Publications

2024

• Statistical Learning of Value-at-Risk and Expected Shortfall
  
  • Barrera D
  • Crépey S
  • Gobet E
  • Nguyen Hoang-Dung
  • Saadeddine B
  , 2024. We propose a non-asymptotic convergence analysis of a two-step approach to learn a conditional value-at-risk (VaR) and a conditional expected shortfall (ES) using Rademacher bounds, in a non-parametric setup allowing for heavy-tails on the financial loss. Our approach for the VaR is extended to the problem of learning at once multiple VaRs corresponding to different quantile levels. This results in efficient learning schemes based on neural network quantile and least-squares regressions. An a posteriori Monte Carlo procedure is introduced to estimate distances to the ground-truth VaR and ES. This is illustrated by numerical experiments in a Student-$t$ toy model and a financial case study where the objective is to learn a dynamic initial margin.
• An approximation of the squared Wasserstein distance and an application to Hamilton-Jacobi equations
  
  • Bertucci Charles
  • Lions Pierre Louis
  , 2024. We provide a simple $C^{1,1}$ approximation of the squared Wasserstein distance on R^d when one of the two measures is fixed. This approximation converges locally uniformly. More importantly, at points where the differential of the squared Wasserstein distance exists, it attracts the differentials of the approximations at nearby points. Our method relies on the Hilbertian lifting of PL Lions and on the regularization in Hilbert spaces of Lasry and Lions. We then provide an application of this result by using it to establish a comparison principle for an Hamilton-Jacobi equation on the set of probability measures.
• A charged liquid drop model with Willmore energy
  
  • Goldman Michael
  • Novaga Matteo
  • Berardo Ruffini
  , 2024. We consider a variational model of electrified liquid drops, involving competition between surface tension and charge repulsion. Since the natural model happens to be ill-posed, we show that by adding to the perimeter a Willmore-type energy, the problem turns back to be well-posed. We also prove that for small charge the droplets is spherical.
• Multi-scale Finite Element Method for incompressible flows in heterogeneous media : Implementation and Convergence analysis.
  
  • Balazi Atchy Nillama Loïc
  , 2024. This thesis is concerned with the application of a Multi-scale Finite Element Method (MsFEM) to solve incompressible flow in multi-scale media. Indeed, simulating the flow in a multi-scale media with numerous obstacles, such as nuclear reactor cores, is a highly challenging endeavour. In order to accurately capture the finest scales of the flow, it is necessary to use a very fine mesh. However, this often leads to intractable simulations due to the lack of computational resources. To address this limitation, this thesis develops an enriched non-conforming MsFEM to solve viscous incompressible flows in heterogeneous media, based on the classical non-conforming Crouzeix--Raviart finite element method with high-order weighting functions. The MsFEM employs a coarse mesh on which new basis functions are defined. These functions are not the classical polynomial basis functions of finite elements, but rather solve fluid mechanics equations on the elements of the coarse mesh. These functions are themselves numerically approximated on a fine mesh, taking into account all the geometric details, which gives the multi-scale aspect of this method. A theoretical investigation of the proposed MsFEM is conducted at both the continuous and discrete levels. Firstly, the well-posedness of the discrete local problems involved in the MsFEM was demonstrated using new families of finite elements. To achieve this, a novel non-conforming finite element family in three dimensions on tetrahedra was developed. Furthermore, the first error estimate for the approximation of the Stokes problem in periodic perforated media using this MSFEM is derived, demonstrating its convergence. This is based on homogenization theory of the Stokes problem in periodic domains and on usual finite element theory. At the numerical level, the MsFEM to solve the Stokes and the Oseen problems in two and three dimensions is implemented in a massively parallel framework in FreeFEM. Furthermore, a methodology to solve the Navier–Stokes problem is provided.
• CMA-ES
  
  • Hansen Nikolaus
  , 2024.
• LB+IC-CMA-ES: Two Simple Modifications of CMA-ES to Handle Mixed-Integer Problems
  
