Publications

2018

• Analytical approximations of non-linear SDEs of McKean-Vlasov type
  
  • Gobet Emmanuel
  • Pagliarani Stefano
  Journal of Mathematical Analysis and Applications, Elsevier, 2018, 466 (1), pp.71-106. We provide analytical approximations for the law of the solutions to a certain class of scalar McKean-Vlasov stochastic differential equations (MKV-SDEs) with random initial datum. " Propagation of chaos " results ([Szn91]) connect this class of SDEs with the macroscopic limiting behavior of a particle, evolving within a mean-field interaction particle system, as the total number of particles tends to infinity. Here we assume the mean-field interaction only acting on the drift of each particle, this giving rise to a MKV-SDE where the drift coefficient depends on the law of the unknown solution. By perturbing the non-linear forward Kolmogorov equation associated to the MKV-SDE, we perform a two-steps approximating procedure that decouples the McKean-Vlasov interaction from the standard dependence on the state-variables. The first step yields an expansion for the marginal distribution at a given time, whereas the second yields an expansion for the transition density. Both the approximating series turn out to be asymptotically convergent in the limit of short times and small noise, the convergence order for the latter expansion being higher than for the former. The resulting approximation formulas are expressed in semi-closed form and can be then regarded as a viable alternative to the numerical simulation of the large-particle system, which can be computationally very expensive. Moreover, these results pave the way for further extensions of this approach to more general dynamics and to high-dimensional settings. (10.1016/j.jmaa.2018.05.059)
  DOI : 10.1016/j.jmaa.2018.05.059
• Statistical and probabilistic modeling of a cloud of particles coupled with a turbulent fluid
  
  • Goudenège Ludovic
  • Larat Adam
  • Llobell Julie
  • Massot Marc
  • Mercier David
  • Thomine Olivier
  • Vié Aymeric
  ESAIM: Proceedings and Surveys, EDP Sciences, 2018, 65, pp.401-424. This paper exposes a novel exploratory formalism, which end goal is the numerical simulation of the dynamics of a cloud of particles weakly or strongly coupled with a turbulent fluid. Given the large panel of expertise of the list of authors, the content of this paper scans a wide range of connex notions, from the physics of turbulence to the rigorous definition of stochastic processes. Our approach is to develop reduced-order models for the dynamics of both carrying and carried phases which remain consistant within this formalism, and to set up a numerical process to validate these models. The novelties of this paper lie in the gathering of a large panel of mathematical and physical definitions and results within a common framework and an agreed vocabulary (sections 1 and 2), and in some preliminary results and achievements within this context, section 3. While the first three sections have been simplified to the context of a gas field providing that the disperse phase only retrieves energy through drag, the fourth section opens this study to the more complex situation when the disperse phase interacts with the continuous phase as well, in an energy conservative manner. This will allow us to expose the perspectives of the project and to conclude. (10.1051/proc/201965401)
  DOI : 10.1051/proc/201965401
• Linear regression and learning : contributions to regularization and aggregation methods
  
  • Deswarte Raphaël
  , 2018. This thesis tackles the topic of linear regression, within several frameworks, mainly linked to statistical learning. The first and second chapters present the context, the results and the mathematical tools of the manuscript. In the third chapter, we provide a way of building an optimal regularization function, improving for instance, in a theoretical way, the LASSO estimator. The fourth chapter presents, in the field of online convex optimization, speed-ups for a recent and promising algorithm, MetaGrad, and shows how to transfer its guarantees from a so-called “online deterministic setting" to a “stochastic batch setting". In the fifth chapter, we introduce a new method to forecast successive intervals by aggregating predictors, without intermediate feedback nor stochastic modeling. The sixth chapter applies several aggregation methods to an oil production dataset, forecasting short-term precise values and long-term intervals.
• A Class of Finite-Dimensional Numerically Solvable McKean-Vlasov Control Problems
  
