Publications

2026

• Surrogate-Based Strategies for Accelerated Bayesian Calibration of Computer Codes With Complete Maximum a Posteriori Estimation of Model Error
  
  • Kahol Omar
  • Le Maître Olivier
  • Congedo Pietro M
  • Denimal Goy Enora
  Journal of Mechanical Design, American Society of Mechanical Engineers, 2026, 148 (9). The calibration of a computer code is a process that reduces the uncertainty of model parameters by matching the code’s predictions to experimental observations of a quantity of interest. A more faithful representation of the global uncertainty is achieved by including a model error term, a discrepancy between the physical system and the computer code. The recently proposed complete maximum a posteriori (CMP) method is able to infer both a posterior distribution of the model parameters and a model error term, improving upon traditional frameworks. On the other hand, the CMP method relies on an optimization step which increases the cost of complex calibration problems. This article proposes a surrogate-based strategy to reduce the computational cost of the CMP method. First, we build a surrogate model of the model error’s hyperparameters using Gaussian processes. Second, we propose an iterative algorithm that builds a training set in regions of the parameter space that are more likely, reducing the overall cost of the algorithm and improving the accuracy of the surrogate. The proposed strategy is applied to four different examples, including a design problem in solid mechanics and a complex test case in fluid dynamics. The results show that the proposed strategy is able to accelerate the CMP method without losing accuracy, making it suitable for real-world applications. In an industrial application, we demonstrate a speed-up of almost 100 compared to the original CMP method. (10.1115/1.4071071)
  DOI : 10.1115/1.4071071
• On the simulation of extreme events with neural networks
  
  • Allouche Michaël
  • Girard Stéphane
  • Gobet Emmanuel
  , 2026. This article aims at investigating the use of generative methods based on neural networks to simulate extreme events. Although very popular, these methods are mainly invoked in empirical works. Therefore, providing theoretical guidelines for using such models in extreme values context is of primal importance. To this end, we propose an overview of most recent generative methods dedicated to extremes, giving some theoretical and practical tips on their tail behaviour thanks to both extreme-value and copula tools.
• A characterization of Generalized functions of Bounded Deformation
  
  • Chambolle Antonin
  • Crismale Vito
  Journal of Functional Analysis, Elsevier, 2026, 290 (9), pp.111391. <div><p>We show that Dal Maso's GBD space, introduced for tackling crack growth in linearized elasticity, can be defined by simple conditions in a finite number of directions of slicing.</p></div> (10.1016/j.jfa.2026.111391)
  DOI : 10.1016/j.jfa.2026.111391
• Numerical approximation of ergodic BSDEs using non linear Feynman-Kac formulas
  
  • Gobet Emmanuel
  • Richou Adrien
  • Szpruch Lukasz
  Stochastic Processes and their Applications, Elsevier, 2026, 195, pp.104871. 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. (10.1016/j.spa.2026.104871)
  DOI : 10.1016/j.spa.2026.104871
• On phase retrieval for continuous and discrete Fourier transforms
  
  • Novikov Roman
  • Xu Tianli
  , 2026. We continue studies on phase retrieval for continuous and discrete Fourier transforms in multidimensions. Using finite difference operators, we give a large class of unexpected examples of non-uniqueness for this problem, including examples with the sparsity condition. A prototype of this construction in the continuous case is given in the work Novikov, Xu (JFAA, 2026), using linear differential operators. The construction of the present work also yields a large class of non-trivial Pauli partners, i.e., different functions with the same intensities in both configuration and Fourier domains. Besides, our construction yields examples that solve an old open question in phase retrieval with background information arising in many areas including Fourier holography.
• A Rank-Based Reward between a Principal and a Field of Agents: Application to Energy Savings
  
  • Alasseur Clémence
  • Bayraktar Erhan
  • Dumitrescu Roxana
  • Jacquet Quentin
  , 2026. In this paper, we consider the problem of a Principal aiming at designing a reward function for a population of heterogeneous agents. We construct an incentive based on the ranking of the agents, so that a competition among the latter is initiated. We place ourselves in the limit setting of mean-field type interactions and prove the existence and uniqueness of the equilibrium distribution for a given reward, for which we can find an explicit representation. Focusing first on the homogeneous setting, we characterize the optimal reward function using a convex reformulation of the problem and provide an interpretation of its behaviour. We then show that this characterization still holds for a sub-class of heterogeneous populations. For the general case, we propose a convergent numerical method which fully exploits the characterization of the mean-field equilibrium. We develop a case study related to the French market of Energy Saving Certificates based on the use of realistic data, which shows that the ranking system allows to achieve the sobriety target imposed by the regulation.
• Malliavin calculus for signatures with applications to finance
  