  • Marty Tristan
  • Hansen Nikolaus
  • Auger Anne
  • Semet Yann
  • Héron Sébastien
  , 2024, 15149, pp.284-299. <div><p>We present LB+IC-CMA-ES, a variant of CMA-ES that handles mixed-integer problems. The algorithm uses two simple mechanisms to handle integer variables: (i) a lower bound (LB) on the variance of integer variables and (ii) integer centering (IC) of variables to their domain middle depending on their value. After presenting the algorithm, we evaluate the different variants ensuing from these modifications on the BBOB mixed-integer testbed and compare the performance with the recently introduced CMA-ES with margin.</p></div> (10.1007/978-3-031-70068-2_18)
  DOI : 10.1007/978-3-031-70068-2_18
• Accelerated Convergence of Error Quantiles using Robust Randomized Quasi Monte Carlo Methods
  
  • Gobet Emmanuel
  • Lerasle Matthieu
  • Métivier David
  , 2024. We aim to calculate an expectation $\mu(F)=\Esp{F(\U)}$ for functions $F:[0,1]^d \mapsto R$ using a family of estimators $(\widehat \mu_B)_B$ with a budget of $B$ evaluation points. The standard Monte Carlo method achieves a root mean squared risk of order $1/\sqrt B$, both for a fixed square integrable function $F$ and for the worst-case risk over the class $\mathcal{F}$ of functions with $\|F\|_{L_2}\leq 1$. Using a sequence of Randomized Quasi Monte Carlo (RQMC) methods, in constrast, we have a faster convergence $\sigma_B \ll 1/\sqrt B$ for the risk $\sigma_B=\sigma_B(F)$ when fixing a function $F$, as a difference with the worst-case risk which is still of order $1/\sqrt B$. We address the convergence of quantiles of the absolute error, namely, for a given confidence level $1-\delta$ this is the minimal $\varepsilon$ such that $\P(|\widehat \mu_B(F)-\mu(F)|&gt; \varepsilon)\leq \delta$ holds. We show that a judicious choice of a robust aggregation method coupled with RQMC methods allows reaching improved convergence rates for $\varepsilon$ depending on $\delta$ and $B$, while fixing a function $F$. This study includes a review on concentration bounds for the empirical mean as well as sub-Gaussian mean estimates and is supported by numerical experiments, ranging from bounded $F$ to heavy-tailed $F(\U)$, the latter being well suited to functions $F$ with a singularity. The different methods we have tested are available in a Julia package.}
• Quantitative effects of the stress response to DNA damage in the cell size control of Escherichia coli
  
  • Canales Ignacio Madrid
  • Broughton James
  • Méléard Sylvie
  • El Karoui Meriem
  , 2024. <div><p>In Escherichia coli the response to DNA damage shows strong cell-to-cell-heterogenity. This results in a random delay in cell division and asymmetrical binary fission of single cells, which can compromise the size homeostasis of the population. To quantify the effect of the heterogeneous response to genotoxic stress (called SOS response in E. coli) on the growth of the bacterial population, we propose a flexible time-continuous parametric model of individual-based population dynamics. We construct a stochastic model based on the "adder" size-control mechanism, extended to incorporate the dynamics of the SOS response and its effect on cell division. The model is fitted to individual lineage data obtained in a 'mother machine' microfluidic device. We show that the heterogeneity of the SOS response can bias the observed division rate. In particular, we show that the adder division rate is decreased by SOS induction and that this perturbative effect is stronger in fast-growing conditions.</p></div>
• Asymptotics for Random Quadratic Transportation Costs
  
  • Huesmann Martin
  • Goldman Michael
  • Trevisan Dario
  , 2024. We establish the validity of asymptotic limits for the general transportation problem between random i.i.d.\ points and their common distribution, with respect to the squared Euclidean distance cost, in any dimension larger than three. Previous results were essentially limited to the two (or one) dimensional case, or to distributions whose absolutely continuous part is uniform. The proof relies upon recent advances in the stability theory of optimal transportation, combined with functional analytic techniques and some ideas from quantitative stochastic homogenization. The key tool we develop is a quantitative upper bound for the usual quadratic optimal transportation problem in terms of its boundary variant, where points can be freely transported along the boundary. The methods we use are applicable to more general random measures, including occupation measure of Brownian paths, and may open the door to further progress on challenging problems at the interface of analysis, probability, and discrete mathematics.
• Robust topology optimization accounting for uncertain micro-structural changes
  
