explains the previously observed discrepancy by bridging the gap between the continuous and discrete processes. The insight conferred by this broader notion of ``continuum approximation'', where we do not require an underlying stochastic process, prompts us to revisit previously expressed doubts regarding continuum approximations. In establishing the utility of higher-order truncations, this approach also contributes to the extensive discussion in the literature regarding the second-order truncation: while recognising the appealing features of an associated stochastic process, in certain cases it may be advantageous to dispense of the process in exchange for the increased approximation accuracy guaranteed by higher-order truncations. (10.1137/20M1332293)
  DOI : 10.1137/20M1332293
• Fast incremental expectation-maximization algorithm: $\sqrt{N}$ iterations for an epsilon-stationary point ?
  
  • Fort Gersende
  • Moulines Eric
  • Gach Pierre
  , 2021. Fast Incremental Expectation Maximization (FIEM) is an iterative algorithm, based on the Expectation Maximization (EM) algorithm, which was introduced to design EM for the large scale learning framework by avoiding the full data set to be processed at each iteration. In this paper, we first recast this algorithm in the Stochastic Approximation (SA) within EM framework. Then, we provide non asymptotic convergence rates as a function of the batch size n and of the maximal number of iterations Kmax fixed by the user. This allows a complexity analysis: in order to reach an-approximate solution, how does Kmax depend upon n and ?
• Application of contract theory to energy regulation problems, and study of the joint laws of a martingale and its running maximum
  
  • Farhat Heythem
  , 2021. This dissertation treats two independent topics. The first one is the application of stochastic differential games with non zero sum; the Principal-Agent models (c.f. Cvitanic & Zhang (2013) and Cvitanic et al.(2018)) to solve some contemporary challenges of energy markets regulation. The second concerns the study of the dynamics of the joint law of a continuous martingale and its running maximum.The first work is about Capacity Remuneration Mechanisms (CRM) in the electricity market. Given the growing share of renewable energies in the production of electricity, "conventional" power plants (gas or coal-fired) are less and less used, which makes them not viable economically. However, shutting down these power plants would expose consumers to a risk of shortage or blackout in the event of a peak demand for electricity. This is due to the fact that electricity can hardly be stored, and so the production capacity should always be maintained at a level above demand. This explains the necessity of a "Capacity Remuneration Mechanism" (CRM) to pay for rarely used power plants, which can be understood as an insurance against electrical shortages and blackouts.We address then the issue of the incentives for decarbonation. The goal is to propose a model of an instrument that can be used by a public agent (the state) in order to incentivize different sectors to reduce their carbon emissions in a context of moral hazard (where the state does not observe the action of the actors and therefore cannot know whether a reduction in emissions comes from a reduction in production and consumption, or from a management effort towards a less polluting production (for example investment in research and development). This provides an alternative to the carbon tax, and does not require perfect information as the latter.The second part of this thesis deals with a completely independent subject, motivated by model calibration and arbitrage in a financial market with barrier options. We provides a result on the joint laws between a martingale and its running maximum. We consider a family of 2 dimensional probabilities and we provide necessary and sufficient conditions for the existence of a martingale such that it's marginal laws coupled with those of its running maximum match the given probabilities.We follow the methodology of Hirsch & Roynette (2012) where they construct a martingale using an SDE corresponding to a wellposed Fokker-Planck PDE satisfied by the marginal laws of this martingale under smoothness assumptions, then using a regularization in the general case.
• What Tropical Geometry Tells Us about the Complexity of Linear Programming
  
  • Allamigeon Xavier
  • Benchimol Pascal
  • Gaubert Stéphane
  • Joswig Michael
  SIAM Review, Society for Industrial and Applied Mathematics, 2021, 63 (1), pp.123-164. Tropical geometry has been recently used to obtain new complexity results in convex optimization and game theory. In this paper, we present an application of this approach to a famous class of algorithms for linear programming, i.e., log-barrier interior point methods. We show that these methods are not strongly polynomial, by constructing a family of linear programs with 3r + 1 inequalities in dimension 2r for which the number of iterations performed is in Ω(2 r). The total curvature of the central path of these linear programs is also exponential in r, disproving a continuous analogue of the Hirsch conjecture proposed by Deza, Terlaky, and Zinchenko. These results are obtained by analyzing the tropical central path, which is the piecewise linear limit of the central paths of parameterized families of classical linear programs viewed through "logarithmic glasses." This allows us to provide combinatorial lower bounds for the number of iterations and the total curvature, in a general setting. (10.1137/20M1380211)
  DOI : 10.1137/20M1380211
• On the curved exponential family in the Stochatic Approximation Expectation Maximization Algorithm
  
  • Debavelaere Vianney
  • Allassonnière Stéphanie
  , 2021. The Expectation-Maximization (EM) Algorithm is a widely used method allowing to estimate the maximum likelihood of models involving latent variables. When the Expectation step cannot be computed easily, one can use stochastic versions of the classical EM such as the Stochastic Approximation EM (SAEM). This algorithm, however, has the disadvantage to require the joint likelihood to belong to the curved exponential family. This hypothesis is a bottleneck in a lot of practical situations where it is not verified. To overcome this problem, Kuhn and Lavielle (2005) introduce a rewriting of the model which ``exponentializes'' it. It consists in considering the parameter as an additional latent variable following a Normal distribution centered on the newly defined parameters and with fixed variance. The likelihood of this new exponentialized model now belongs to the curved exponential family and stochastic EM algorithms apply. Although often used, there is no guarantee that the estimated mean will be close to the maximum likelihood estimate of the initial model. In this paper, we will quantify the error done in this estimation while considering the exponentialized model instead of the initial one. More precisely, we will show that this error tends to 0 as the variance of the new Gaussian distribution goes to 0 while computing an upper bound. By verifying those results on an example, we will see that a compromise must be made in the choice of the variance between the speed of convergence and the tolerated error. Finally, we will propose a new algorithm allowing a better estimation of the parameter in a reasonable computation time.