  • Abi Jaber Eduardo
  • Rey Clément
  • Sotnikov Dimitri
  , 2026. Malliavin calculus is a powerful and general framework for the analysis of square-integrable random variables, but it often suffers from a lack of tractability and explicit representations. To address this limitation, we focus on a subclass of random variables given by finite linear combinations of time-extended Brownian motion signatures. The class remains rich due to the universal approximation properties of signatures. Leveraging the algebraic structure of signatures, we first derive explicit formulas for the Malliavin derivative of signatures of continuous Itô processes. As a consequence, we obtain closed-form expressions for the Clark-Ocone representation, the Ornstein-Uhlenbeck semigroup and its generator, as well as the integration-by-parts formula within the class of Brownian signature variables. These results provide purely algebraic formulations of the classical operators of Malliavin calculus. As an application, we compute Greeks for general path-dependent options under signature volatility models, and numerically compare different choices of Malliavin weights.
• Numerical approximation of Markovian BSDEs in infinite horizon and elliptic PDEs
  
  • Gobet Emmanuel
  • Richou Adrien
  • Shardul Charu
  , 2026. <div><p>We study backward stochastic differential equations (BSDEs) in infinite horizon and design efficient numerical schemes for solving them. We establish a probabilistic representation of the solution of the BSDE using Malliavin derivative and prove results for contraction of a Picard scheme. We develop three numerical schemes, of which the first two are based on a fixed point argument using contraction, imposing additional assumptions compared to what is needed for existence and uniqueness of the solution. The first scheme is a space grid based approximation where we establish tight numerical error bounds using a growth truncation argument; it performs well in low dimensions but computational times increase exponentially with dimension. The second scheme uses neural network approximations for which we have proved a convergence result. Using neural networks alleviates the curse of dimensionality, giving good accuracy in very high dimensions. The third scheme also uses neural networks but does not rely on contraction arguments, showcasing good performance even for larger z-Lipschitz dependence outside the domain of contraction.</p></div>
• Asymptotic Stability of Equilibrium States for Cohesive Fluids
  
  • Giovangigli Vincent
  , 2026. We investigate asymptotic stability of constant equilibrium states for compressible nonisothermal cohesive fluids also termed capillary fluids or diffuse interface fluids. The density gradient is added as an extra variable and the augmented system of equations is recast into a normal form with symmetric transport first order terms, symmetric dissipative second order terms and antisymmetric cohesive second order terms. Global existence and asymptotic stability of constant equilibrium states are established by using new dissipative conditions for such augmented hyperbolic-parabolic-dispersive systems of equations. Decay estimates are obtained in all spatial dimensions by using the augmented formulation as well as estimates in Fourier spaces.
• Erratum for Symmetrization and Local Existence of Strong Solutions for Diffuse Interface Fluid Models
  
  • Giovangigli Vincent
  , 2023. An inaccurate assumption has been identified in Vincent Giovangigli, Yoann Le Calvez and Flore Nabet ``Symmetrization and Local Existence of Strong Solutions for Diffuse Interface Fluid Models'', J.~Math.~Fluid Dyn., Volume 25, 82, (2023),https://doi.org/10.1007/s00021-023-00825-4 and is corrected here.
• Growing random planar network with oriented branching and fusion
  
  • Bansaye Vincent
  • Raoul Gaël
  • Tomasevic Milica
  , 2026. We consider a growing planar network where a tip grows at constant speed, branches at constant rate and inactivates when it meets a branch already created. We only consider here orthogonal branching occurring always in the same direction. This yields a spatial branching property to the growing network. The connected components of the network then form a branching process of rectangles with double immigration. Using a spine approach for a typical rectangle and coupling arguments, the study is boiled down to a one dimensional stick breaking model with aging. We can then prove long time convergence of empirical measure of the family of rectangles after polynomial rescaling. The limiting distribution and speed of convergence can be explicitly described. The proofs also rely on the description of common ancestor of rectangles in the branching structure with double immigration.
• From stochastic individual-based models to free-boundary Hamilton-Jacobi equations
  
