Publications

2020

• Forward Event-Chain Monte Carlo: Fast sampling by randomness control in irreversible Markov chains
  
  • Michel Manon
  • Durmus Alain
  • Sénécal Stéphane
  Journal of Computational and Graphical Statistics, Taylor & Francis, 2020. Irreversible and rejection-free Monte Carlo methods, recently developed in Physics under the name Event-Chain and known in Statistics as Piecewise Deterministic Monte Carlo (PDMC), have proven to produce clear acceleration over standard Monte Carlo methods, thanks to the reduction of their random-walk behavior. However, while applying such schemes to standard statistical models, one generally needs to introduce an additional randomization for sake of correctness. We propose here a new class of Event-Chain Monte Carlo methods that reduces this extra-randomization to a bare minimum. We compare the efficiency of this new methodology to standard PDMC and Monte Carlo methods. Accelerations up to several magnitudes and reduced dimensional scalings are exhibited. (10.1080/10618600.2020.1750417)
  DOI : 10.1080/10618600.2020.1750417
• A Convex Variational Model for Learning Convolutional Image Atoms from Incomplete Data
  
  • Chambolle Antonin
  • Holler M.
  • Pock T.
  Journal of Mathematical Imaging and Vision, Springer Verlag, 2020, 62 (3), pp.417-444. (10.1007/s10851-019-00919-7)
  DOI : 10.1007/s10851-019-00919-7
• Inexact First-Order Primal-Dual Algorithms
  
  • Rasch Julian
  • Chambolle Antonin
  Computational Optimization and Applications, Springer Verlag, 2020, 76, pp.381--430. We investigate the convergence of a recently popular class of first-order primal-dual algorithms for saddle point problems under the presence of errors in the proximal maps and gradients. We study several types of errors and show that, provided a sufficient decay of these errors, the same convergence rates as for the error-free algorithm can be established. More precisely, we prove the (optimal) O(1/N) convergence to a saddle point in finite dimensions for the class of non-smooth problems considered in this paper, and prove a O(1/N^2) or even linear O(θ^N) convergence rate if either the primal or dual objective respectively both are strongly convex. Moreover we show that also under a slower decay of errors we can establish rates, however slower and directly depending on the decay of the errors. We demonstrate the performance and practical use of the algorithms on the example of nested algorithms and show how they can be used to split the global objective more efficiently. (10.1007/s10589-020-00186-y)
  DOI : 10.1007/s10589-020-00186-y
• No‐arbitrage implies power‐law market impact and rough volatility
  
  • Jusselin Paul
  • Rosenbaum Mathieu
  Mathematical Finance, Wiley, 2020, 30 (4), pp.1309-1336. Abstract Market impact is the link between the volume of a (large) order and the price move during and after the execution of this order. We show that in a quite general framework, under no‐arbitrage assumption, the market impact function can only be of power‐law type. Furthermore, we prove this implies that the macroscopic price is diffusive with rough volatility, with a one‐to‐one correspondence between the exponent of the impact function and the Hurst parameter of the volatility. Hence, we simply explain the universal rough behavior of the volatility as a consequence of the no‐arbitrage property. From a mathematical viewpoint, our study relies, in particular, on new results about hyper‐rough stochastic Volterra equations. (10.1111/mafi.12254)
  DOI : 10.1111/mafi.12254
• Consistent transport properties in multicomponent two-temperature magnetized plasmas: Application to the Sun chromosphere
  
