Publications

2023

• Interpolating between Clustering and Dimensionality Reduction with Gromov-Wasserstein
  
  • Van Assel Hugues
  • Vincent-Cuaz Cédric
  • Vayer Titouan
  • Flamary Rémi
  • Courty Nicolas
  , 2023. We present a versatile adaptation of existing dimensionality reduction (DR) objectives, enabling the simultaneous reduction of both sample and feature sizes. Correspondances between input and embedding samples are computed through a semi-relaxed Gromov-Wasserstein optimal transport (OT) problem. When the embedding sample size matches that of the input, our model recovers classical popular DR models. When the embedding's dimensionality is unconstrained, we show that the OT plan delivers a competitive hard clustering. We emphasize the importance of intermediate stages that blend DR and clustering for summarizing real data and apply our method to visualize datasets of images.
• Experimental Investigation of Mechanical Properties of Additively Manufactured Fibre-Reinforced Composite Structures for Robotic Applications
  
  • Bisoi Arnav
  • Tüfekci Mertol
  • Öztekin Vehbi
  • Denimal Goy Enora
  • Salles Loïc
  Applied Composite Materials, Springer Verlag (Germany), 2023, pp.1-26. Abstract This study explores the variation in mechanical properties of additively manufactured composite structures for robotic applications with different infill densities and layer heights using fused deposition modelling (FDM). Glass fibre-reinforced polyamide (GFRP), and carbon fibre-reinforced polyamide (CFRP) filaments are used, and the specimens are printed with 20%, 40%, 60% and 100% infill density lattice structures for tensile and three-point bending tests. These printed samples are examined in the microscope to gain more understanding of the microstructure of the printed composites. To characterise the mechanical properties, a set of tensile and three-point bend tests are conducted on the manufactured composite samples. Test results indicate the variations in tensile strength and Young’s modulus of specimens based on the printing parameters and reveal the tensile and bending behaviour of those printed composite structures against varying infill ratios and reinforcing fibres. The experimental findings are also compared to analytical and empirical modelling approaches. Finally, based on the results, the applications of the additively manufactured structure to the robotic components are presented. (10.1007/s10443-023-10179-9)
  DOI : 10.1007/s10443-023-10179-9
• Model-free Posterior Sampling via Learning Rate Randomization
  
  • Tiapkin Daniil
  • Belomestny Denis
  • Calandriello Daniele
  • Moulines Eric
  • Munos Remi
  • Naumov Alexey
  • Perrault Pierre
  • Valko Michal
  • Menard Pierre
  , 2023. In this paper, we introduce Randomized Q-learning (RandQL), a novel randomized model-free algorithm for regret minimization in episodic Markov Decision Processes (MDPs). To the best of our knowledge, RandQL is the first tractable model-free posterior sampling-based algorithm. We analyze the performance of RandQL in both tabular and non-tabular metric space settings. In tabular MDPs, RandQL achieves a regret bound of order $\widetilde{O}(\sqrt{H^{5}SAT})$, where $H$ is the planning horizon, $S$ is the number of states, $A$ is the number of actions, and $T$ is the number of episodes. For a metric state-action space, RandQL enjoys a regret bound of order $\widetilde{O}(H^{5/2} T^{(d_z+1)/(d_z+2)})$, where $d_z$ denotes the zooming dimension. Notably, RandQL achieves optimistic exploration without using bonuses, relying instead on a novel idea of learning rate randomization. Our empirical study shows that RandQL outperforms existing approaches on baseline exploration environments. (10.48550/arXiv.2310.18186)
  DOI : 10.48550/arXiv.2310.18186
• On the contraction properties of a pseudo-Hilbert projective metric
  
  • Ligonnière Maxime
  , 2023. In this note, we define a bounded variant on the Hilbert projective metric on an infinite dimensional space E and study the contraction properties of the projective maps associated with positive linear operators on E. More precisely, we prove that any positive linear operator acts projectively as a 1-Lipschitz map relatively to this distance. We also show that for a positive linear operator, strict projective contraction is equivalent to a property called uniform positivity.
• Optimal Transport with Adaptive Regularisation
  