  • Champagnat Nicolas
  • Méléard Sylvie
  • Mirrahimi Sepideh
  • Tran Chi
  , 2026. We study a stochastic branching model for a population structured by a quantitative phenotypic trait and subject to births, deaths, and mutations. In a regime of large population and small mutations, and in logarithmic scales of size and time, we derive a certain class of free boundary Hamilton-Jacobi equations with state constraints from the stochastic individual-based system. This goes beyond the classical Hamilton-Jacobi equations obtained from deterministic models by taking into account the possible extinction of the system in certain regions of the trait space. The proof is obtained by combining methods for the analysis of Hamilton-Jacobi equations with probabilistic tools from the theory of large deviations and branching processes.
• Acute Smurf mortality and age-dependence in a two-phase ageing model.
  
  • Breuil Luce
  • Doumic Marie
  • Kaakaï Sarah
  • Rera Michaël
  , 2026. Ageing is traditionally conceived as a continuous process of progressive physiological decline. However, recent evidence across species suggests that ageing may instead proceed through distinct phases. Using state-of-the-art statistical methods, we develop a rigorous analysis of longitudinal survival data from 1,159 individually tracked female Drosophila melanogaster. This data-driven analysis leads us to introduce a new parametric model of transition rates within the two-phase ageing framework. Flies were monitored using the Smurf assay, which detects increased intestinal permeability through leakage of an ingested blue dye, and is a strong biological marker of ageing. The Smurf phenotype identifies a sharp transition from a non-Smurf state to a Smurf state that precedes death. Our results yield three key findings. First, the Smurf transition rate follows a Gompertz-Makeham law, increasing exponentially with age. Second, contrary to previous constant-rate assumptions, newly transitioned Smurf flies exhibit remarkably high mortality - approximately 40% die within 24 hours - followed by an exponential decline in death rate that stabilises to a lower constant baseline. Third, we identified a mild but statistically significant negative dependence between time spent non-Smurf and subsequent Smurf lifespan. Our best-fit model captures a potential bimodal nature of mortality curves using simple, biologically interpretable functions. Validation using data from two mouse strains confirms the broader applicability of this framework. These results establish a quantitative foundation for the two-phase ageing paradigm and highlight a critical period of vulnerability immediately following the physiological transition to frailty. (10.64898/2026.02.18.706552)
  DOI : 10.64898/2026.02.18.706552
• Scattering and inverse scattering for multipoint potentials at high energies
  
  • Kuo Pei-Cheng
  • Novikov Roman
  , 2026. We consider the Schrödinger equation with a multipoint potential of Bethe-Peierls-Thomas-Fermi type. For this singular potential, we develop scattering and inverse scattering at high energies. In particular, in this framework, our results include analogs of the "regular" Born-Faddeev formula for the scattering amplitude and analogs of related "regular" inverse scattering reconstructions at high energies. Related results for scattering solutions at high energies are also presented.
• Pareto Set Characterization in Constrained Multiobjective Optimization and the COBI Problem Generator
  
  • Auger Anne
  • Brockhoff Dimo
  • Opravš Luka
  • Tušar Tea
  , 2026. Benchmark problems play a central role in assessing the performance of numerical optimization algorithms. However, many existing constrained multiobjective optimization benchmark problems rely on overly restricted constructions or lack formal analysis of their optimal solution sets, limiting their relevance for systematic algorithm evaluation. In this work, we introduce a class of analytically tractable constrained multiobjective optimization problems whose Pareto sets can be formally characterized. The construction is based on convex-quadratic functions with positive definite Hessians, combined through multipeak formulations in which each objective is defined as the minimum over several convex-quadratic components. This approach preserves analytical structure while enabling multimodality (non-convexity), ill-conditioning and non-separability. The constraints are built as sublevel sets of multipeak functions giving rise to problems with potentially disconnected feasible regions. Building on these results, we propose COBI, a scalable generator of constrained bi-objective test problems designed for benchmarking derivative-free optimization algorithms. We provide a reference Python implementation that enables straightforward integration of COBI instances into benchmarking workflows.
• Error-Based mesh selection for efficient numerical simulations with variable parameters
  
