Publications

2023

• Computer-assisted proof of shear-induced chaos in stochastically perturbed Hopf systems
  
  • Breden Maxime
  • Engel Maximilian
  The Annals of Applied Probability, Institute of Mathematical Statistics (IMS), 2023, 33 (2), pp.1252-1294. We confirm a long-standing conjecture concerning shear-induced chaos in stochastically perturbed systems exhibiting a Hopf bifurcation. The method of showing the main chaotic property, a positive Lyapunov exponent, is a computer-assisted proof. Using the recently developed theory of conditioned Lyapunov exponents on bounded domains and the modified Furstenberg-Khasminskii formula, the problem boils down to the rigorous computation of eigenfunctions of the Kolmogorov operators describing distributions of the underlying stochastic process. (10.1214/22-AAP1841)
  DOI : 10.1214/22-AAP1841
• Waveform Inversion with a Data Driven Estimate of the Internal Wave
  
  • Borcea Liliana
  • Garnier Josselin
  • Mamonov Alexander
  • Zimmerling Jörn
  SIAM Journal on Imaging Sciences, Society for Industrial and Applied Mathematics, 2023, 16 (1), pp.280-312. We study an inverse problem for the wave equation, concerned with estimating the wave speed from data gathered by an array of sources and receivers that emit probing signals and measure the resulting waves. The typical approach to solving this problem is a nonlinear least squares minimization of the data misfit, over a search space. There are two main impediments to this approach, which manifest as multiple local minima of the objective function: The nonlinearity of the mapping from the wave speed to the data, which accounts for multiple scattering effects, and poor knowledge of the kinematics (smooth part of the wave speed), which causes cycle skipping. We show that the nonlinearity can be mitigated using a data driven estimate of the wave field at points inside the medium, also known as the “internal wave field.” This leads to improved performance of the inversion for a reasonable initial guess of the kinematics. (10.1137/22M1517342)
  DOI : 10.1137/22M1517342
• Multipoint formulas in inverse problems and their numerical implementation
  
  • Novikov Roman
  • Sivkin Vladimir
  • Sabinin Grigory
  Inverse Problems, IOP Publishing, 2023, 39 (12), pp.125016. We present the first numerical study of multipoint formulas for finding leading coefficients in asymptotic expansions arising in potential and scattering theories. In particular, we implement different formulas for finding the Fourier transform of potential from the scattering amplitude at several high energies. We show that the aforementioned approach can be used for essential numerical improvements of classical results including the slowly convergent Born-Faddeev formula for inverse scattering at high energies. The approach of multipoint formulas can be also used for recovering the X-ray transform of potential from boundary values of the scattering wave functions at several high energies. Determination of total charge (electric or gravitational) from several exterior measurements is also considered. In addition, we show that the aforementioned multipoint formulas admit an efficient regularization for the case of random noise. In particular, we proceed from theoretical works [Novikov, 2020, 2021]. (10.1088/1361-6420/ad06e6)
  DOI : 10.1088/1361-6420/ad06e6
• Speckle Memory Effect in the Frequency Domain and Stability in Time-Reversal Experiments
  
  • Garnier Josselin
  • Sølna Knut
  Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal, Society for Industrial and Applied Mathematics, 2023, 21 (1), pp.80-118. When waves propagate through a complex medium like the turbulent atmosphere the wave field becomes incoherent and the wave intensity forms a complex speckle pattern. In this paper we study a speckle memory effect in the frequency domain and some of its consequences. This effect means that certain properties of the speckle pattern produced by wave transmission through a randomly scattering medium is preserved when shifting the frequency of the illumination. The speckle memory effect is characterized via a detailed novel analysis of the fourth-order moment of the random paraxial Green’s function at four different frequencies. We arrive at a precise characterization of the frequency memory effect and what governs the strength of the memory. As an application we quantify the statistical stability of time-reversal wave refocusing through a randomly scattering medium in the paraxial or beam regime. Time reversal refers to the situation when a transmitted wave field is recorded on a time-reversal mirror then time reversed and sent back into the complex medium. The re-emitted wave field then refocuses at the original source point. We compute the mean of the refocused wave and identify a novel quantitative description of its variance in terms of the radius of the time-reversal mirror, the size of its elements, the source bandwidth, and the statistics of the random medium fluctuations. (10.1137/22M1470414)
  DOI : 10.1137/22M1470414
• Coupling Bertoin's and Aldous-Pitman's representations of the additive coalescent
  