  • Van Assel Hugues
  • Vayer Titouan
  • Flamary Remi
  • Courty Nicolas
  , 2023. Regularising the primal formulation of optimal transport (OT) with a strictly convex term leads to enhanced numerical complexity and a denser transport plan. Many formulations impose a global constraint on the transport plan, for instance by relying on entropic regularisation. As it is more expensive to diffuse mass for outlier points compared to central ones, this typically results in a significant imbalance in the way mass is spread across the points. This can be detrimental for some applications where a minimum of smoothing is required per point. To remedy this, we introduce OT with Adaptive RegularIsation (OTARI), a new formulation of OT that imposes constraints on the mass going in or/and out of each point. We then showcase the benefits of this approach for domain adaptation.
• Coupling by reflection for controlled diffusion processes: Turnpike property and large time behavior of Hamilton–Jacobi–Bellman equations
  
  • Conforti Giovanni
  The Annals of Applied Probability, Institute of Mathematical Statistics (IMS), 2023, 33 (6A). (10.1214/22-AAP1927)
  DOI : 10.1214/22-AAP1927
• On convex numerical schemes for inelastic contacts with friction
  
  • Bloch Hélène
  • Lefebvre-Lepot Aline
  ESAIM: Proceedings and Surveys, EDP Sciences, 2023, 75, pp.24-59. This paper reviews the different existing Contact Dynamics schemes for the simulation of granular media, for which the discrete incremental problem is based on the resolution of convex problems. This type of discretization has the great advantage of allowing the use of standard convex optimization algorithms. In the case of frictional contacts, we consider schemes based on a convex relaxation of the constraint as well as a fixed point scheme. The model and the computations leading to the discrete problems are detailed in the case of convex, regular but not necessarily spherical particles. We prove, using basic tools of convex analysis, that the discrete optimization problem can be seen as a minimization problem of a global discrete energy for the system, in which the velocity to be considered is an average of the pre-and post-impact velocities. A numerical study on an academic test case is conducted, illustrating for the first time the convergence with order 1 in the time step of the different schemes. We also discuss the influence of the convex relaxation of the constraint on the behavior of the system. We show in particular that, although it induces numerical dilatation, it does not significantly modify the macrosopic behavior of a column collapse en 2d. The numerical tests are performed using the code SCoPI. (10.1051/proc/202375024)
  DOI : 10.1051/proc/202375024
• Non-Parametric Measure Approximations for Constrained Multi-Objective Optimisation under Uncertainty
  
  • Rivier Mickael
  • Razaaly Nassim
  • Congedo Pietro Marco
  International Journal for Numerical Methods in Engineering, Wiley, 2023. In this paper, we propose non-parametric estimations of robustness and reliability measures approximation error, employed in the context of constrained multi-objective optimisation under uncertainty. These approximations with tunable accuracy permit to capture the Pareto front in a parsimonious way, and can be exploited within an adaptive refinement strategy. First, we illustrate an efficient approach for obtaining joint representations of the robustness and reliability measures, allowing sharper discrimination of Pareto-optimal designs. A specific surrogate model of these objectives and constraints is then proposed to accelerate the optimisation process. Secondly, we propose an adaptive refinement strategy, using these tunable accuracy approximations to drive the computational effort towards the computation of the optimal area. To this extent, an adapted Pareto dominance rule and Pareto optimal probability computation are formulated. The performance of the proposed strategy is assessed on several analytical test-cases against classical approaches. We also illustrate the method on an engineering application, performing shape optimisation under uncertainty of an Organic Rankine Cycle turbine. (10.1002/nme.7403)
  DOI : 10.1002/nme.7403
• A mean field model for the development of renewable capacities
  
  • Alasseur Clémence
  • Basei Matteo
  • Bertucci Charles
  • Cecchin Alekos
  Mathematics and Financial Economics, Springer Verlag, 2023, 17 (4), pp.695-719. We propose a model based on a large number of small competitive producers of renewable energies, to study the effect of subsidies on the aggregate level of capacity, taking into account a cannibalization effect. We first derive a model to explain how long-time equilibrium can be reached on the market of production of renewable electricity and compare this equilibrium to the case of monopoly. Then we consider the case in which other capacities of production adjust to the production of renewable energies. The analysis is based on a master equation and we get explicit formulae for the long-time equilibria. We also provide new numerical methods to simulate the master equation and the evolution of the capacities. Thus we find the optimal subsidies to be given by a central planner to the installation and the production in order to reach a desired equilibrium capacity. (10.1007/s11579-023-00348-6)
  DOI : 10.1007/s11579-023-00348-6
• Mixed volumes and the Blaschke-Lebesgue theorem
  
