Publications

2025

• Tight analyses of first-order methods with error feedback
  
  • Thomsen Daniel Berg
  • Taylor Adrien
  • Dieuleveut Aymeric
  , 2025. Communication between agents often constitutes a major computational bottleneck in distributed learning. One of the most common mitigation strategies is to compress the information exchanged, thereby reducing communication overhead. To counteract the degradation in convergence associated with compressed communication, error feedback schemes -- most notably $\mathrm{EF}$ and $\mathrm{EF}^{21}$ -- were introduced. In this work, we provide a tight analysis of both of these methods. Specifically, we find the Lyapunov function that yields the best possible convergence rate for each method -- with matching lower bounds. This principled approach yields sharp performance guarantees and enables a rigorous, apples-to-apples comparison between $\mathrm{EF}$, $\mathrm{EF}^{21}$, and compressed gradient descent. Our analysis is carried out in the simplified single-agent setting, which allows for clean theoretical insights and fair comparison of the underlying mechanisms. (10.48550/arXiv.2506.05271)
  DOI : 10.48550/arXiv.2506.05271
• Optimization and Generalization of Linear Models with Power-Law Spectrum
  
  • Velikanov Maksim
  , 2025. This thesis develops a phenomenological approach to linear models with power-law spectrum, compressing data and architecture details into the shape of the spectrum while keeping training dynamics analytically tractable. In Power-law spectrum (Ch. 3), we derive the power-law asymptotics of the eigenvalues and coefficients of the infinite-width neural tangent kernel (NTK) for ReLU MLPs on generic low-dimensional data; the power laws originate from a diagonal NTK singularity caused by activation non-smoothness, and the NTK eigenfunctions connect to Fourier harmonics. In Non-stochastic optimization (Ch. 4), to characterize problems with power-law spectrum, we introduce a new condition based on the cumulative spectral measure that is tighter than the classical source condition, and is associated with a procedure yielding accelerated algorithms. This leads to matching upper and lower bounds for six first-order algorithms: plain gradient descent (GD), Heavy Ball (HB), their predefined-schedule variants, Steepest Descent, and Conjugate Gradients. In Stochastic optimization (Ch. 5), we develop two approaches—generating functions and propagator expansions—to derive precise loss asymptotics and stability conditions. We find that the full-batch acceleration strategy taking the momentum parameter β → 1 becomes unstable. To overcome this instability, we consider a family of algorithms that linearly combine M velocity vectors with the gradient and identify a region of parameter space allowing for stable acceleration. In Generalization error (Ch. 6), we express the test loss, for both noisy and noiseless targets, as a quadratic functional of the algorithm’s spectral profile h(λ), yielding a unified description of classical algorithms such as kernel ridge regression and gradient flow, and beyond. Our observations include (i) localization of the loss on a certain spectral scale, (ii) reduction to a simple universal functional for noisy observations, and (iii) an overlearning transition for sufficiently hard targets.
• Graph Alignment via Birkhoff Relaxation
  
  • Varma Sushil Mahavir
  • Waldspurger Irène
  • Massoulié Laurent
  , 2025. We consider the graph alignment problem, wherein the objective is to find a vertex correspondence between two graphs that maximizes the edge overlap. The graph alignment problem is an instance of the quadratic assignment problem (QAP), known to be NP-hard in the worst case even to approximately solve. In this paper, we analyze Birkhoff relaxation, a tight convex relaxation of QAP, and present theoretical guarantees on its performance when the inputs follow the Gaussian Wigner Model. More specifically, the weighted adjacency matrices are correlated Gaussian Orthogonal Ensemble with correlation $\frac{1}{1+\sigma^2}$ .
• Thresholds for sensitive optimality and Blackwell optimality in stochastic games
  