  • Kortchemski Igor
  • Thévenin Paul
  Random Structures and Algorithms, Wiley, 2023, 65 (1), pp.17-45. We construct a coupling between two seemingly very di erent constructions of the standard additive coalescent, which describes the evolution of masses merging pairwise at rates proportional to their sums. The rst construction, due to Aldous & Pitman, involves the components obtained by logging the Brownian Continuum Random Tree (CRT) by a Poissonian rain on its skeleton as time increases. The second one, due to Bertoin, involves the excursions above its running in mum of a linear-drifted standard Brownian excursion as its drift decreases. Our main tool is the use of an exploration algorithm of the so-called cut-tree of the Brownian CRT, which is a tree that encodes the genealogy of the fragmentation of the CRT. (10.1002/rsa.21206)
  DOI : 10.1002/rsa.21206
• Modélisation fine des dynamiques épidémiques - le projet ICI
  
  • Colomb Maxime
  • Graham Carl
  • Perret Julien
  • Talay Denis
  • Garnier Josselin
  , 2023. Le projet ICI est un simulateur de propagation d’épidémies, particulièrement adapté au SARS-COV-2, reposant sur une modélisation très détaillée d’un espace urbain et de sa population. Cette précision permet de simuler des chaînes de contaminations détaillées ainsi que l’effet de différentes politiques sanitaires sur la diffusion de l’épidémie. La méthodologie mise en œuvre pour créer le jumeau numérique de présence d’individus au sein d’activités est particulièrement présentée dans cet exposé.
• The condensation of a trapped dilute Bose gas with three-body interactions
  
  • Nam Phan Thành
  • Ricaud Julien
  • Triay Arnaud
  Probability and Mathematical Physics, MSP, 2023, 4 (1), pp.91-149. We consider a trapped dilute gas of $N$ bosons in $\mathbb{R}^3$ interacting via a three-body interaction potential of the form $N V(N^{1/2}(x-y,x-z))$. In the limit $N\to \infty$, we prove that the ground state of the system exhibits the complete Bose--Einstein condensation, and that the condensate is the unique minimizer of the 3D energy-critical nonlinear Schr\"odinger functional where the nonlinear term is coupled with the scattering energy of the interaction potential. (10.2140/pmp.2023.4.91)
  DOI : 10.2140/pmp.2023.4.91
• Contributions to Federated Learning and First-Order Optimization
  
  • Dieuleveut Aymeric
  , 2023.
• Discrete potential mean field games: duality and numerical resolution
  
  • Bonnans J. Frédéric
  • Lavigne Pierre
  • Pfeiffer Laurent
  Mathematical Programming, Springer Verlag, 2023, 202 (1-2), pp.241-278. We propose and investigate a general class of discrete time and finite state space mean field game (MFG) problems with potential structure. Our model incorporates interactions through a congestion term and a price variable. It also allows hard constraints on the distribution of the agents. We analyze the connection between the MFG problem and two optimal control problems in duality. We present two families of numerical methods and detail their implementation: (i) primal-dual proximal methods (and their extension with nonlinear proximity operators), (ii) the alternating direction method of multipliers (ADMM) and a variant called ADM-G. We give some convergence results. Numerical results are provided for two examples with hard constraints. (10.1007/s10107-023-01934-8)
  DOI : 10.1007/s10107-023-01934-8
• On the Blaschke-Lebesgue theorem for the Cheeger constant via areas and perimeters of inner parallel sets
  
  • Bogosel Beniamin
  , 2023. The first main result presented in the paper shows that the perimeters of inner parallel sets of planar shapes having a given constant width are minimal for the Reuleaux triangles. This implies that the areas of inner parallel sets and, consequently, the inverse of the Cheeger constant are also minimal for the Reuleaux triangles. Proofs use elementary geometry arguments and are based on direct comparisons between general constant width shapes and the Reuleaux triangle.
• Generative modeling of extremes with neural networks
  
  • Allouche Michaël
  • Girard Stéphane
  • Gobet Emmanuel
  , 2023. We investigate new parametrizations based on neural networks in order to approximate and sample multi-variate extreme values, especially in the case of heavy-tailed distributions. We discuss two approaches. First, transformations of Feedforward neural networks based on Rectified linear units (ReLU) are used. An analysis of the uniform error between the extreme quantile and its GAN approximation is provided, and shows that second-order parameters of the marginal data distributions play an important role. Second, eLU based NN are used, to efficiently get rid of the bias term in tails approximation, in the presence of arbitrary high-order parameters. These results are illustrated on synthetic and real data.
• Large population limit of the spectrum of killed birth-and-death processes
  