  • Balata Alessandro
  • Huré Côme
  • Laurière Mathieu
  • Pham Huyên
  • Pimentel Isaque
  , 2018. We address a class of McKean-Vlasov (MKV) control problems with common noise, called polynomial conditional MKV, and extending the known class of linear quadratic stochastic MKV control problems. We show how this polynomial class can be reduced by suitable Markov embedding to finite-dimensional stochastic control problems, and provide a discussion and comparison of three probabilistic numerical methods for solving the reduced control problem: quantization, regression by control randomization, and regress later methods. Our numerical results are illustrated on various examples from portfolio selection and liquidation under drift uncertainty, and a model of interbank systemic risk with partial observation.
• Random Uniform Permutations
  
  • Gerin Lucas
  , 2018. This course is at the interplay between Probability and Combinatorics. It is intended for Master students with a background in Probability (random variables, expectation, conditional probability). The question we will adress is ”What can we say about a typical large permutation?”: the number of cycles, their lengths, the number of fixed points,... This is also a pretext to present some universal phenomena in Probability: reinforcement, the Poisson paradigm, size-bias,...
• Oscillating behaviour of the spectrum for a plasmonic problem in a domain with a rounded corner
  
  • Chesnel Lucas
  • Claeys Xavier
  • Nazarov Sergei A.
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2018. We investigate the eigenvalue problem −div(σ∇u) = λu (P) in a 2D domain Ω divided into two regions Ω±. We are interested in situations where σ takes positive values on Ω+ and negative ones on Ω−. Such problems appear in time harmonic electromagnetics in the modeling of plasmonic technologies. In a recent work [15], we highlighted an unusual instability phenomenon for the source term problem associated with (P): for certain configurations, when the interface between the subdo-mains Ω± presents a rounded corner, the solution may depend critically on the value of the rounding parameter. In the present article, we explain this property studying the eigenvalue problem (P). We provide an asymptotic expansion of the eigenvalues and prove error estimates. We establish an oscillatory behaviour of the eigenvalues as the rounding parameter of the corner tends to zero. We end the paper illustrating this phenomenon with numerical experiments. (10.1051/m2an/2016080)
  DOI : 10.1051/m2an/2016080
• Central limit theorem for discretization errors based on stopping time sampling
  
  • Gobet Emmanuel
  • Landon Nicolas
  • Stazhynski Uladzislau
  , 2018. We study the convergence in distribution of the renormalized error arising from the discretization of a Brownian semimartingale sampled at stopping times. Our mild assumptions on the form of stopping times allow the time grid to be a combination of hitting times of stochastic domains and of Poisson-like random times. Remarkably, a Functional Central Limit Theorem holds under great generality on the semimartingale and on the form of stopping times. Furthermore, the asymptotic characteristics are quite explicit. Along the derivation of such results, we also establish some key estimates related to approximations and sensitivities of hitting time/position with respect to model and domain perturbations.
• Anisotropic osmosis filtering for shadow removal in images
  
  • Parisotto Simone
  • Calatroni Luca
  • Caliari Marco
  • Schönlieb Carola-Bibiane
  • Weickert Joachim
  , 2018. We present an anisotropic extension of the isotropic osmosis model that has been introduced by Weickert et al. [38] for visual computing applications, and we adapt it specifically to shadow removal applications. We show that in the integrable setting, linear anisotropic osmosis minimises an energy that involves a suitable quadratic form which models local directional structures. In our shadow removal applications we estimate the local structure via a modified tensor voting approach [24] and use this information within an anisotropic diffusion inpainting that resembles edge-enhancing anisotropic diffusion inpainting [39, 13]. Our numerical scheme combines the nonnegativity preserving stencil of Fehrenbach and Mirebeau [10] with an exact time stepping based on highly accurate polynomial approximations of the matrix exponential. The resulting anisotropic model is tested on several synthetic and natural images corrupted by constant shadows. We show that it outperforms isotropic osmosis, since it does not suffer from blurring artefacts at the shadow boundaries.
• Multi-phase flow with two velocities modelling for jet atomization simulations
  