  • Bogosel Beniamin
  Acta Mathematica Hungarica, Springer Verlag, 2023, 173 (1), pp.122-138. The mixed area of a Reuleaux polygon and its symmetric with respect to the origin is expressed in terms of the mixed area of two explicit polygons. This gives a geometric explanation of a classical proof due to Chakerian. Mixed areas and volumes are also used to reformulate the minimization of the volume under constant width constraint as isoperimetric problems. In the two dimensional case, the equivalent formulation is solved, providing another proof of the Blaschke-Lebesgue theorem. In the three dimensional case the proposed relaxed formulation involves the mean width, the area and inclusion constraints. (10.1007/s10474-024-01435-w)
  DOI : 10.1007/s10474-024-01435-w
• Weak convergence of continuous-state branching processes with large immigration
  
  • Foucart Clément
  • Yuan Linglong
  , 2023. Functional limit theorems are established for continuous-state branching processes with immigration (CBIs) whose reproduction laws have finite mean and immigration laws are heavy-tailed. Different regimes of immigration are identified for which the limiting processes are either subordinators, CBIs, extremal processes or extremal shot noise processes.
• Computer-assisted proofs for the many steady states of a chemotaxis model with local sensing
  
  • Breden Maxime
  • Payan Maxime
  , 2023. We study the steady states of a system of cross-diffusion equations arising from the modeling of chemotaxis with local sensing, where the motility is a decreasing function of the concentration of the chemical. In order to capture the many different equilibria that sometimes co-exist, we use computer-assisted proofs: Given an approximate solution obtained numerically, we apply a fixed-point argument in a small neighborhood of this approximate solution to prove the existence of an exact solution nearby. This allows us to rigorously study the steady states of this crossdiffusion system much more extensively than what previously possible with purely pen-and-paper techniques. Our computer-assisted argument makes use of Fourier series decomposition, which is common in the literature, but usually restricted to systems with polynomial nonlinearities. This is not the case for the model considered in this paper, and we develop a new way of dealing with some nonpolynomial nonlinearities in the context of computer-assisted proofs with Fourier series.
• Advanced implied volatility modeling for risk management and central clearing
  
  • Mingone Arianna
  , 2023. In the first part of this thesis we address the non trivial task of building arbitrage-free implied volatility surfaces which could be used by market operators for practical purposes. We study in depth static arbitrage constrains for option portfolios and apply them to notorious implied volatility models. We firstly fully characterize the absence of Butterfly arbitrage in the SVI model by Gatheral, and study the case of some 3-parameter sub-SVIs models, such as the Symmetric SVI, the Vanishing Upward/Downward SVI, and SSVI. We then reconsider the latter model, extended to multiple maturity slices, and combine the so identified conditions of no Butterfly arbitrage with the already known conditions of no Calendar Spread arbitrage by [Hendriks and Martini, The extended SSVI volatility surface, Journal of Comp Finance, 2019]. As a result, we identify a global calibration algorithm for the eSSVI model ensuring the absence of arbitrage.Secondly, we study the characterization of a weaker notion of absence of Butterfly arbitrage (which we call "weak no arbitrage condition"), i.e. the two monotonicity requirements of the functions d_1 and d_2 in the Black-Scholes formula, identified by [Fukasawa, The normalizing transformation of the implied volatility smile, Math Fin, 2012]. We consider the framework of smiles parameterized in delta (following the typical convention on FX markets), and, as a result, we characterize the set of volatility smiles satisfying this weak condition of no static arbitrage.Finally, based on the -- simple but, to our knowledge, not yet exploited -- remark that Call options can be seen as Calls written on other Calls, we study the dynamic properties of these contracts.In the second part, we consider the problem of quantifying the counterparty risk for option portfolios that Central Clearing Counterparties face daily. We identify a new model-free formula for the short-term VaR of option portfolios which performs better than the classical approach of Filtered Historical Simulation in our numerical tests. Finally, we look at the notion of Expected Shortfall, and compare different types of backtesting measures.
• Wellposedness of the cubic Gross-Pitaevskii equation with spatial white noise on R^2
  
  • Mackowiak Pierre
  , 2023. In this paper, we prove the global wellposedness of the Gross-Pitaevskii equation with white noise potential, i.e. a cubic nonlinear Schrödinger equation with harmonic confining potential and spatial white noise multiplicative term. This problem is ill-defined and a Wick renormalization is needed in order to give a meaning to solutions. In order to do this, we introduce a change of variables which transforms the original equation into one with less irregular terms. We construct a solution as a limit of solutions of the same equation but with a regularized noise. This convergence is shown by interpolating between a diverging bound in a high regularity Hermite-Sobolev space and a Cauchy estimate in L^2(R^2).
• Assessment and Evaluation of Empirical and Scientific Data
  