  • Masson Hugo
  • Peigney Michaël
  • Denimal Goy Enora
  , 2024.
• A hypothesis test for the domain of attraction of a random variable
  
  • Olivero Héctor
  • Talay Denis
  ESAIM: Probability and Statistics, EDP Sciences, 2024, 28, pp.292-328. In this work we address the problem of detecting wether a sampled probability distribution has infinite expectation. This issue is notably important when the sample results from complex numerical simulation methods. For example, such a situation occurs when one simulates stochastic particle systems with complex and singular McKean-Vlasov interaction kernels. As stated, the detection problem is ill-posed. We thus propose and analyze an asymptotic hypothesis test for independent copies of a given random variable~$X$ which is supposed to belong to an unknown domain of attraction of a stable law. The null hypothesis $\mathbf{H_0}$ is: `$X$ is in the domain of attraction of the Normal law' and the alternative hypothesis is $\mathbf{H_1}$: `$X$ is in the domain of attraction of a stable law with index smaller than 2'. Our key observation is that~$X$ cannot have a finite second moment when $\mathbf{H_0}$ is rejected (and therefore $\mathbf{H_1}$ is accepted). Surprisingly, we find it useful to derive our test from the statistics of random processes. More precisely, our hypothesis test is based on a statistic which is inspired by statistical methodologies to determine whether a semimartingale has jumps from the observation of one single path at discrete times. We justify our test by proving asymptotic properties of discrete time functionals of Brownian bridges. We also discuss many numerical experiments which allow us to illustrate satisfying properties of the proposed test. (10.1051/ps/2024010)
  DOI : 10.1051/ps/2024010
• Numerical approximation of ergodic BSDEs using non linear Feynman-Kac formulas
  
  • Gobet Emmanuel
  • Richou Adrien
  • Szpruch Lukasz
  , 2024. In this work we study the numerical approximation of a class of ergodic Backward Stochastic Differential Equations. These equations are formulated in an infinite horizon framework and provide a probabilistic representation for elliptic Partial Differential Equations of ergodic type. In order to build our numerical scheme, we put forward a new representation of the PDE solution by using a classical probabilistic representation of the gradient. Then, based on this representation, we propose a fully implementable numerical scheme using a Picard iteration procedure, a grid space discretization and a Monte-Carlo approximation. Up to a limiting technical condition that guarantees the contraction of the Picard procedure, we obtain an upper bound for the numerical error. We also provide some numerical experiments that show the efficiency of this approach for small dimensions.
• Multicomponent thermodynamics with instabilities and diffuse interfaces fluids
  
  • Giovangigli Vincent
  • Le Calvez Yoann
  • Ribert Guillaume
  AIMS Mathematics, AIMS Press, 2024, 9, pp.25979 - 26034. We investigated the mathematical structure of Gibbsian multicomponent thermodynamics with instabilities. We analyzed the construction of such thermodynamics from a pressure law using ideal gases as the low density limit. The fluid mixtures were allowed to have mechanically and chemically unstable states that were excluded in previous work on supercritical fluids, and the Soave-Redlich-Kwong cubic equation of state was specifically considered. We also investigated the mathematical structure of extended thermodynamics in the presence of cohesive forces-capillary effects-for a simplified diffuse interface fluid model. The thermodynamic formalism was validated by comparison with experimental data for mixtures of ethane and nitrogen. Very good agreement with experimental data was obtained for specific heats, multiphase equilibrium, and critical points, and we also analyzed the structure of strained jets of ethane. (10.3934/math.20241270)
  DOI : 10.3934/math.20241270
• An inverse problem: recovering the fragmentation kernel from the short-time behaviour of the fragmentation equation
  