  • Wargnier Quentin
  • Alvarez-Laguna Alejandro
  • Scoggins James B
  • Mansour Nagi N
  • Massot Marc
  • Magin Thierry E.
  Astronomy & Astrophysics - A&A, EDP Sciences, 2020, 635, pp.A87. A fluid model is developed for multicomponent two-temperature magnetized plasmas in chemical non-equilibrium from the partially- to fully-ionized collisional regimes. We focus on transport phenomena aiming at representing the chromosphere of the Sun. Graille et al. [M3AS 19(04):527-599, 2009] have derived an asymptotic fluid model for multicomponent plamas from kinetic theory, yielding a rigorous description of the dissipative effects. The governing equations and consistent transport properties are obtained using a multiscale Chapman-Enskog perturbative solution to the Boltzmann equation based on a non-dimensional analysis. The mass disparity between the electrons and heavy particles is accounted for, as well as the influence of the electromagnetic field. We couple this model to the Maxwell equations for the electromagnetic field and derive the generalized Ohm's law for multicomponent plasmas. The model inherits a well-identified mathematical structure leading to an extended range of validity for the Sun chromosphere conditions. We compute consistent transport properties by means of a spectral Galerkin method using the Laguerre-Sonine polynomial approximation. Two non-vanishing polynomial terms are used when deriving the transport systems for electrons, whereas only one term is retained for heavy particles. In a simplified framework where the plasma is fully ionized, we compare the transport properties for the Sun chromosphere to conventional expressions for magnetized plasmas due to Braginskii, showing a good agreement between both results. For more general partially ionized conditions, representative of the Sun chromosphere, we compute the muticomponent transport properties corresponding to the species diffusion velocities, heavy-particle and electron heat fluxes, and viscous stress tensor of the model, for a Helium-Hydrogen mixture in local thermodynamic equilibrium. The model is assessed for the 3D radiative magnetohydrodynamic simulation of a pore, in the highly turbulent upper layer of the solar convective zone. The resistive term is found to dominate mainly the dynamics of the electric field at the pore location. The battery term for heavy particles appears to be higher at the pore location and at some intergranulation boundaries. (10.1051/0004-6361/201834686)
  DOI : 10.1051/0004-6361/201834686
• Space adaptive methods with error control based on adaptive multiresolution for the simulation of low-Mach reactive flows
  
  • N'Guessan Marc-Arthur
  , 2020. We address the development of new numerical methods for the efficient resolution of stiff Partial Differential Equations modelling multi-scale time/space physical phenomena. We are more specifically interested in low Mach reacting flow processes, that cover various real-world applications such as flame dynamics at low gas velocity, buoyant jet flows or plasma/flow interactions. It is well-known that the numerical simulation of these problems is a highly difficult task, due to the large spectrum of spatial and time scales caused by the presence of nonlinear The adaptive spatial discretization is coupled to a new 3rd-order additive Runge-Kutta method for the incompressible Navier-Stokes equations, combining a 3rd-order, A-stable, stiffly accurate, 4-stage ESDIRK method for the algebraic linear part of these equations, and a 4th-order explicit Runge-Kutta scheme for the nonlinear convective part. This numerical strategy is implemented from scratch in the in-house numerical code mrpy. This software is written in Python, and relies on the PETSc library, written in C, for linear algebra operations. We assess the capabilities of this mechanisms taking place into dynamic fronts. In this general context, this work introduces dedicated numerical tools for the resolution of the incompressible Navier-Stokes equations, an important first step when designing an hydrodynamic solver for low Mach flows. We build a space adaptive numerical scheme to solve incompressible flows in a finite-volume context, that relies on multiresolution analysis with error control. To this end, we introduce a new collocated finite-volume method on adaptive rectangular grids, with an original treatment of the spurious pressure and velocity modes that does not alter the precision of the discretization technique. new hydrodynamic solver in terms of speed and efficiency, in the context of scalar transport on adaptive grids. Hence, this study presents a new high-order hydrodynamics solver for incompressible flows, with grid adaptation by multiresolution, that can be extended to the more general low-Mach flow configuration.
• On quasi-reversibility solutions to the Cauchy problem for the Laplace equation: regularity and error estimates
  
  • Bourgeois Laurent
  • Chesnel Lucas
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2020. We are interested in the classical ill-posed Cauchy problem for the Laplace equation. One method to approximate the solution associated with compatible data consists in considering a family of regularized well-posed problems depending on a small parameter ε > 0. In this context, in order to prove convergence of finite elements methods, it is necessary to get regularity results of the solutions to these regularized problems which hold uniformly in ε. In the present work, we obtain these results in smooth domains and in 2D polygonal geometries. In presence of corners, due to the particular structure of the regularized problems, classical techniques à la Grisvard do not work and instead, we apply the Kondratiev approach. We describe the procedure in detail to keep track of the dependence in ε in all the estimates. The main originality of this study lies in the fact that the limit problem is ill-posed in any framework. (10.1051/m2an/2019073)
  DOI : 10.1051/m2an/2019073
• Stability of piecewise deterministic Markovian metapopulation processes on networks
  