  • Dornier Hugo
  • Le Maître Olivier P
  • Congedo Pietro Marco
  • Salah El Din Itham
  • Marty Julien
  • Bourasseau Sébastien
  Computers and Fluids, Elsevier, 2026, 313. Advanced numerical simulations often depend on mesh refinement techniques to manage discretization errors in complex models and reduce computational costs. This work concentrates on Adaptive Mesh Refinement (AMR) for steady-state solutions, which uses error estimators to iteratively refine the mesh locally and gradually tailor it to the solution. AMR requires evaluating the solution across a series of meshes. When solving the model for multiple operating conditions, such as in uncertainty quantification studies, full systematic adaptation can cause significant computational overhead. To mitigate this, the Error-based Mesh Selection (EMS) method is introduced to decrease the cost of adaptation. For each operating condition, EMS seeks to choose, from a library of pre-adapted meshes, the one that minimizes the discretization error. A key feature of this approach is the use of Gaussian Process models to predict the solution errors for each mesh in the library. These error models are built solely from the library's meshes and their solutions, using restriction errors as proxies for discretization errors, thereby avoiding additional model evaluations. The EMS method is tested on an analytical shock problem and a supersonic scramjet configuration, showing near-optimal mesh selection. The influence of library size on the resulting error level is also examined. (10.1016/j.compfluid.2026.107081)
  DOI : 10.1016/j.compfluid.2026.107081
• Molecular Scattering Distributions under New Boundary Conditions for the Boltzmann Equation
  
  • Kosuge Shingo
  • Aoki Kazuo
  • Giovangigli Vincent
  • Golse François
  , 2026. The scattering properties of kinetic boundary conditions for the Boltzmann equation recently proposed by Aoki et al. [see, e.g., Phys. Rev. E 106, 035306 (2022)] are investigated. Scattering patterns of reflected molecules are obtained for molecular-beam and Maxwellian incident distributions, and representative numerical results are presented.
• Do you precondition on the left or on the right?
  
  • Spillane Nicole
  • Matalon Pierre
  • Szyld Daniel B
  , 2026. This work is a follow-up to a poster that was presented at the DD29 conference. Participants were asked the question: “Do you precondition on the left or on the right?”. Here we report on the results of this social experiment. We also provide context on left, right and split preconditioning, share our literature review on the topic, and analyze some of the finer points. Two examples illustrate that convergence bounds can sometimes lead to misleading conclusions.
• Accelerating Nash learning from human feedback via Mirror Prox
  
  • Tiapkin Daniil
  • Calandriello Daniele
  • Belomestny Denis
  • Moulines Eric
  • Naumov Alexey
  • Rasul Kashif
  • Valko Michal
  • Ménard Pierre
  , 2025.
• A class of optimal virtual fields for inverse problems in elasticity
  
  • Chibli Nagham
  • Genet Martin
  • Imperiale Sébastien
  , 2026. This work addresses the identification of nonhomogeneous constitutive parameters from full-field measurements in both linear and nonlinear elasticity, considering incompressible as well as compressible materials. The inverse identification procedure relies on the Virtual Fields Method (VFM), which is based on the principle of virtual work with specifically chosen virtual fields. We propose an optimal class of virtual fields, designed to optimize the reconstruction stability with respect to measurement noise. A series of numerical experiments illustrate the effectiveness of the proposed approach. The method exhibits moderate sensitivity to measurement noise and remains robust even when the boundary conditions are only partially known.
• Autoregressive Multiplier Bootstrap for In-situ Error Estimation and Quality Monitoring of Finite Time Averages in Turbulent Flow Simulations
  
  • Papagiannis Christos
  • Balarac Guillaume
  • Congedo Pietro Marco
  • Le Maître Olivier P
  Computer Methods in Applied Mechanics and Engineering, Elsevier, 2026, 452, pp.118664. In Computational Fluid Dynamics (CFD), and particularly within Direct Numerical Simulation (DNS) and Large Eddy Simulation (LES), the computational cost is largely dictated by the effort required to obtain statistically converged quantities such as time-averaged fields and higher-order moments. Despite the importance of accurately quantifying statistical uncertainty in unsteady simulations, no continuous and cost-effective, on-line method currently exists for monitoring the convergence quality of such statistics during runtime. This work introduces a novel, fully on-line bootstrapping approach to estimate the variance of finite-time averages without requiring the estimation of the flows Auto-Correlation Function (ACF). Unlike existing methods that rely on ACF estimation, which are often impractical due to excessive storage demands in large-scale simulations, or require off-line processing or a priori modeling assumptions, our method operates entirely during the simulation and incurs minimal overhead. The proposed technique employs a recursive update of bootstrap replicates of the time average, using correlated random weights generated via an autoregressive model. This formulation is computationally efficient: the update cost scales linearly with the number of bootstrap replicates and the dimensionality of the flow field, and the autoregressive model is inexpensive to evaluate. The method only requires storage of a small number of fields, making it suitable for large-scale CFD applications. We demonstrate the effectiveness of the approach on synthetic data from the Ornstein-Uhlenbeck process and on two canonical LES cases: a turbulent pipe flow and a round jet. We further discuss the methods applicability to simulations with non-uniform time stepping, highlighting its flexibility and robustness. (10.1016/j.cma.2025.118664)
  DOI : 10.1016/j.cma.2025.118664
• Semi-discrete convergence analysis of a numerical method for waves in nearly-incompressible media with spectral finite elements
  