  • Doumic Marie
  • Escobedo Miguel
  • Tournus Magali
  Annales Henri Lebesgue, UFR de Mathématiques - IRMAR, 2024, 7, pp.621-671. Given a phenomenon described by a self-similar fragmentation equation, how to infer the fragmentation kernel from experimental measurements of the solution ? To answer this question at the basis of our work, a formal asymptotic expansion suggested us that using short-time observations and initial data close to a Dirac measure should be a well-adapted strategy. As a necessary preliminary step, we study the direct problem, i.e. we prove existence, uniqueness and stability with respect to the initial data of non negative measure-valued solutions when the initial data is a compactly supported, bounded, non negative measure. A representation of the solution as a power series in the space of Radon measures is also shown. This representation is used to propose a reconstruction formula for the fragmentation kernel, using short-time experimental measurements when the initial data is close to a Dirac measure. We prove error estimates in Total Variation and Bounded Lipshitz norms; this gives a quantitative meaning to what a ”short” time observation is. For general initial data in the space of compactly supported measures, we provide estimates on how the short-time measurements approximate the convolution of the fragmentation kernel with a suitably-scaled version of the initial data. The series representation also yields a reconstruction formula for the Mellin transform of the fragmentation kernel κ and an error estimate for such an approximation. Our analysis is complemented by a numerical investigation. (10.5802/ahl.207)
  DOI : 10.5802/ahl.207
• An Efficient SSP-based Methodology for Assessing Climate Risks of a Large Credit Portfolio
  
  • Bourgey Florian
  • Gobet Emmanuel
  • Jiao Ying
  , 2024.
• Reconciling rough volatility with jumps
  
  • Abi Jaber Eduardo
  • de Carvalho Nathan
  SIAM Journal on Financial Mathematics, Society for Industrial and Applied Mathematics, 2024, 15 (3), pp.785-823. We reconcile rough volatility models and jump models using a class of reversionary Heston models with fast mean reversions and large vol-of-vols. Starting from hyper-rough Heston models with a Hurst index H ∈ (−1/2, 1/2), we derive a Markovian approximating class of one dimensional reversionary Hestontype models. Such proxies encode a trade-off between an exploding vol-of-vol and a fast mean-reversion speed controlled by a reversionary timescale ϵ &gt; 0 and an unconstrained parameter H ∈ R. Sending ϵ to 0 yields convergence of the reversionary Heston model towards different explicit asymptotic regimes based on the value of the parameter H. In particular, for H ≤ −1/2, the reversionary Heston model converges to a class of Lévy jump processes of Normal Inverse Gaussian type. Numerical illustrations show that the reversionary Heston model is capable of generating at-the-money skews similar to the ones generated by rough, hyper-rough and jump models.
• On Lipschitz solutions of mean field games master equations
  
  • Bertucci Charles
  • Lasry Jean-Michel
  • Lions Pierre-Louis
  Journal of Functional Analysis, Elsevier, 2024, 287 (5), pp.110486. We develop a theory of existence and uniqueness of solutions of MFG master equations when the initial condition is Lipschitz continuous. Namely, we show that as long as the solution of the master equation is Lipschitz continuous in space, it is uniquely defined. Because we do not impose any structural assumptions, such as monotonicity for instance, there is a maximal time of existence for the notion of solution we provide. We analyze three cases: the case of a finite state space, the case of master equation set on a Hilbert space, and finally on the set of probability measures, all in cases involving common noises. In the last case, the Lipschitz continuity we refer to is on the gradient of the value function with respect to the state variable of the player. (10.1016/j.jfa.2024.110486)
  DOI : 10.1016/j.jfa.2024.110486
• Transparent scatterers and transmission eigenvalues
  