  • Montagnon Pierre
  Stochastic Processes and their Applications, Elsevier, 2020, 130 (3), pp.1515-1544. The purpose of this paper is to study a Markovian metapopulation model on a directed graph with edge-supported transfers and deterministic intra-nodal population dynamics. We first state tractable stability conditions for two typical frameworks motivated by applications: constant jump rates with multiplicative transfer amplitudes, and coercive jump rates with unitary transfers. More general criteria for boundedness, petiteness and ergodicity are then given. ( (10.1016/j.spa.2019.05.012)
  DOI : 10.1016/j.spa.2019.05.012
• Regulation of Renewable Resource Exploitation
  
  • Kharroubi Idris
  • Lim Thomas
  • Mastrolia Thibaut
  SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2020, 58 (1), pp.551-579. (10.1137/19M1265740)
  DOI : 10.1137/19M1265740
• Scaling Limit of Sub-ballistic 1D Random Walk among Biased Conductances: a Story of Wells and Walls
  
  • Berger Quentin
  • Salvi Michele
  Electronic Journal of Probability, Institute of Mathematical Statistics (IMS), 2020, 25, pp.p. 1-43. We consider a one-dimensional random walk among biased i.i.d. conductances, in the case where the random walk is transient but sub-ballistic: this occurs when the conductances have a heavy-tail at $+\infty$ or at $0$. We prove that the scaling limit of the process is the inverse of an $\alpha$-stable subordinator, which indicates an aging phenomenon, expressed in terms of the generalized arcsine law. In analogy with the case of an i.i.d. random environment studied in details in [Enriquez, Sabot, Zindy, Bull. Soc. Math. 2009; Enriquez, Sabot, Tournier, Zindy, Ann. Appl. Probab. 2013], some `traps' are responsible for the slowdown of the random walk. However, the phenomenology is somehow different (and richer) here. In particular, three types of traps may occur, depending on the fine properties of the tails of the conductances: (i) a very large conductance (a well in the potential); (ii) a very small conductance (a wall in the potential); (iii) the combination of a large conductance followed shortly after by a small conductance (a well-and-wall in the potential). (10.1214/20-EJP427)
  DOI : 10.1214/20-EJP427
• Optimization of a Sequential Decision Making Problem for a Rare Disease Diagnostic Application
  
  • Besson Rémi
  • Le Pennec Erwan
  • Spaggiari Emmanuel
  • Neuraz Antoine
  • Stirnemann Julien
  • Allassonnière Stéphanie
  , 2020, pp.475-482. In this work, we propose a new optimization formulation for a sequential decision making problem for a rare disease diagnostic application. We aim to minimize the number of medical tests necessary to achieve a state where the uncertainty regarding the patient’s disease is less than a predetermined threshold. In doing so, we take into account the need in many medical applications, to avoid as much as possible, any misdiagnosis. To solve this optimization task, we investigate several reinforcement learning algorithms and make them operable in our high-dimensional setting: the strategies learned are much more efficient than classical greedy strategies. (10.5220/0008938804750482)
  DOI : 10.5220/0008938804750482
• A quantitative Mc Diarmid's inequality for geometrically ergodic Markov chains
  
  • Havet Antoine
  • Lerasle Matthieu
  • Moulines Éric
  • Vernet Elodie
  Electronic Communications in Probability, Institute of Mathematical Statistics (IMS), 2020. We state and prove a quantitative version of the bounded difference inequality for geometrically ergodic Markov chains. Our proof uses the same martingale decomposition as \cite{MR3407208} but, compared to this paper, the exact coupling argument is modified to fill a gap between the strongly aperiodic case and the general aperiodic case.
• STRATEGIC ADVANTAGES IN MEAN FIELD GAMES WITH A MAJOR PLAYER
  