  • Gaubert Stéphane
  • Grand-Clément Julien
  • Katz Ricardo D.
  , 2025. We investigate refinements of the mean-payoff criterion in two-player zero-sum perfect-information stochastic games. A strategy is Blackwell optimal if it is optimal in the discounted game for all discount factors sufficiently close to $1$. The notion of $d$-sensitive optimality interpolates between mean-payoff optimality (corresponding to the case $d=-1$) and Blackwell optimality ($d=+\infty$). The Blackwell threshold $α_{\sf Bw} \in [0,1[$ is the discount factor above which all optimal strategies in the discounted game are guaranteed to be Blackwell optimal. The $d$-sensitive threshold $α_{\sf d} \in [0,1[$ is defined analogously. Bounding $α_{\sf Bw}$ and $α_{\sf d}$ are fundamental problems in algorithmic game theory, since these thresholds control the complexity for computing Blackwell and $d$-sensitive optimal strategies, by reduction to discounted games which can be solved in $O\left((1-α)^{-1}\right)$ iterations. We provide the first bounds on the $d$-sensitive threshold $α_{\sf d}$ beyond the case $d=-1$, and we establish improved bounds for the Blackwell threshold $α_{\sf Bw}$. This is achieved by leveraging separation bounds on algebraic numbers, relying on Lagrange bounds and more advanced techniques based on Mahler measures and multiplicity theorems.
• The quest for the GRAph Level autoEncoder (GRALE)
  
  • Krzakala Paul
  • Melo Gabriel
  • Laclau Charlotte
  • d'Alché-Buc Florence
  • Flamary Rémi
  , 2025. Although graph-based learning has attracted a lot of attention, graph representation learning is still a challenging task whose resolution may impact key application fields such as chemistry or biology. To this end, we introduce GRALE, a novel graph autoencoder that encodes and decodes graphs of varying sizes into a shared embedding space. GRALE is trained using an Optimal Transport-inspired loss that compares the original and reconstructed graphs and leverages a differentiable node matching module, which is trained jointly with the encoder and decoder. The proposed attention-based architecture relies on Evoformer, the core component of AlphaFold, which we extend to support both graph encoding and decoding. We show, in numerical experiments on simulated and molecular data, that GRALE enables a highly general form of pre-training, applicable to a wide range of downstream tasks, from classification and regression to more complex tasks such as graph interpolation, editing, matching, and prediction.
• A holographic non-uniqueness for the Helmholtz equation
  
  • Novikov Roman
  • Sivkin Vladimir
  , 2025. We consider a plane wave, a radiation solution, and the sum of these solutions (total solution) for the Helmholtz equation in $\mathbb{R}^d$ , $d \geq 2$, outside the origin. We consider a sphere $S$ centered at the origin in $\mathbb{R}^d$ . We show that the radiation solution on $S$ is not uniquely determined by the intensity of the total solution on $S$. Extensions of this result to the case of other surfaces in place of $S$ are also mentioned. Our construction involves and develops technique of scattering on obstacles with Dirichlet boundary condition.
• Relu and softplus neural nets as zero-sum turn-based games
  
  • Gaubert Stephane
  • Vlassopoulos Yiannis
  , 2025. We show that the output of a ReLU neural network can be interpreted as the value of a zero-sum, turn-based, stopping game, which we call the ReLU net game. The game runs in the direction opposite to that of the network, and the input of the network serves as the terminal reward of the game. In fact, evaluating the network is the same as running the Shapley-Bellman backward recursion for the value of the game. Using the expression of the value of the game as an expected total payoff with respect to the path measure induced by the transition probabilities and a pair of optimal policies, we derive a discrete Feynman-Kac-type path-integral formula for the network output. This game-theoretic representation can be used to derive bounds on the output from bounds on the input, leveraging the monotonicity of Shapley operators, and to verify robustness properties using policies as certificates. Moreover, training the neural network becomes an inverse game problem: given pairs of terminal rewards and corresponding values, one seeks transition probabilities and rewards of a game that reproduces them. Finally, we show that a similar approach applies to neural networks with Softplus activation functions, where the ReLU net game is replaced by its entropic regularization.
• Random Lévy Looptrees and Lévy Maps
  