  • Cordesse Pierre
  • Massot Marc
  • Murrone Angelo
  , 2018. Jet atomization is at the core of many industrial applications such as in cryogenic combustion chambers. Since Direct Numerical Simulations (DNS) of these two-phase flows in real engines are still out of reach, reduced-order models must be built to develop predictive numerical tools. However great care must be taken on the choice of these models in order to reach sound mathematical properties and predictive simulations after a validation process. The contribution of this talk is three-fold. First, we present an Euler-Euler modelling strategy. It uses a novel hierarchy of diffuse interface models, with a proper description of various disequilibrium levels of the mixture inspired from [1,2] and a special attention devoted to the choice of these models to respect the entropy inequality. These diffuse interface models are then coupled to an element of the Kinetic-Based Moment Method (KBMM) for the dispersed flow [3]. Secondly, to cope with the strong discontinuities encountered in jet atomization, a robust and accurate numerical method using multi-slope MUSCL technique is applied [4]. Finally, relying on the previous two points, simulations of a jet atomization in a cryogenic combustion chamber in subcritical conditions have been conducted and results show that physical properties are recovered.
• Singularités en géométrie sous-riemannienne
  
  • Sacchelli Ludovic
  , 2018. Nous étudions les relations qui existent entre des aspects de la géométrie sous-riemannienne et une diversité de singularités typiques dans ce contexte.Avec les théorèmes de Whitney sous-riemanniens, nous conditionnons l’existence de prolongements globaux de courbes horizontales définies sur des fermés à des hypothèses de non-singularité de l’application point-final dans l’approximation nilpotente de la variété.Nous appliquons des méthodes perturbatives pour obtenir des asymptotiques sur la longueur de courbes localement minimisantes perdant leur optimalité proche de leur point de départ dans le cas des variétés sous-riemanniennes de contact de dimension arbitraire. Nous décrivons la géométrie du lieu singulier et prouvons sa stabilité dans le cas des variétés de dimension 5.Nous introduisons une construction permettant de définir des champs de directions à l’aide de couples de champs de vecteurs. Ceci fournit une topologie naturelle pour analyser la stabilité des singularités de champs de directions sur des surfaces.
• A Comparative Study of Large-scale Variants of CMA-ES
  
  • Varelas Konstantinos
  • Auger Anne
  • Brockhoff Dimo
  • Hansen Nikolaus
  • Elhara Ouassim Ait
  • Semet Yann
  • Kassab Rami
  • Barbaresco Frédéric
  , 2018, 11101, pp.3-15. The CMA-ES is one of the most powerful stochastic numerical optimizers to address difficult black-box problems. Its intrinsic time and space complexity is quadratic-limiting its applicability with increasing problem dimensionality. To circumvent this limitation, different large-scale variants of CMA-ES with subquadratic complexity have been proposed over the past ten years. To-date however, these variants have been tested and compared only in rather restrictive settings, due to the lack of a comprehensive large-scale testbed to assess their performance. In this context, we introduce a new large-scale testbed with dimension up to 640, implemented within the COCO benchmarking platform. We use this testbed to assess the performance of several promising variants of CMA-ES and the standard limited-memory L-BFGS. In all tested dimensions, the best CMA-ES variant solves more problems than L-BFGS for larger budgets while L-BFGS outperforms the best CMA-ES variant for smaller budgets. However, over all functions, the cumulative runtime distributions between L-BFGS and the best CMA-ES variants are close (less than a factor of 4 in high dimension). Our results illustrate different scaling behaviors of the methods, expose a few defects of the algorithms and reveal that for dimension larger than 80, LM-CMA solves more problems than VkD-CMA while in the cumulative runtime distribution over all functions the VkD-CMA dominates or shows almost equal success rate with LM-CMA for budgets up to 10 4 times dimension and for all budgets up to dimension 80. (10.1007/978-3-319-99253-2_1)
  DOI : 10.1007/978-3-319-99253-2_1
• Rejuvenating Functional Responses with Renewal Theory
  
  • Billiard Sylvain
  • Bansaye Vincent
  • Chazottes J.-R
  Journal of the Royal Society Interface, the Royal Society, 2018, 15 (146). (10.1098/rsif.2018.0239)
  DOI : 10.1098/rsif.2018.0239
• An Augmented Reality Audio Device Helping Blind People Navigation
  
  • Ferrand Sylvain
  • Alouges François
  • Aussal Matthieu
  , 2018, pp.28-35. (10.1007/978-3-319-94274-2_5)
  DOI : 10.1007/978-3-319-94274-2_5
• Uncertainty Quantification of Stochastic Approximation Limits
  
