Publications

2025

• From energy bounds to dimensional estimates in a branched transport model for type-I superconductors
  
  • De Philippis Guido
  • Goldman Michael
  • Berardo Ruffini
  , 2025. We consider a branched transport type problem which describes the magnetic flux through type-I superconductors in a regime of very weak applied fields. At the boundary of the sample, deviation of the magnetization from being uniform is penalized through a negative Sobolev norm. It was conjectured by S. Conti, F. Otto and S. Serfaty that as a result, the trace of the magnetization on the boundary should be a measure of Hausdorff dimension 8/5. We prove that this conjecture is equivalent to the proof of local energy bounds with an optimal exponent. We then obtain local bounds which are however not optimal. These yield improved lower bounds on the dimension of the irrigated measure but unfortunately does not improve on the trivial upper bound. In order to illustrate the dependence of this dimension on the choice of penalization, we consider in the last part of the paper a toy model where the boundary energy is given by a Wasserstein distance to Lebesgue. In this case minimizers are finite graphs and thus the trace is atomic.
• Asymptotic Analysis of a bi-monomeric nonlinear Becker-Döring system
  
  • Doumic Marie
  • Fellner Klemens
  • Mezache Mathieu
  • Velázquez Juan J L
  Nonlinearity, IOP Publishing, 2025. To provide a mechanistic explanation of sustained then damped oscillations observed in a depolymerisation experiment, a bi-monomeric variant of the seminal Becker-D\"oring system has been proposed in~(Doumic, Fellner, Mezache, Rezaei, J. of Theor. Biol., 2019). When all reaction rates are constant, the equations are the following: \begin{align*} \frac{dv}{dt} & =-vw+v\sum_{j=2}^{\infty}c_{j}, \qquad \frac{dw}{dt} =vw-w\sum_{j=1}^{\infty}c_{j}, \\ \frac{dc_{j}}{dt} & =J_{j-1}-J_{j}\ \ ,\ \ j\geq1\ \ ,\ \ \ J_{j}=wc_{j}-vc_{j+1}\ \ ,\ \ j\geq1\ \ ,\ J_{0}=0, \end{align*} where $v$ and $w$ are two distinct unit species, and $c_i$ represents the concentration of clusters containing $i$ units. We study in detail the mechanisms leading to such oscillations and characterise the different phases of the dynamics, from the initial high-amplitude oscillations to the progressive damping leading to the convergence towards the unique positive stationary solution. We give quantitative approximations for the main quantities of interest: period of the oscillations, size of the damping (corresponding to a loss of energy), number of oscillations characterising each phase. We illustrate these results by numerical simulation, in line with the theoretical results, and provide numerical methods to solve the system. (10.1088/1361-6544/adc3e5)
  DOI : 10.1088/1361-6544/adc3e5
• Model Updating of Rotating Wind Turbines Using Operational Modal Analysis and Floquet Mode Decomposition
  
  • Delette Nina
  • Denimal Goy Enora
  • Pfister Jean-Lou
  • El Amri Reda
  • Mevel Laurent
  , 2025, pp.1-7. The structural complexity of modern wind turbines, combined with numerous uncertain or unknown parameters, presents significant challenges for accurate predictive modeling. Model updating, which refines numerical model parameters using measurement data, offers a means to mitigate these discrepancies. While extensively applied to stationary structures, its extension to rotating wind turbines remains limited, as their time-periodic dynamics violate key assumptions underlying conventional methods. This study develops a numerical framework for model updating of rotating wind turbines based on an equivalent Linear Time-Invariant (LTI) approximation, derived through a Fourier decomposition of the system’s Floquet modes. A simplified 5 Degrees of Freedom (DoF) turbine model is employed to evaluate the effectiveness of a deterministic model updating strategy leveraging this approximation. Synthetic vibration data, generated from the model using a predefined parameter set, serve as reference measurements for assessing parameter recovery accuracy. Modal features extracted via Operational Modal Analysis (OMA) are used to construct the cost function that quantifies discrepancies between predicted and observed modes. The results underscore the potential of equivalent LTI representations in facilitating model updating for rotating systems, as they effectively capture the modal characteristics identified via OMA. This study establishes a foundation for extending this methodology to more complex, industrial-scale wind turbine models, provided that the computational cost of model evaluation remains manageable.
• The equilibrium price of bubble assets
  