  • Hansen Nikolaus
  , 2023.
• Large-scale nonconvex optimization: randomization, gap estimation, and numerical resolution
  
  • Bonnans J. Frédéric
  • Liu Kang
  • Oudjane Nadia
  • Pfeiffer Laurent
  • Wan Cheng
  SIAM Journal on Optimization, Society for Industrial and Applied Mathematics, 2023, 33 (4), pp.3083-3113. We address a large-scale and nonconvex optimization problem, involving an aggregative term. This term can be interpreted as the sum of the contributions of N agents to some common good, with N large. We investigate a relaxation of this problem, obtained by randomization. The relaxation gap is proved to converge to zeros as N goes to infinity, independently of the dimension of the aggregate. We propose a stochastic method to construct an approximate minimizer of the original problem, given an approximate solution of the randomized problem. McDiarmid's concentration inequality is used to quantify the probability of success of the method. We consider the Frank-Wolfe (FW) algorithm for the resolution of the randomized problem. Each iteration of the algorithm requires to solve a subproblem which can be decomposed into N independent optimization problems. A sublinear convergence rate is obtained for the FW algorithm. In order to handle the memory overflow problem possibly caused by the FW algorithm, we propose a stochastic Frank-Wolfe (SFW) algorithm, which ensures the convergence in both expectation and probability senses. Numerical experiments on a mixed-integer quadratic program illustrate the efficiency of the method. (10.1137/22M1488892)
  DOI : 10.1137/22M1488892
• Approximation and simulation of reflected Backward Stochastic Differential Equations, applications in Finance
  
  • Wang Wanqing
  , 2023. In this thesis, one concentrates on solving the reflected backward stochastic differential equations via penalization approach and its applications in finance. A complete non-asymptotic convergence result is investigated around penalized BSDE. The rate of penalized solution converging to reflected solution is presented in the first place and it follows by the rate of discrete penalized solution converging to the continuous penalized one. For solving numerically PBSDEs, one provides an implicit scheme using least-squares regression Monte-Carlo method. The non-asymptotic error analysis is deduced for this numerical scheme, in which both linear and non-linear least-squares regression are considered as optimization method. The thesis is completed by an application of RBSDEs on American Put/Call options in the non-linear market. The change of num'eraire is investigated on reflected diffusions. One finds, as in the perfect market, the pricing equivalence between an American Put option and an American Call option is achieved by exchanging the interest rates with dividend rates, and swapping the spot price with the strike price, in a imperfect market with 2 interest rates and 2 dividend rates.
• Critical trees are neither too short nor too fat
  
  • Addario-Berry Louigi
  • Donderwinkel Serte
  • Kortchemski Igor
  Annales Henri Lebesgue, UFR de Mathématiques - IRMAR, 2023, 8, pp.113-149. We establish lower tail bounds for the height, and upper tail bounds for the width, of critical size-conditioned Bienaym\'e trees. Our bounds are optimal at this level of generality. We also obtain precise asymptotics for offspring distributions within the domain of attraction of a Cauchy distribution, under a local regularity assumption. Finally, we pose some questions on the possible asymptotic behaviours of the height and width of critical size-conditioned Bienaym\'e trees. (10.5802/ahl.231)
  DOI : 10.5802/ahl.231
• Monotone solutions for mean field games master equations : continuous state space and common noise
  
  • Bertucci Charles
  Communications in Partial Differential Equations, Taylor & Francis, 2023, pp.1-41. We present the notion of monotone solution of mean field games master equations in the case of a continuous state space. We establish the existence, uniqueness and stability of such solutions under standard assumptions. This notion allows us to work with solutions which are merely continuous in the measure argument, in the case of first order master equations. We study several structures of common noises, in particular ones in which common jumps (or aggregate shocks) can happen randomly, and ones in which the correlation of randomness is carried by an additional parameter. (10.1080/03605302.2023.2276564)
  DOI : 10.1080/03605302.2023.2276564
• Numerical simulations of granular dam break: comparison between discrete element, Navier-Stokes and thin-layer models
  