  • Kortchemski Igor
  • Marzouk Cyril
  The Annals of Applied Probability, Institute of Mathematical Statistics (IMS), 2025, 35 (6). Motivated by scaling limits of random planar maps in random geometry, we introduce and study Lévy looptrees and Lévy maps. They are defined using excursions of general Lévy processes with no negative jump and extend the known stable looptrees and stable maps, associated with stable processes. We compute in particular their fractal dimensions in terms of the upper and lower Blumenthal–Getoor exponents of the coding Lévy process. In a second part, we consider excursions of stable processes with a drift and prove that the corresponding looptrees interpolate continuously as the drift varies from $-\infty$ to $\infty$, or as the self-similarity index varies from $1$ to $2$, between a circle and the Brownian tree, whereas the corresponding maps interpolate between the Brownian tree and the Brownian sphere. (10.1214/25-AAP2212)
  DOI : 10.1214/25-AAP2212
• The Nullstellensatz and Positivstellensatz for Sparse Tropical Polynomial Systems
  
  • Akian Marianne
  • Béreau Antoine
  • Gaubert Stéphane
  Foundations of Computational Mathematics, Springer Verlag, 2025. Grigoriev and Podolskii (2018) have established a tropical analogue of the effective Nullstellensatz, showing that a system of tropical polynomial equations is solvable if and only if a linearized system obtained from a truncated Macaulay matrix is solvable. They provided an upper bound of the minimal admissible truncation degree, as a function of the degrees of the tropical polynomials. We establish a tropical Nullstellensatz adapted to sparse tropical polynomial systems. Our approach is inspired by a construction of Canny-Emiris (1993), refined by Sturmfels (1994). This leads to an improved bound of the truncation degree, which coincides with the classical Macaulay degree in the case of n + 1 equations in n unknowns. We also establish a tropical Positivstellensatz, allowing one to decide the inclusion of tropical basic semialgebraic sets. This allows one to reduce decision problems for tropical semi-algebraic sets to the solution of systems of tropical linear equalities and inequalities. (10.1007/s10208-025-09708-8)
  DOI : 10.1007/s10208-025-09708-8
• Optimal observer theory in manifolds - from formulations to applications
  
  • Le Ruz Gaël
  , 2025. Data assimilation is the process of optimally combining dynamical models with observational data in order to estimate the evolving state and parameters of a physical system. In the deterministic optimal observer framework, this is achieved by defining an observer: a modified version of the original dynamics, augmented with a gain term acting on the discrepancy between the observations and the model output. The optimal observer is then designed to minimize a criterion that penalizes deviations from the model dynamics and discrepancies with the observations. The aim of this thesis is to develop an optimal observer theory from a deterministic perspective, in the context of constrained physical systems. More specifically, we focus on cases where the constrained space admits a formulation as a Riemannian manifold. In the first part, we derive an exact formulation of the deterministic optimal observer on compact manifolds, extending the Mortensen observer originally defined in the Euclidean setting. Since such a formulation inherits the high computational cost and the curse of dimensionality of its Euclidean counterpart, we then develop approximation strategies that preserve the theoretical foundations of the optimal observer in Riemannian manifolds while reducing its computational burden. In particular, we propose and theoretically justify extensions of the Extended Kalman Filter (EKF) and the Unscented Kalman Filter (UKF) to the manifold setting, by approximating the value functions of the Mortensen observer with quadratic functions. We also develop an approximation of the fully space and time discretized Mortensen observer using dimension-reduction techniques, and in particular low-rank matrix representations of the associated expansion coefficients. For each proposed observer, we provide numerical illustrations on submanifolds of R³ and Cartesian products thereof. Finally, we assess the established framework in a real-world application: the monitoring of wildland fire propagation. By endowing the abstract space of shapes with a Riemannian structure, we are able to design an observer for this front-propagation problem using observational data of shape type. This work thus bridges rigorous theoretical advances in observer design with practical strategies for data assimilation in a complex, constrained physical systems. (10.70675/213d34cez5fa3z42c6z89a1z02e6e689b868)
  DOI : 10.70675/213d34cez5fa3z42c6z89a1z02e6e689b868
• Signature approach for pricing and hedging path-dependent options with frictions
  