  • Crépey Stéphane
  • Fort Gersende
  • Gobet Emmanuel
  • Stazhynski Uladzislau
  , 2018.
• Efficient algorithm for optimizing spectral partitions
  
  • Bogosel Beniamin
  Applied Mathematics and Computation, Elsevier, 2018, 333, pp.61-75. We present an amelioration of current known algorithms for minimizing functions depending on the eigenvalues corresponding to a partition of a given domain. The idea is to use the advantage of a representation using density functions on a fixed grid while decreasing the computational time. This is done by restricting the computation to neighbourhoods of regions where the associated densities are above a certain threshold. The algorithm extends and improves known methods in the plane and on surfaces in dimension 3. It also makes possible to make computations of optimal volumic 3D spectral partitions on sufficiently important discretizations. (10.1016/j.amc.2018.03.087)
  DOI : 10.1016/j.amc.2018.03.087
• Layer potential approach for fast eigenvalue characterization of the Helmholtz equation with mixed boundary conditions
  
  • Dupré Matthieu
  • Fink Mathias
  • Garnier Josselin
  • Lerosey Geoffroy
  Computational & Applied Mathematics, Springer Verlag, 2018, 37 (4), pp.4675-4685. (10.1007/s40314-018-0591-9)
  DOI : 10.1007/s40314-018-0591-9
• Imaging through a scattering medium by speckle intensity correlations
  
  • Garnier Josselin
  • Solna Knut
  Inverse Problems, IOP Publishing, 2018, 34 (9), pp.094003. (10.1088/1361-6420/aacfb0)
  DOI : 10.1088/1361-6420/aacfb0
• A Bayesian nonparametric approach for generalized Bradley-Terry models in random environment
  
  • Le Corff Sylvain
  • Lerasle Matthieu
  • Vernet Elodie
  , 2018. This paper deals with the estimation of the unknown distribution of hidden random variables from the observation of pairwise comparisons between these variables. This problem is inspired by recent developments on Bradley-Terry models in random environment since this framework happens to be relevant to predict for instance the issue of a championship from the observation of a few contests per team. This paper provides three contributions on a Bayesian nonparametric approach to solve this problem. First, we establish contraction rates of the posterior distribution. We also propose a Markov Chain Monte Carlo algorithm to approximately sample from this posterior distribution inspired from a recent Bayesian nonparametric method for hidden Markov models. Finally, the performance of this algorithm are appreciated by comparing predictions on the issue of a championship based on the actual values of the teams and those obtained by sampling from the estimated posterior distribution.
• Invisibility and perfect reflectivity in waveguides with finite length branches
  
  • Chesnel Lucas
  • Nazarov Sergei A
  • Pagneux Vincent
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2018. We consider a time-harmonic wave problem, appearing for example in water-waves and in acoustics, in a setting such that the analysis reduces to the study of a 2D waveguide problem with a Neumann boundary condition. The geometry is symmetric with respect to an axis orthogonal to the direction of propagation of waves. Moreover, the waveguide contains one branch of finite length. We analyse the behaviour of the complex scattering coefficients R, T as the length of the branch increases and we exhibit situations where non reflectivity (R = 0, |T| = 1), perfect reflectivity (|R| = 1, T = 0) or perfect invisibility (R = 0, T = 1) hold. Numerical experiments allow us to illustrate the different results.
• Tropical compound matrix identities
  
  • Akian Marianne
  • Gaubert Stéphane
  • Niv Adi
  Linear Algebra and its Applications, Elsevier, 2018, 551, pp.162-206. We prove identities on compound matrices in extended tropical semirings. Such identities include analogues to properties of conjugate matrices, powers of matrices and $\operatorname{adj}(A)\det(A)^{ -1}$, all of which have implications on the eigenvalues of the corresponding matrices. A tropical Sylvester-Franke identity is provided as well. (10.1016/j.laa.2018.04.006)
  DOI : 10.1016/j.laa.2018.04.006
• Dynamically Orthogonal Numerical Schemes for Efficient Stochastic Advection and Lagrangian Transport
  