  • Bertucci Charles
  • Lasry Jean-Michel
  • Lions Pierre Louis
  , 2020. This note is concerned with a modeling question arising from the mean field games theory. We show how to model mean field games involving a major player which has a strategic advantage, while only allowing closed loop markovian strategies for all the players. We illustrate this property through three examples.
• Plasma-sheath transition in multi-fluid models with inertial terms under low pressure conditions: Comparison with the classical and kinetic theory
  
  • Alvarez-Laguna Alejandro
  • Magin Thierry E.
  • Massot Marc
  • Bourdon Anne
  • Chabert Pascal
  Plasma Sources Science and Technology, IOP Publishing, 2020. (10.1088/1361-6595/ab6242)
  DOI : 10.1088/1361-6595/ab6242
• Crouzeix-Raviart approximation of the total variation on simplicial meshes
  
  • Chambolle Antonin
  • Pock Thomas
  Journal of Mathematical Imaging and Vision, Springer Verlag, 2020, 62 (6-7), pp.872--899. We propose an adaptive implementation of a Crouzeix-Raviart based discretization of the total variation, which has the property of approximating from below the total variation, with metrication errors only depending on the local curvature, rather than on the orientation as is usual for other approaches. (10.1007/s10851-019-00939-3)
  DOI : 10.1007/s10851-019-00939-3
• Parking 3-sphere swimmer: II. The long-arm asymptotic regime
  
  • Alouges François
  • Di Fratta Giovanni
  European Physical Journal E: Soft matter and biological physics, EDP Sciences: EPJ / Springer Nature, 2020, 43 (2). Abstract. The paper carries on our previous investigations on the complementary version of Purcell’s rotator ( sPr 3 ): a low-Reynolds-number swimmer composed of three balls of equal radii. In the asymptotic regime of very long arms, the Stokes-induced governing dynamics is derived, and then experimented in the context of energy-minimizing self-propulsion characterized in the first part of the paper. Graphical abstract (10.1140/epje/i2020-11932-5)
  DOI : 10.1140/epje/i2020-11932-5
• Factorization Method for Imaging a Local Perturbation in Inhomogeneous Periodic Layers from Far Field Measurements
  
  • Haddar Houssem
  • Konschin Alexander
  Inverse Problems and Imaging, AIMS American Institute of Mathematical Sciences, 2020, 14 (1), pp.133-152. We analyze the Factorization method to reconstruct the geometry of a local defect in a periodic absorbing layer using almost only incident plane waves at a fixed frequency. A crucial part of our analysis relies on the consideration of the range of a carefully designed far field operator, which characterizes the geometry of the defect. We further provide some validating numerical results in a two dimensional setting. (10.3934/ipi.2019067)
  DOI : 10.3934/ipi.2019067
• Tropical spectrahedra
  
  • Allamigeon Xavier
  • Gaubert Stéphane
  • Skomra Mateusz
  Discrete and Computational Geometry, Springer Verlag, 2020, 63, pp.507–548. We introduce tropical spectrahedra, defined as the images by the nonarchimedean valuation of spectrahedra over the field of real Puiseux series. We provide an explicit characterization of generic tropical spectrahedra, involving principal tropical minors of size at most 2. To do so, we show that the nonarchimedean valuation maps semialgebraic sets to semilinear sets that are closed. We also prove that, under a regularity assumption, the image by the valuation of a basic semialgebraic set is obtained by tropicalizing the inequalities which define it. (10.1007/s00454-020-00176-1)
  DOI : 10.1007/s00454-020-00176-1
• Motion of a solid particle in a bounded viscous flow using the Sparse Cardinal Sine Decomposition
  