  • Abi Jaber Eduardo
  • Hainaut Donatien
  • Motte Edouard
  , 2025. We introduce a novel signature approach for pricing and hedging path-dependent options with instantaneous and permanent market impact under a mean-quadratic variation criterion. Leveraging the expressive power of signatures, we recast an inherently nonlinear and non-Markovian stochastic control problem into a tractable form, yielding hedging strategies in (possibly infinite) linear feedback form in the time-augmented signature of the control variables, with coefficients characterized by non-standard infinite-dimensional Riccati equations on the extended tensor algebra. Numerical experiments demonstrate the effectiveness of these signature-based strategies for pricing and hedging general path-dependent payoffs in the presence of frictions. In particular, market impact naturally smooths optimal trading strategies, making low-truncated signature approximations highly accurate and robust in frictional markets, contrary to the frictionless case.
• Multi-source and test-time domain adaptation on multivariate signals using Spatio-Temporal Monge Alignment
  
  • Gnassounou Theo
  • Collas Antoine
  • Flamary Rémi
  • Lounici Karim
  • Gramfort Alexandre
  , 2024. Machine learning applications on signals such as computer vision or biomedical data often face significant challenges due to the variability that exists across hardware devices or session recordings. This variability poses a Domain Adaptation (DA) problem, as training and testing data distributions often differ. In this work, we propose Spatio-Temporal Monge Alignment (STMA) to mitigate these variabilities. This Optimal Transport (OT) based method adapts the cross-power spectrum density (cross-PSD) of multivariate signals by mapping them to the Wasserstein barycenter of source domains (multi-source DA). Predictions for new domains can be done with a filtering without the need for retraining a model with source data (test-time DA). We also study and discuss two special cases of the method, Temporal Monge Alignment (TMA) and Spatial Monge Alignment (SMA). Non-asymptotic concentration bounds are derived for the mappings estimation, which reveals a bias-plus-variance error structure with a variance decay rate of $\mathcal{O}(n_\ell^{-1/2})$ with $n_\ell$ the signal length. This theoretical guarantee demonstrates the efficiency of the proposed computational schema. Numerical experiments on multivariate biosignals and image data show that STMA leads to significant and consistent performance gains between datasets acquired with very different settings. Notably, STMA is a pre-processing step complementary to state-of-the-art deep learning methods. (10.48550/arXiv.2407.14303)
  DOI : 10.48550/arXiv.2407.14303
• PSDNORM: TEST-TIME TEMPORAL NORMALIZATION FOR DEEP LEARNING IN SLEEP STAGING
  
  • Gnassounou Theo
  • Collas Antoine
  • Flamary Rémi
  • Gramfort Alexandre
  , 2025. <div><p>Distribution shift poses a significant challenge in machine learning, particularly in biomedical applications using data collected across different subjects, institutions, and recording devices, such as sleep data. While existing normalization layers, BatchNorm, LayerNorm and InstanceNorm, help mitigate distribution shifts, when applied over the time dimension they ignore the dependencies and auto-correlation inherent to the vector coefficients they normalize. In this paper, we propose PSDNorm that leverages Monge mapping and temporal context to normalize feature maps in deep learning models for signals. Notably, the proposed method operates as a test-time domain adaptation technique, addressing distribution shifts without additional training. Evaluations with architectures based on U-Net or transformer backbones trained on 10K subjects across 10 datasets, show that PSDNorm achieves state-of-the-art performance on unseen left-out datasets while being 4-times more data-efficient than BatchNorm.</p></div> (10.48550/arXiv.2503.04582)
  DOI : 10.48550/arXiv.2503.04582
• Certificates of nonnegativity of multivariate polynomials with zero-dimensional gradient ideal and infimum attained
  