  • Feppon Florian
  • Lermusiaux Pierre F J
  SIAM Review, Society for Industrial and Applied Mathematics, 2018, 60 (3), pp.595 - 625. Quantifying the uncertainty of Lagrangian motion can be performed by solving a large number of ordinary differential equations with random velocities, or equivalently a stochastic transport partial differential equation (PDE) for the ensemble of flow-maps. The Dynamically Orthogonal (DO) decomposition is applied as an efficient dynamical model order reduction to solve for such stochastic advection and Lagrangian transport. Its interpretation as the method that applies instantaneously the truncated SVD on the matrix discretization of the original stochastic PDE is used to obtain new numerical schemes. Fully linear, explicit central advection schemes stabilized with numerical filters are selected to ensure efficiency, accuracy, stability, and direct consistency between the original deterministic and stochastic DO advections and flow-maps. Various strategies are presented for selecting a time-stepping that accounts for the curvature of the fixed rank manifold and the error related to closely singular coefficient matrices. Efficient schemes are developed to dynamically evolve the rank of the reduced solution and to ensure the orthogonality of the basis matrix while preserving its smooth evolution over time. Finally, the new schemes are applied to quantify the uncertain Lagrangian motions of a 2D double gyre flow with random frequency and of a stochastic flow past a cylinder. (10.1137/16m1109394)
  DOI : 10.1137/16m1109394
• MCMC and nested extreme risks
  
  • Fort Gersende
  • Gobet Emmanuel
  • Moulines Éric
  , 2018.
• From a monotone probabilistic scheme to a probabilistic max-plus algorithm for solving Hamilton-Jacobi-Bellman equations
  
  • Akian Marianne
  • Fodjo Eric
  , 2018, 21. In a previous work, we introduced a lower complexity probabilistic max-plus numerical method for solving fully nonlinear Hamilton-Jacobi-Bellman equations associated to diffusion control problems involving a finite set-valued (or switching) control and possibly a continuum-valued control. This method was based on the idempotent expansion properties obtained by McEneaney, Kaise and Han (2011) and on the numerical probabilistic method proposed by Fahim, Touzi and Warin (2011) for solving some fully nonlinear parabolic partial differential equations. A difficulty of the latter algorithm is in the critical constraints imposed on the Hamiltonian to ensure the monotonicity of the scheme, hence the convergence of the algorithm. Here, we propose a new "probabilistic scheme" which is monotone under rather weak assumptions, including the case of strongly elliptic PDE with bounded derivatives. This allows us to apply our probabilistic max-plus method in more general situations. We illustrate this on the evaluation of the superhedging price of an option under uncertain correlation model with several underlying stocks, and consider in particular the case of 5 stocks leading to a PDE in dimension 5.
• Optimal Battery Aging : an Adaptive Weights Dynamic Programming Algorithm
  
  • Heymann Benjamin
  • Martinon Pierre
  Journal of Optimization Theory and Applications, Springer Verlag, 2018. We present an algorithm to handle the optimization over a long horizon of an electric microgrid including a battery energy storage system. While the battery is an important and costly component of the microgrid, its aging process is often not taken into account by the Energy Management System, mostly because of modeling and computing challenges. We address the computing aspect by a new approach combining dynamic programming, decomposition and relaxation techniques. We illustrate this ’adaptive weight’ method with numerical simulations for a toy microgrid model. Compared to a straightforward resolution by dynamic programming, our algorithm decreases the computing time by more than one order of magnitude, can be parallelized, and allows for online implementations. We believe that this approach can be used for other applications presenting fast and slow variables. (10.1007/s10957-018-1371-9)
  DOI : 10.1007/s10957-018-1371-9
• Optimal discretization of stochastic integrals driven by general Brownian semimartingale
  
  • Gobet Emmanuel
  • Stazhynski Uladzislau
  Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, Institut Henri Poincaré (IHP), 2018, 54 (3). We study the optimal discretization error of stochastic integrals, driven by a multidimensional continuous Brownian semimartingale. In this setting we establish a pathwise lower bound for the renormalized quadratic variation of the error and we provide a sequence of discretiza- tion stopping times, which is asymptotically optimal. The latter is defined as hitting times of random ellipsoids by the semimartingale at hand. In comparison with previous available results, we allow a quite large class of semimartingales (relaxing in particular the non degeneracy conditions usually requested) and we prove that the asymptotic lower bound is attainable. (10.1214/17-AIHP848)
  DOI : 10.1214/17-AIHP848