  • Chazottes J.-R
  • Collet P
  • Méléard S
  Journal of Functional Analysis, Elsevier, 2023, 285 (9), pp.110092. We consider a general class of birth-and-death processes with state space {0, 1, 2, 3,. . .} which describes the size of a population going eventually to extinction with probability one. We obtain the complete spectrum of the generator of the process killed at 0 in the large population limit, that is, we scale the process by a parameter K, and take the limit K → +∞. We assume that the di erential equation dx/dt = b(x)−d(x) describing the infinite population limit (in any finite-time interval) has a repulsive fixed point at 0, and an attractive fixed point x * > 0. We prove that, asymptotically, the spectrum is the superposition of two spectra. One is the spectrum of the generator of an Ornstein-Uhlenbeck process, which is n(b (x *) − d (x *)), n ≥ 0. The other one is the spectrum of a continuous-time binary branching process conditioned on nonextinction, and is given by n(d (0) − b (0)), n ≥ 1. A major di culty is that di erent scales and function spaces are involved. We work at the level of the eigenfunctions that we split over di erent regions, and study their asymptotic dependence on K in each region. In particular, we prove that the spectral gap goes to min b (0) − d (0), d (x *) − b (x *). This work complements a previous work of ours in which we studied the approximation of the quasi-stationary distribution and of the mean time to extinction. (10.1016/j.jfa.2023.110092)
  DOI : 10.1016/j.jfa.2023.110092
• Deep learning solutions to some prediction problems in finance
  
  • Chen Qinkai
  , 2023. This thesis consists of two connected parts that examen respectively two prediction problems in finance, stock return prediction and short-term volatility prediction. It also has another additional part which examens a related issue in contemporary artists connection discovery with some methods derived from the two first parts.In the first part, we present a univariate and a multivariate deep learning solution to the problem of stock return prediction with financial news. We first introduce a univariate prediction procedure that predicts the short-term return of a stock after the publication of news associated with this stock (Chapter 2). In this procedure, we first use a transfer learning based method to generate contextualized embeddings of the words in a news’ headline, a recurrent neural network is then used to make predictions from the generated embeddings. Through extensive experiments, we show that this approach outperforms other baseline models. We then extend our univariate approach to a multivariate model (Chapter 3), in which a single news can not only impact one stock but also all other related stocks. Through an innovative multi-graph convolutional network structure, we can model the information transmission process from one stock to others based on stock relationships built from different sources. We demonstrate the effectiveness of this approach with a similar experiment setup as the first study.In the second part of this thesis, we are interested in predicting short-term realized volatility from limit order book with a multivariate model (Chapter 4). To achieve this goal, we design a graph neural network containing both temporal and cross-sectional relations. Graph transformer operators are integrated into the model for better accuracy and computing efficiency on this large graph. Through experiments based on more than 500 stocks, we demonstrate that a graph-based multivariate approach has better predictive power than commonly used univariate baselines.In the third part, we introduce a solution to discovering the relations among the contemporary artists through their biographies (Chapter 5). For this purpose, we design a general NLP framework, in which we first continue to pre-train an existing general language model with unlabeled art-related texts. We then fine-tune this new pre-trained model with labeled biography pairs. We demonstrate that our approach achieves more than 85% accuracy in identifying the connection between two artists and outperforms other baseline models in the experiments.
• Improved convergence rate for Reflected BSDEs by penalization method
  
  • Gobet Emmanuel
  • Wang Wanqing
  , 2023. We investigate the convergence of numerical solution of Reflected Backward Stochastic Differential Equations (RBSDEs) using the penalization approach in a general non-Markovian framework. We prove the convergence between the continuous penalized solution and the reflected one, in full generality, at order 1/2 as a function of the penalty parameter; the convergence order becomes 1 when the increasing process of the RBSDE has a bounded density, which is a mild condition in practice. The convergence is analyzed in a.s.-sense and Lp-sense (p \geq 2). To achieve these new results, we have developed a refined analysis of the behavior of the process close to the barrier. Then we propose an implicit scheme for computing the discrete solution of the penalized equation and we derive that the global convergence order is 3/8 as a function of time discretization under mild regularity assumptions. This convergence rate is verified in the case of American Put options and some numerical tests illustrate these results.
• flowMC: Normalizing flow enhanced sampling package for probabilistic inference in JAX
  
  • Wong Kaze
  • Gabrié Marylou
  • Foreman-Mackey Daniel
  Journal of Open Source Software, Open Journals, 2023, 8 (83), pp.5021. (10.21105/joss.05021)
  DOI : 10.21105/joss.05021
• Drosophilids with darker cuticle have higher body temperature under light
  