  • Bender Matías
  • Tsigaridas Elias
  • Zhu Chaoping
  , 2025. We study the problem of certifying nonnegativity of multivariate polynomials with rational coefficients, under the assumptions that the polynomial attains its infimum and its gradient ideal is zero-dimensional. We introduce a new certificate, based on Rational Univariate Representations (RUR), that is both perfectly complete, it certifies every nonnegative polynomial, and perfectly sound, it correctly identifies any polynomial that is not nonnegative. The certificate reduces the problem to checking whether a univariate polynomial admits a weighted sum-of-squares decomposition modulo an ideal defined by the RUR, thereby avoiding the radicality assumption that limits other existing approaches. We present a Monte Carlo algorithm, \sosrur, that produces such a certificate or returns a rational witness point where the polynomial takes a negative value. For a polynomial in $n$ variables of degree~$d$ and maximum coefficient bitsize~$\tau$, \sosrur runs in bit complexity $\widetilde{\mathcal{O}}_B(e^{\omega n + \mathcal{O}(\lg n)} d^{(\omega+2)n}(d+\tau))$, and the polynomials in the certificate have coefficients of bitsize at most $\widetilde{\mathcal{O}}(d^{2n+1}(d+n\tau))$, when we ignore polylogarithmic factors. If the polynomial can take negative values, then the algorithm outputs a rational witness point with coordinates of bitsize $\widetilde{\mathcal{O}}(nd^{2n+2}(d+n\tau))$. In the special case where the gradient ideal is radical, in addition to the improvement in the bit size, our certificate specializes in a representation with improved degree bounds, better by an additive factor of $d$, and lower bit complexity compared to previous results, together with an explicit validation of the RUR, which had previously been overlooked. We also consider sparse certificates, that their structure, complexity, and (bitsize) bounds depend on the Newton polytope~$Q$ of the input polynomial and not on the total degree $d$. To compute them, we rely on a Monte Carlo algorithm to obtain a sparse RUR for unmixed systems, which is of independent interest. In particular, we compute the sparse certificate in {$\widetilde{\mathcal{O}}_B( n^{6n} \, \vol(Q)^6 \, \tau)$} bit operations, where $\vol(Q)$ is the volume of $Q$. The certificate involves sparse polynomials of degree at most $2^{n+2} \, n!^2 \, \vol(Q)^2$ and coefficient bitsize {$\widetilde{\mathcal{O}}(n^{4n} \, \vol(Q)^4 \, \tau)$}. These bounds provide sharper guaranties in the presence of sparsity, as they are input sensitive. This result contributes to the ongoing effort to encode polynomial optimization problems through compact descriptions, in our case by exploiting the sparsity of the input, thereby extending the current frontier of tractability from a computational point of view. Overall, our results extend the scope of certificates modulo the gradient ideal by eliminating the radicality requirement, introduce a variant that exploits the sparsity of the input polynomial, improve existing bitsize complexity bounds by at least a factor of~$d^n$, and provide efficient, verifiable algebraic certificates of nonnegativity for polynomial optimization.
• Optimal exit from Uniswap v3 and best expected return for a liquidity provider
  
  • Agarwal Ankush
  • Gobet Emmanuel
  , 2025. We analyze the profitability of liquidity providers’ (LPs) positions in Uniswap v3 by aggregating fee income and impermanent loss within an optimal stopping framework. Our first result shows that the liquidity burn should be optimized over one range at a time, rather than simultaneously. Second, without discounting future fees, there is no finite optimal liquidity burn time and indefinite liquidity provision is optimal. In this case, we derive closed-form expressions for the value of LP positions according to different price levels of liquidity burn. Third, with a discount factor, we introduce an equivalent rate of return and demonstrate that under a Black-Scholes model with volatility σ, the optimal return is approximately 0.425·σ2 (i.e. about 10% return for 50% volatility), and it is achieved by choosing the at-the-money range of liquidity. These results provide explicit formulas and strategic insights for LPs in Uniswap v3, and complement recent works on Uniswap v2 and fee modelling by highlighting the distinct impact of concentrated liquidity.
• AdaCap: An Adaptive Contrastive Approach for Small-Data Neural Networks
  
  • Belucci Bruno
  • Lounici Karim
  • Meziani Katia
  , 2025. Neural networks struggle on small tabular datasets, where tree-based models remain dominant. We introduce Adaptive Contrastive Approach (AdaCap ), a training scheme that combines a permutationbased contrastive loss with a Tikhonov-based closed-form output mapping. Across 85 real-world regression datasets and multiple architectures, AdaCap yields consistent and statistically significant improvements in the small-sample regime, particularly for residual models. A meta-predictor trained on dataset characteristics (size, skewness, noise) accurately anticipates when AdaCap is beneficial. These results show that AdaCap acts as a targeted regularization mechanism, strengthening neural networks precisely where they are most fragile. All results and code are publicly available at https://github.com/BrunoBelucci/adacap.
• A Distributionally Robust Optimization Approach to Model Risk in Mathematical Finance
  