  • Martin Hugo A.
  • Peruzzetto Marc
  • Viroulet Sylvain
  • Mangeney Anne
  • Lagrée Pierre-Yves
  • Popinet Stéphane
  • Maury Bertrand
  • Lefebvre-Lepot Aline
  • Maday Yvon
  • Bouchut François
  Physical Review E, American Physical Society (APS), 2023, 108 (5), pp.054902-054927. Granular flows occur in various contexts including laboratory experiments, industrial processes and natural geophysical flows. In order to investigate their dynamics, different kinds of physically-based models have been developed. These models can be characterized by the length scale at which dynamic processes are described. Discrete models use a microscopic scale to model individually each grain, Navier-Stokes models use a mesoscopic scale to consider elementary volumes of grains, and thin-layer models use a macroscopic scale to model the dynamics of elementary columns of fluids. In each case, the derivation of the associated equations is well known. However, few studies focus on the extent to which these modeling solutions yield mutually coherent results. In this work, we compare the simulations of a granular dam break on a horizontal or inclined plane, for the discrete model COCD, the Navier-Stokes model Basilisk, and the thin-layer model SHALTOP. We show that, although all three models allow reproducing the temporal evolution of the free surface in the horizontal case (except for SHALTOP at the initiation), the modeled flow dynamics are significantly different, and in particular during the stopping phase. The pressures measured at the flow's bottom are in relatively good agreement, but significant variations are obtained with the COCD model due to complex and fast-varying granular lattices. Similar conclusions are drawn using the same rheological parameters to model a dam break on an inclined plane. This comparison exercise is essential for assessing the limits and uncertainties of granular flow modeling. (10.1103/PhysRevE.108.054902)
  DOI : 10.1103/PhysRevE.108.054902
• Fourth-order moments analysis for partially coherent electromagnetic beams in random media
  
  • Garnier Josselin
  • Sølna Knut
  Waves in Random and Complex Media, Taylor & Francis, 2023, 33 (5-6), pp.1346-1365. (10.1080/17455030.2022.2096272)
  DOI : 10.1080/17455030.2022.2096272
• A mesh-independent method for second-order potential mean field games
  
  • Liu Kang
  • Pfeiffer Laurent
  , 2023. This article investigates the convergence of the Generalized Frank-Wolfe (GFW) algorithm for the resolution of potential and convex second-order mean field games. More specifically, the impact of the discretization of the mean-field-game system on the effectiveness of the GFW algorithm is analyzed. The article focuses on the theta-scheme introduced by the authors in a previous study. A sublinear and a linear rate of convergence are obtained, for two different choices of stepsizes. These rates have the mesh-independence property: the underlying convergence constants are independent of the discretization parameters.
• Models of boundary condition for the Boltzmann equation based on kinetic approach
  
  • Kosuge Shingo
  • Aoki Kazuo
  • Giovangigli Vincent
  , 2023. Models of the boundary conditions for the Boltzmann equation were constructed systematically in a recent paper by the present authors (K. Aoki et al., in: Phys. Rev. E 106:035306, 2022) using an iteration scheme for the half-space problem of a linear kinetic equation describing the behavior of gas and physisorbed molecules in a thin layer adjacent to a solid surface (physisorbate layer). In the present paper, special attention is focused on the model based on the second iteration that was only touched on in the above reference. The model is presented in an explicit form, and its properties are investigated numerically. In particular, it is shown by the comparison with the numerical solution of the half-space problem that the model is a significant improvement compared with the model based on the first iteration and is accurate and useful.
• Statistical dynamics of a hard sphere gas: fluctuating Boltzmann equation and large deviations
  
  • Bodineau Thierry
  • Gallagher Isabelle
  • Saint-Raymond Laure
  • Sergio Simonella
  Annals of Mathematics, Princeton University, Department of Mathematics, 2023, 198 (3). (10.4007/annals.2023.198.3.3)
  DOI : 10.4007/annals.2023.198.3.3
• Finite Volume Approximations for Non-Linear Parabolic Problems with Stochastic Forcing
  
  • Bauzet Caroline
  • Nabet Flore
  • Schmitz Kerstin
  • Zimmermann Aleksandra
  , 2023, 432, pp.157-166. We propose a two-point flux approximation finite-volume scheme for a stochastic non-linear parabolic equation with a multiplicative noise. The time discretization is implicit except for the stochastic noise term in order to be compatible with stochastic integration in the sense of Itô. We show existence and uniqueness of solutions to the scheme and the appropriate measurability for stochastic integration follows from the uniqueness of approximate solutions. (10.1007/978-3-031-40864-9_10)
  DOI : 10.1007/978-3-031-40864-9_10