  • Freoa Laurent
  • Chevin Luis-Miguel
  • Christol Philippe
  • Méléard Sylvie
  • Rera Michael
  • Véber Amandine
  • Gibert Jean-Michel
  Scientific Reports, Nature Publishing Group, 2023, 13 (1), pp.3513. Cuticle pigmentation was shown to be associated with body temperature for several relatively large species of insects, but it was questioned for small insects. Here we used a thermal camera to assess the association between drosophilid cuticle pigmentation and body temperature increase when individuals are exposed to light. We compared mutants of large effects within species ( Drosophila melanogaster ebony and yellow mutants). Then we analyzed the impact of naturally occurring pigmentation variation within species complexes ( Drosophila americana/Drosophila novamexicana and Drosophila yakuba/Drosophila santomea ). Finally we analyzed lines of D. melanogaster with moderate differences in pigmentation. We found significant differences in temperatures for each of the four pairs we analyzed. The temperature differences appeared to be proportional to the differently pigmented area: between Drosophila melanogaster ebony and yellow mutants or between Drosophila americana and Drosophila novamexicana, for which the whole body is differently pigmented, the temperature difference was around 0.6 °C ± 0.2 °C. By contrast, between D. yakuba and D. santomea or between Drosophila melanogaster Dark and Pale lines, for which only the posterior abdomen is differentially pigmented, we detected a temperature difference of about 0.14 °C ± 0.10 °C. This strongly suggests that cuticle pigmentation has ecological implications in drosophilids regarding adaptation to environmental temperature. (10.1038/s41598-023-30652-6)
  DOI : 10.1038/s41598-023-30652-6
• MULTI-STEP VARIANT OF THE PARAREAL ALGORITHM: CONVERGENCE ANALYSIS AND NUMERICS
  
  • Ait-Ameur Katia
  • Maday Yvon
  , 2023. In this paper, we consider the problem of accelerating the numerical simulation of time dependent problems involving a multi-step time scheme by the parareal algorithm. The parareal method is based on combining predictions made by a coarse and cheap propagator, with corrections computed with two propagators: the previous coarse and a precise and expensive one used in a parallel way over the time windows. A multi-step time scheme can potentially bring higher approximation orders than plain one-step methods but the initialisation of each time window needs to be appropriately chosen. Our main contribution is the design and analysis of an algorithm adapted to this type of discretisation without being too much intrusive in the coarse or fine propagators. At convergence, the parareal algorithm provides a solution that coincides with the solution of the fine solver. In the classical version of parareal, the local initial condition of each time window is corrected at every iteration. When the fine and/or coarse propagators is a multi-step time scheme, we need to choose a consistent approximation of the solutions involved in the initialisation of the fine solver at each time windows. Otherwise, the initialisation error will prevent the parareal algorithm to converge towards the solution with fine solver's accuracy. In this paper, we develop a variant of the algorithm that overcome this obstacle. Thanks to this, the parareal algorithm is more coherent with the underlying time scheme and we recover the properties of the original version. We show both theoretically and numerically that the accuracy and convergence of the multi-step variant of parareal algorithm are preserved when we choose carefully the initialisation of each time window.
• Market Making and Incentives Design in the Presence of a Dark Pool: A Stackelberg Actor–Critic Approach
  
  • Baldacci Bastien
  • Manziuk Iuliia
  • Mastrolia Thibaut
  • Rosenbaum Mathieu
  Operations Research, INFORMS, 2023, 71 (2), pp.727-749. A Stackelberg actor–critic approach to optimal market making and incentives design with dark pools. We consider the issue of a market maker acting at the same time in the lit and dark pools of an exchange. The exchange wishes to establish a suitable make–take fee policy to attract transactions on its venues. We first solve the stochastic control problem of the market maker without the intervention of the exchange. Then, we derive the equations defining the optimal contract to be set between the market maker and the exchange. This contract depends on the trading flows generated by the market maker’s activity on the two venues. In both cases, we show existence and uniqueness, in the viscosity sense, of the solutions of the Hamilton–Jacobi–Bellman equations associated to the market maker and exchange’s problems. We finally design an actor–critic algorithm inspired by deep reinforcement learning methods, enabling us to approximate efficiently the optimal controls of the market maker and the optimal incentives to be provided by the exchange. (10.1287/opre.2022.2406)
  DOI : 10.1287/opre.2022.2406
• Waveform inversion via reduced order modeling
  