  • Sauldubois Nathan
  , 2025. This thesis addresses a series of problems related to model risk.In the first part, we conduct a sensitivity analysis of Wasserstein distributionally robust optimization (W-DRO) and adapted Wasserstein distributionally robust optimization (AW-DRO), in the presence of infinitely many constraints, such as conditional moment restrictions.We begin by examining W-DRO and AW-DRO formulations that incorporate martingale constraints.Building upon the literature on martingale optimal transport, we then move to the case of the martingale coupling constraint.To address this latter case, we introduce a novel family of martingale couplings, derived through an application of the Implicit Function Theorem.It turns out that this method is applicable to other constrained versions of W-DRO and AW-DRO problems.Subsequently, we investigate higher-order expansion of the W-DRO and AW-DRO problems, by proving that it can be reduce to a functional optimization problem.This reformulation enables more efficient initialization of neural networks when numerically solving both W-DRO and AW-DRO.Finally, we turn our attention to the sensitivity analysis of non-markovian stochastic control and optimal stopping problems. (10.70675/d584991cza57cz4ea2zbb74zd1e5e6e22fe3)
  DOI : 10.70675/d584991cza57cz4ea2zbb74zd1e5e6e22fe3
• New dimensional bounds for a branched transport problem
  
  • Cosenza Alessandro
  • Goldman Michael
  • Koser Melanie
  , 2024. We consider a branched transport problem with weakly imposed boundary conditions. This problem arises as a reduced model for pattern formation in type-I superconductors. For this model, it is conjectured that the dimension of the boundary measure is non-integer. We prove this conjecture in a simplified 2D setting, under the (strong) assumption of Ahlfors regularity of the irrigated measure. This work is the first rigorous proof of a singular behaviour for irrigated measures resulting from minimality.
• Gaussian process regression for non-Euclidean inputs : graphs and categorical variables
  
  • Carpintero Perez Raphaël
  , 2025. In the design phases of aircraft engine parts, numerical simulation is traditionally used to identify the right characteristics (geometry, materials, etc.) to meet desired specifications. Meta-models like Gaussian processes can achieve this objective at lower cost by learning from physics-based simulations, with the added benefit of uncertainty quantification. The goal of this thesis is to improve the predictivity of Gaussian process meta-models for non-Euclidean spaces. We propose the Sliced Wasserstein Weisfeiler-Lehman (SWWL) graph kernel combining continuous Weisfeiler-Lehman iterations and an optimal transport for inputs that are graphs with continuous node attributes. This new kernel is used to learn scalar outputs with Gaussian processes when the inputs are finite element meshes. In the context of numerical simulations, it is also customary to work with output fields defined on meshes, which are challenging to handle due to changes between input geometries in terms of both size and adjacency structure. We develop a new methodology to address such limitations by relying on the combination of regularized optimal transport, dimension reduction techniques, and the use of Gaussian processes indexed by graphs. We illustrate the efficiency of the method to solve real problems in fluid dynamics and solid mechanics with uncertainty quantification. A last axis focuses on Gaussian process regression with categorical input variables. We carry out a reproducible comparative study of existing categorical kernels with novel evaluation metrics. We also introduce a new clustering-based strategy using target encodings of categorical variables in order to identify groups of levels when they are unknown. (10.70675/77a7bcecz92acz4927zba00zab095f33e741)
  DOI : 10.70675/77a7bcecz92acz4927zba00zab095f33e741
• Path-dependent processes from signatures
  