  • Grinevich Petr
  • Novikov Roman
  , 2024. We give a short review of old and recent results on scatterers with transmission eigenvalues of infinite multiplicity, including transparent scatterers. Historically, these studies go back to the publications: Regge (Nuovo Cimento 14, 1959), Newton (J. Math. Phys. 3, 1962) and Sabatier (J. Math. Phys. 7, 1966). Our review is based on the works: Grinevich, Novikov (Commun. Math. Phys. 174, 1995; Eurasian Journal of Mathematical and Computer Applications 9(4), 2021; Russian Math. Surveys, 77(6), 2022). Results of the first of these works include examples of transparent at fixed energy potentials from the Schwartz class in two dimensions. The two others works include the result that, for compactly supported multipoint potentials of Bethe - Peierls - Thomas type in two and three dimensions, any positive energy is a transmission eigenvalue of infinite multiplicity.
• Conditioning the logistic continuous-state branching process on non-extinction via its total progeny
  
  • Foucart Clément
  • Rivero Víctor
  • Winter Anita
  , 2024. The problem of conditioning a continuous-state branching process with quadratic competition (logistic CB process) on non-extinction is investigated. We first establish that non-extinction is equivalent to the total progeny of the population being infinite. The conditioning we propose is then designed by requiring the total progeny to exceed arbitrarily large exponential random variables. This is related to a Doob's $h$-transform with an explicit excessive function $h$. The $h$-transformed process, i.e. the conditioned process, is shown to have a finite lifetime almost surely (it is either killed or it explodes continuously). When starting from positive values, the conditioned process is furthermore characterized, up to its lifetime, as the solution to a certain stochastic equation with jumps. The latter superposes the dynamics of the initial logistic CB process with an additional density-dependent immigration term. Last, it is established that the conditioned process can be starting from zero. Key tools employed are a representation of the logistic CB process through a time-changed generalized Ornstein-Uhlenbeck process, as well as Laplace and Siegmund duality relationships with auxiliary diffusion processes.
• Supplementary Material to ``A hypothesis test for the domain of attraction of a random variable
  
  • Olivero Héctor
  • Talay Denis
  , 2024. In the paper ``A hypothesis test for the domain of attraction of a random variable'' we address theoreticals aspects of testing whether a sampled probability distribution of a random variable $V$ belongs to the domain of attraction of the Normal law or in the domain of attraction of a stable law with index smaller than 2. In this supplementary paper we present and discuss numerical results which allow us to illustrate satisfying properties of the proposed test.
• Escape Rate Games
  
  • Akian Marianne
  • Gaubert Stéphane
  • Marchesini Loïc
  , 2024. We consider a new class of repeated zero-sum games in which the payo↵ of one player is the escape rate of a dynamical system which evolves according to a nonexpansive nonlinear operator depending on the actions of both players. Considering order preserving finite dimensional linear operators over the positive cone endowed with Hilbert's projective (hemi-)metric, we recover the matrix multiplication games, introduced by Asarin et al., which generalize the joint spectral radius of sets of nonnegative matrices and arise in some population dynamics problems (growth maximization and minimization). We establish a two-player version of Mañé's lemma characterizing the value of the game in terms of a nonlinear eigenproblem. This generalizes to the two-player case the characterization of joint spectral radii in terms of extremals norms. This also allows us to show the existence of optimal strategies of both players.
• On a multi-dimensional McKean-Vlasov SDE with memorial and singular interaction associated to the parabolic-parabolic Keller-Segel model
  
  • Tomašević Milica
  • Woessner Guillaume
  Stochastic Analysis and Applications, Taylor & Francis: STM, Behavioural Science and Public Health Titles, 2024, 42 (4), pp.767-796. In this work we firstly prove the well-posedness of the non-linear martingale problem related to a McKean-Vlasov stochastic differential equation with singular interaction kernel in $\mathbb{R}^d$ for $d\geq 3$. The particularity of our setting is that the McKean-Vlasov process we study interacts at each time with all its past time marginal laws by means of a singular space-time kernel. Secondly, we prove that our stochastic process is a probabilistic interpretation for the parabolic-parabolic Keller-Segel system in $\mathbb{R}^d$. We thus obtain a well-posedness result to the latter under explicit smallness condition on the parameters of the model. (10.1080/07362994.2024.2381768)
  DOI : 10.1080/07362994.2024.2381768
• Kernel density estimation for a stochastic process with values in a Riemannian manifold
  