  • Borcea Liliana
  • Garnier Josselin
  • Mamonov Alexander
  • Zimmerling Jörn
  Geophysics, Society of Exploration Geophysicists, 2023, 88 (2), pp.R175-R191. We introduce a novel approach to waveform inversion based on a data-driven reduced order model (ROM) of the wave operator. The presentation is for the acoustic wave equation, but the approach can be extended to elastic or electromagnetic waves. The data are time resolved measurements of the pressure wave gathered by an acquisition system that probes the unknown medium with pulses and measures the generated waves. We propose to solve the inverse problem of velocity estimation by minimizing the square misfit between the ROM computed from the recorded data and the ROM computed from the modeled data, at the current guess of the velocity. We give a step by step computation of the ROM, which depends nonlinearly on the data and yet can be obtained from them in a noniterative fashion, using efficient methods from linear algebra. We also explain how to make the ROM robust to data inaccuracy. The ROM computation requires the full array response matrix gathered with colocated sources and receivers. However, we find that the computation can deal with an approximation of this matrix, obtained from towed-streamer data using interpolation and reciprocity on-the-fly. Although the full-waveform inversion approach of nonlinear least-squares data fitting is challenging without low-frequency information, due to multiple minima of the data fit objective function, we find that the ROM misfit objective function has better behavior, even for a poor initial guess. We also find by explicit computation of the objective functions in a simple setting that the ROM misfit objective function has convexity properties, whereas the least-squares data fit objective function displays multiple local minima. (10.1190/geo2022-0070.1)
  DOI : 10.1190/geo2022-0070.1
• Transform MCMC schemes for sampling intractable factor copula models
  
  • Bénézet Cyril
  • Gobet Emmanuel
  • Targino Rodrigo
  Methodology and Computing in Applied Probability, Springer Verlag, 2023, 25 (1), pp.13. In financial risk management, modelling dependency within a random vector X is crucial, a standard approach is the use of a copula model. Say the copula model can be sampled through realizations of Y having copula function C: had the marginals of Y been known, sampling X^(i) , the i-th component of X, would directly follow by composing Y^(i) with its cumulative distribution function (c.d.f.) and the inverse c.d.f. of X^(i). In this work, the marginals of Y are not explicit, as in a factor copula model. We design an algorithm which samples X through an empirical approximation of the c.d.f. of the Y marginals. To be able to handle complex distributions for Y or rare-event computations, we allow Markov Chain Monte Carlo (MCMC) samplers. We establish convergence results whose rates depend on the tails of X, Y and the Lyapunov function of the MCMC sampler. We present numerical experiments confirming the convergence rates and also revisit a real data analysis from financial risk management. (10.1007/s11009-023-09983-4)
  DOI : 10.1007/s11009-023-09983-4
• Paraxial Wave Propagation in Random Media with Long-Range Correlations
  
  • Borcea Liliana
  • Garnier Josselin
  • Sølna Knut
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2023, 83 (1), pp.25-51. We study the paraxial wave equation with a randomly perturbed index of refraction, which can model the propagation of a wave beam in a turbulent medium. The random perturbation is a stationary and isotropic process with a general form of the covariance that may be integrable or not. We focus attention mostly on the non-integrable case, which corresponds to a random perturbation with long-range correlations, that is relevant for propagation through a cloudy turbulent atmosphere. The analysis is carried out in a high-frequency regime where the forward scattering approximation holds. It reveals that the randomization of the wave field is multiscale: The travel time of the wave front is randomized at short distances of propagation and it can be described by a fractional Brownian motion. The wave field observed in the random travel time frame is affected by the random perturbations at long distances, and it is described by a Schr¨odinger-type equation driven by a standard Brownian field. We use these results to quantify how scattering leads to decorrelation of the spatial and spectral components of the wave field and to a deformation of the pulse emitted by the source. These are important questions for applications like imaging and free space communications with pulsed laser beams through a turbulent atmosphere. We also compare the results with those used in the optics literature, which are based on the Kolmogorov model of turbulence. (10.1137/22M149524X)
  DOI : 10.1137/22M149524X
• Procédé de synthèse d'images
  
  • Prenat Michel
  • Le Pennec Erwan
  • Berginc Gérard
  , 2023.
• A multiscale algorithm for computing realistic image transformations in the LDDMM framework -Application to the modelling of fetal brain growth
  
  • Gaudfernau Fleur
  • Allassonière Stéphanie
  • Le Pennec Erwan
  , 2023.
• Discretization of the Ergodic Functional Central Limit Theorem
  
  • Pagès Gilles
  • Rey Clément
  Journal of Theoretical Probability, Springer, 2023, 36 (4), pp.2359-2402. (10.1007/s10959-023-01237-w)
  DOI : 10.1007/s10959-023-01237-w
• On Lipschitz solutions of mean field games master equations
  
  • Bertucci Charles
  • Lasry Jean-Michel
  • Lions Pierre-Louis
  , 2023. 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. (10.48550/arXiv.2302.05218)
  DOI : 10.48550/arXiv.2302.05218