  • Abi Jaber Eduardo
  • Gérard Louis-Amand
  • Huang Yuxing
  , 2024. We provide explicit series expansions to certain stochastic path-dependent integral equations in terms of the path signature of the time augmented driving Brownian motion. Our framework encompasses a large class of stochastic linear Volterra and delay equations and in particular the fractional Brownian motion with a Hurst index H ∈ (0, 1). Our expressions allow to disentangle an infinite dimensional Markovian structure and open the door to straightforward and simple approximation schemes, that we illustrate numerically. A key application is the derivation of explicit series representations for both conditional and unconditional moments.
• Continuous-time modeling and bootstrap for chain ladder reserving
  
  • Baradel Nicolas
  , 2024. We revisit the famous Mack's model which gives an estimate for the mean square error of prediction of the chain ladder claims reserves. We introduce a stochastic dierential equation driven by a Brownian motion to model accumulated total claims amount for the chain ladder method. Within this continuous-time framework, we propose a bootstrap technique for estimating the distribution of claims reserves. It turns out that our approach leads to inherently capturing asymmetry and non-negativity, eliminating the necessity for additional assumptions. We conclude with a case study and comparative analysis against alternative methodologies based on Mack's model. (10.1016/j.insmatheco.2025.103176)
  DOI : 10.1016/j.insmatheco.2025.103176
• From Kinetic Theory to Hyperbolic-Parabolic Fluid Systems of Equations
  
  • Giovangigli Vincent
  , 2025. Fluid systems of equations can be derived from the kinetic theory by using the Chapman-Enskog asymptotic method. We investigate the links between the hyperbolic-parabolic structure of such systems and the kinetic framework. We also discuss the mathematical structure of linear systems associated with transport property evaluation.
• Mathematical modeling of fluids from the kinetic theory
  
  • Giovangigli Vincent
  , 2025. A synthesis of mathematical fluid models derived from the kinetic theory of gases is presented. Kinetic theory yields well structured set of partial equations with entropies compatible with convection, diffusion and sources. It also yields properly structured nonlinear source terms as well as the transport coefficients. We first address multicomponent flow models including detailed transport and chemical reactions. Symmetrization, hyperbolic-parabolic structure and asymptotic stability of constant equilibrium states are established. We next investigate relaxation issues for a two temperature model that leads to symmetrized hyperbolic-parabolic systems with stiff source terms. Local existence of solutions is established and the apparition of the bulk viscosity coefficient is justified in the fast relaxation limit. We finally consider nonideal fluids and their cohesive properties that lead to diffuse interfaces models with capillary effects. Using an augmented formulation-with density gradient added as an extra unknown-a new type of hyperbolic-parabolic-dispersive structure is obtained with antisymmetric second order coupling terms arising from capillarity. Local existence of solutions is established as well as asymptotic stability of constant states.
• Inf-sup stable non-conforming finite elements on tetrahedra with second and third order accuracy
  
  • Balazi Loïc
  • Allaire Grégoire
  • Jolivet Pierre
  • Omnes Pascal
  , 2025. We introduce a family of scalar non-conforming finite elements with second and third order accuracy with respect to the $H^1$-norm on tetrahedra. Their vector-valued versions generate, together with discontinuous pressure approximations of order one and two respectively, inf-sup stable finite element pairs with convergence order two and three for the Stokes problem in energy norm.
• High-order multistep coupling: convergence, stability and PDE application
  
  • Simon Antoine
  • François Laurent
  • Massot Marc
  Comptes Rendus. Mécanique, Académie des sciences (Paris), 2025, 353 (G1), pp.1159-1184. Designing coupling schemes for specialized advanced mono-physics solvers in order to conduct accurate and efficient multiphysics simulations is a key issue that has recently received a lot of attention. A novel high-order adaptive multistep coupling strategy has shown potential to improve the efficiency and accuracy of such simulations, but requires further analysis. The purpose of the present contribution is to conduct the numerical analysis of convergence of the explicit and implicit variants of the method and to provide a first analysis of its absolute stability. A simplified coupled problem is constructed to assess the stability of the method along the lines of the Dahlquist’s test equation for ODEs. We propose a connection with the stability analysis of other methods such as splitting and ImEx schemes. A stability analysis on a representative conjugate heat transfer case is also presented. This work constitutes a first building block to an a priori analysis of the stability of coupled PDEs. (10.5802/crmeca.333)
  DOI : 10.5802/crmeca.333