  • Alouges François
  • Lefebvre-Lepot Aline
  • Sellier Alain
  Meccanica, Springer Verlag, 2020, 55, pp.403-419. This work investigates the Sparse Cardinal Sine Decomposition (SCSD) method ability to efficiently deal with a Stokes flow about a solid particle immersed in a liquid. In contrat to Alouges and Aussal (Numer Algorithms 70:1–22, 2015), the liquid domain is bounded by a solid and motionless wall. The advocated procedure inverts on the particle and truncated wall boundaries the boundary-integral equation governing the stress there. This is numerically achieved by implementing a Galerkin method. The resulting linear system, with fully-populated and non-symmetric influence matrix, is both compressed and solved by the new SCSD method which allows to accurately deal with a large number of unknowns. Both analytical and numerical comparisons are reported for a spherical particle and several bounded liquid domains. Moreover, the rigid-body motion of spheroidal particles settling in a cylindrical tube is examined. (10.1007/s11012-019-00993-6)
  DOI : 10.1007/s11012-019-00993-6
• Phase Field Approach to Optimal Packing Problems and Related Cheeger Clusters
  
  • Bogosel Beniamin
  • Bucur Dorin
  • Fragalà Ilaria
  Applied Mathematics and Optimization, Springer Verlag (Germany), 2020, 81 (1), pp.63-87. This paper stems from the idea of adopting a new appraoch to solve some classical optimal packing problems for balls. In fact, we attack this kind of problems (which are of discrete nature) by means of shape optimization techniques, applied to suitable Γ-converging sequences of energies associated to Cheeger type problems. More precisely, in a first step we prove that different optimal packing problems are limits of sequences of optimal clusters associated to the minimization of energies involving suitable (generalized) Cheeger constants. In a second step, we propose an efficient phase field approach based on a multiphase Γ-convergence result of Modica-Mortola type, in order to compute those generalized Cheeger constants, their optimal clusters and, as a consequence of the asymptotic result, optimal packings. Numerical experiments are carried over in two and three space dimensions. Our continuous shape optimization approach to solve discrete packing problems circumvents the NP-hard character of these ones, and efficiently leads to configurations close to the global minima. (10.1007/s00245-018-9476-y)
  DOI : 10.1007/s00245-018-9476-y
• Recursive computation of invariant distributions of Feller processes
  
  • Pagès Gilles
  • Rey Clément
  Stochastic Processes and their Applications, Elsevier, 2020, 130, pp.328 - 365. (10.1016/j.spa.2019.03.008)
  DOI : 10.1016/j.spa.2019.03.008
• A variational formulation for computing shape derivatives of geometric constraints along rays
  
  • Feppon Florian
  • Allaire Grégoire
  • Dapogny Charles
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2020, 54 (1), pp.181-228. In the formulation of shape optimization problems, multiple geometric constraint functionals involve the signed distance function to the optimized shape Ω. The numerical evaluation of their shape derivatives requires to integrate some quantities along the normal rays to Ω, a challenging operation to implement, which is usually achieved thanks to the method of characteristics. The goal of the present paper is to propose an alternative, variational approach for this purpose. Our method amounts, in full generality, to compute integral quantities along the characteristic curves of a given velocity field without requiring the explicit knowledge of these curves on the spatial discretization; it rather relies on a variational problem which can be solved conveniently by the finite element method. The well-posedness of this problem is established thanks to a detailed analysis of weighted graph spaces of the advection operator β·∇ associated to a C1 velocity field β. One novelty of our approach is the ability to handle velocity fields with possibly unbounded divergence: we do not assume div(β)∈L∞. Our working assumptions are fulfilled in the context of shape optimization of C2 domains Ω, where the velocity field β=∇dΩ is an extension of the unit outward normal vector to the optimized shape. The efficiency of our variational method with respect to the direct integration of numerical quantities along rays is evaluated on several numerical examples. Classical albeit important implementation issues such as the calculation of a shape's curvature and the detection of its skeleton are discussed. Finally, we demonstrate the convenience and potential of our method when it comes to enforcing maximum and minimum thickness constraints in structural shape optimization. (10.1051/m2an/2019056)
  DOI : 10.1051/m2an/2019056
• Learning Multimodal Digital Models of Disease Progression from Longitudinal Data : Methods & Algorithms for the Description, Prediction and Simulation of Alzheimer’s Disease Progression
  