  • Abdillahi Isman Mohamed
  • Nefzi Wiem
  • Mbaye Papa
  • Khardani Salah
  • Yao Anne-Françoise
  Journal of Nonparametric Statistics, American Statistical Association, 2024, pp.1-20. This paper is related to the issue of the density estimation of observations with values in a Riemannian submanifold. In this context, Henry and Rodriguez ((2009), ‘Kernel Density Estimation on Riemannian Manifolds: Asymptotic Results’, Journal of Mathematical Imaging and Vision, 34, 235–239) proposed a kernel density estimator for independent data. We investigate here the behaviour of Pelletier's estimator when the observations are generated from a strictly stationary α-mixing process with values in this submanifold. Our study encompasses both pointwise and uniform analyses of the weak and strong consistency of the estimator. Specifically, we give the rate of convergence in terms of mean square error, probability, and almost sure convergence (a.s.). We also give a central-limit theorem and illustrate our proposal through some simulations and a real data application. (10.1080/10485252.2024.2382442)
  DOI : 10.1080/10485252.2024.2382442
• PEPit: computer-assisted worst-case analyses of first-order optimization methods in Python
  
  • Goujaud Baptiste
  • Moucer Céline
  • Glineur François
  • Hendrickx Julien
  • Taylor Adrien
  • Dieuleveut Aymeric
  Mathematical Programming Computation, Springer, 2024, 16 (3), pp.337-367. PEPit is a Python package aiming at simplifying the access to worst-case analyses of a large family of first-order optimization methods possibly involving gradient, projection, proximal, or linear optimization oracles, along with their approximate, or Bregman variants. In short, PEPit is a package enabling computer-assisted worst-case analyses of first-order optimization methods. The key underlying idea is to cast the problem of performing a worst-case analysis, often referred to as a performance estimation problem (PEP), as a semidefinite program (SDP) which can be solved numerically. For doing that, the package users are only required to write first-order methods nearly as they would have implemented them. The package then takes care of the SDP modelling parts, and the worst-case analysis is performed numerically via a standard solver. (10.1007/s12532-024-00259-7)
  DOI : 10.1007/s12532-024-00259-7
• When Data Driven Reduced Order Modeling Meets Full Waveform Inversion
  
  • Borcea Liliana
  • Garnier Josselin
  • Mamonov Alexander
  • Zimmerling Jörn
  SIAM Review, Society for Industrial and Applied Mathematics, 2024, 66 (3), pp.501-532. Waveform inversion is concerned with estimating a heterogeneous medium, modeled by variable coefficients of wave equations, using sources that emit probing signals and receivers that record the generated waves. It is an old and intensively studied inverse problem with a wide range of applications, but the existing inversion methodologies are still far from satisfactory. The typical mathematical formulation is a nonlinear least squares data fit optimization and the difficulty stems from the non-convexity of the objective function that displays numerous local minima at which local optimization approaches stagnate. This pathological behavior has at least three unavoidable causes: (1) The mapping from the unknown coefficients to the wave field is nonlinear and complicated. (2) The sources and receivers typically lie on a single side of the medium, so only backscattered waves are measured. (3) The probing signals are band limited and with high frequency content. There is a lot of activity in the computational science and engineering communities that seeks to mitigate the difficulty of estimating the medium by data fitting. In this paper we present a different point of view, based on reduced order models (ROMs) of two operators that control the wave propagation. The ROMs are called data driven because they are computed directly from the measurements, without any knowledge of the wave field inside the inaccessible medium. This computation is non-iterative and uses standard numerical linear algebra methods. The resulting ROMs capture features of the physics of wave propagation in a complementary way and have surprisingly good approximation properties that facilitate waveform inversion. (10.1137/23M1552826)
  DOI : 10.1137/23M1552826