  • Ramiche Zineb
  • Imperiale Sébastien
  , 2026. In this work, we present a semi-discrete convergence analysis of a high-order space discretization approach for the computation of elastic field propagation in a nearly incompressible medium. Our approach relies on the use of high-order continuous spectral finite elements with mass-lumping. We present an approach that is valid for full hexahedral and quadrilateral meshes, where the elastic field is sought in the space of Q_k continuous finite elements and the pressure in Q_k-2 discontinuous finite elements. We further provide a proof of the inf-sup stability of the finite element discretization. This allows us to carry out error estimates for the semi-discrete problem in space, accounting in particular for quadrature errors.
• A stability result on optimal Skorokhod embedding
  
  • Guo Gaoyue
  , 2017. Motivated by the model- independent pricing of derivatives calibrated to the real market, we consider an optimization problem similar to the optimal Skorokhod embedding problem, where the embedded Brownian motion needs only to reproduce a finite number of prices of Vanilla options. We derive in this paper the corresponding dualities and the geometric characterization of optimizers. Then we show a stability result, i.e. when more and more Vanilla options are given, the optimization problem converges to an optimal Skorokhod embedding problem, which constitutes the basis of the numerical computation in practice. In addition, by means of different metrics on the space of probability measures, a convergence rate analysis is provided under suitable conditions.
• Constructive existence proofs and stability of stationary solutions to parabolic PDEs using Gegenbauer polynomials
  
  • Breden Maxime
  • Cadiot Matthieu
  • Zurek Antoine
  , 2026. In this paper, we present a computer-assisted framework for constructive proofs of existence for stationary solutions to one-dimensional parabolic PDEs and the rigorous determination of their linear stability. By expanding solutions in Gegenbauer polynomials, we first develop a general approach for boundary value problems (BVPs), corresponding to the stationary part of the PDE. This yields a computationally efficient sparse structure for both differential and multiplication operators. By deriving sharp, explicit and quantitative estimates for the inverse of differential operators, we implement a Newton-Kantorovich approach. Specifically, given a numerical approximation <span>u&#772;</span>, we prove the existence of a true stationary solution u within a small, rigorously quantified neighborhood of <span>u&#772;</span>. A key advantage of this approach is that the sharp control over the defect u -<span>u&#772;</span>, integrated with the spectral properties of the Gegenbauer basis, enables an accurate enclosure of the linearization's spectrum around u. This allows for a definitive conclusion regarding the (in)stability of the verified solution, which is the main contribution of the paper. We demonstrate the efficacy of this method through several applications, capturing both stable and unstable equilibrium states.
• Invasion dynamics for a quasi-critical birth-death process
  
  • Bansaye Vincent
  • Belmabrouk Nadia
  • Erny Xavier
  • Girel Simon
  , 2026. We study the dynamics of invasion of populations which enjoy positive density dependent effect. We start with a single individual and consider a single type birth and death process. The initial individual growth rate is zero and it increases. We prove that the probability that the population reaches macroscopic levels decreases as $\sqrt{K}$, where $K$ is the scaling parameter. We also describe the associated trajectories and show that invasion can be split into three time periods. First, the process needs to escape the neighborhood of $0$, and conditioning on survival, it grows linearly until the order $\sqrt{K}$. The renormalized process is approximated by a diffusion, as for critical branching process, with an additional drift term coming from cooperation. Second, in intermediate scale $\sqrt{K}$, we observe another diffusion, without conditioning. Finally, this diffusion can be approximated by classical fluid limit corresponding to macroscopic approximation by an ODE. The proof of the first phase involves change of probability and characterization of uniform integrability of martingales, while the two other phases need uniform approximations on polynomial time scales.