  • Koval Igor
  , 2020. This thesis focuses on the statistical learning of digital models of neurodegenerative disease progression, especially Alzheimer's disease. It aims at reconstructing the complex and heterogeneous dynamic of evolution of the structure, the functions and the cognitive abilities of the brain, at both an average and individual level. To do so, we consider a mixed-effects model that, based on longitudinal data, namely repeated observations per subjects that present multiple modalities, in parallel recombines the individual spatiotemporal trajectories into a group-average scenario of change, and, estimates the variability of this characteristic progression which characterizes the individual trajectories. This variability results from a temporal un-alignment (in term of pace of progression and age at disease onset) along with a spatial variability that takes the form of a modification in the sequence of events that appear during the course of the disease. The different parts of the thesis are ordered in a coherent sequence: from the medical problematic, followed by the statistical model introduced to tackle the aforementioned challenge and its application to the description of the course of Alzheimer's disease, and, finally, numerical tools developed to make the previous model available to the medical community.
• Quantile-based robust optimization of a supersonic nozzle for Organic Rankine Cycle turbines
  
  • Razaaly Nassim
  • Gori Giulio
  • Persico Giacomo
  • Congedo Pietro Marco
  , 2020. Organic Rankine Cycle (ORC) turbines usually operate in thermodynamic regions characterized by high-pressure ratios and strong non-ideal gas effects, complicating the aerodynamic design significantly. Systematic optimization methods accounting for multiple uncertainties due to variable operating conditions, referred to as Robust Optimization may benefit to ORC turbines aerodynamic design. This study presents an original and fast robust shape optimization approach to overcome the limitation of a deterministic optimization that neglects operating conditions variability, applied to a well-known supersonic turbine nozzle for ORC applications. The flow around the blade is assumed inviscid and adiabatic and it is reconstructed using the opensource SU2 code. The non-ideal gasdynamics is modeled through the Peng-Robinson-Stryjek-Vera equation of state. We propose here a mono-objective formulation which consists in minimizing the a-quantile of the targeted Quantity of Interest (QoI) under a probabilistic constraint, at a low computational cost. This problem is solved by using an efficient robust optimization approach, coupling a state-of-the-art quantile estimation and a classical Bayesian optimization method. First, the advantages of a quantile-based formulation are illustrated with respect to a conventional mean-based robust optimization. Secondly, we demonstrate the effectiveness of applying this robust optimization framework with a low-fidelity inviscid solver by comparing the resulting optimal design with the ones obtained with a deterministic optimization using a fully turbulent solver.
• Modulation of homogeneous and isotropic turbulence by sub-Kolmogorov particles: Impact of particle field heterogeneity
  
  • Letournel Roxane
  • Laurent Frédérique
  • Massot Marc
  • Vié Aymeric
  International Journal of Multiphase Flow, Elsevier, 2020, 125, pp.103233. The modulation of turbulence by sub-Kolmogorov particles has been thoroughly characterized in the literature, showing either enhancement or reduction of kinetic energy at small or large scale depending on the Stokes number and the mass loading. However , the impact of a third parameter, the number density of particles, has not been independently investigated. In the present work, we perform direct numerical simulations of decaying Homogeneous Isotropic Turbulence loaded with monodisperse sub-Kolmogorov particles, varying independently the Stokes number, the mass loading and the number density of particles. Like previous investigators, crossover and modulations of the fluid energy spectra are observed consistently with the change in Stokes number and mass loading. Additionally, DNS results show a clear impact of the particle number density, promoting the energy at small scales while reducing the energy at large scales. For high particle number density, the turbulence statistics and spectra become insensitive to the increase of this parameter, presenting a two-way asymptotic behavior. Our investigation identifies the energy transfer mechanisms, and highlights the differences between the influence of a highly concentrated disperse phase (high particle number density, limit behavior) and that of heterogeneous concentration fields (low particle number density). In particular, a measure of this heterogeneity is proposed and discussed which allows to identify specific regimes in the evolution of turbulence statistics and spectra. (10.1016/j.ijmultiphaseflow.2020.103233)
  DOI : 10.1016/j.ijmultiphaseflow.2020.103233