  • Bertucci Charles
  • Lasry Jean-Michel
  • Lions Pierre Louis
  , 2025. Considering a simple economy, we derive a new Hamilton-Jacobi equation which is satisfied by the value of a "bubble" asset. We then show, by providing a rigorous mathematical analysis of this equation, that a unique non-zero stable solution exists under certain assumptions. The economic interpretation of this result is that, if the bubble asset can produce more stable returns than fiat money, agents protect themselves from hazardous situations through the bubble asset, thus forming a bubble's consensus value. Our mathematical analysis uses different ideas coming from the study of semi-linear elliptic equations.
• Robust topology optimization accounting for uncertain microstructural changes
  
  • Masson Hugo
  • Peigney Michaël
  • Denimal Goy Enora
  , 2025.
• Mean Mesh Adaptation for Efficient CFD Simulations with Operating Conditions Variability
  
  • Dornier Hugo
  • Le Maitre Olivier
  • Congedo Pietro Marco
  • Salah El Din Itham
  • Marty Julien
  • Bourasseau Sébastien
  Computers and Fluids, Elsevier, 2025, 298, pp.106666. When numerically solving partial differential equations, for a given problem and operating condition producing a steady-state, mesh adaptation has proven its efficiency to automatically build a discretization achieving a prescribed error level at low cost. However, with continuously varying operating conditions, such as those encountered in uncertainty quantification, adapting a mesh for each condition and controlling the error level becomes complex and computationally expensive. To enable more effective error and cost control, this work proposes a novel approach to mesh adaptation. The method consists in building a single adapted mesh that aims to minimize the average error for a continuous set of operating conditions. In the proposed implementation, this single mesh is built iteratively, informed by an estimate of the local average interpolation error. The proposed method leverages the iterative nature of mesh adaptation by re-sampling Monte Carlo quadratures to obtain accurate average error estimates over a reduced set of sample conditions, ensuring a low computational cost. This approach is especially effective for localized flow features whose positions change only slightly with operating conditions, such as moving shocks in supersonic flows, as the refinement is confined to smaller areas of the computational domain. The study focuses on evaluating the method’s average error convergence, robustness, and computational cost in comparison to state-of-the-art adaptation techniques. Additionally, the sensitivity of the approach to the choice and size of the quadrature, as well as to the error estimation method, is assessed. For this purpose, the methodology is applied to a one-dimensional variable-step solution of the Burgers equation and a two-dimensional Euler scramjet flow with a variable inlet Mach number. The results show that Mean Mesh Adaptation (MMA) achieves error convergence comparable to specific mesh adaptation while reducing the evaluation cost by up to a factor of five (in the scramjet case). This efficiency gain stems from the reduced dependence on the number of sampled conditions, thanks to robust Monte Carlo re-sampling, as well as the shared computational expense of mesh construction across multiple evaluations. Therefore, the proposed method enables computational efficiency while maintaining error control across varying operating conditions within a prescribed parameter variation range. (10.1016/j.compfluid.2025.106666)
  DOI : 10.1016/j.compfluid.2025.106666
• Asymptotic-preserving IMEX schemes for the Euler equations of non-ideal gases
  
  • Orlando Giuseppe
  • Bonaventura Luca
  Journal of Computational Physics, Elsevier, 2025, 529, pp.113889. We analyze schemes based on a general Implicit-Explicit (IMEX) time discretization for the compressible Euler equations of gas dynamics, showing that they are asymptotic-preserving (AP) in the low Mach number limit. The analysis is carried out for a general equation of state (EOS). We consider both a single asymptotic length scale and two length scales. We then show that, when coupling these time discretizations with a Discontinuous Galerkin (DG) space discretization with appropriate fluxes, a numerical method effective for a wide range of Mach numbers is obtained. A number of benchmarks for ideal gases and their non-trivial extension to non-ideal EOS validate the performed analysis. (10.1016/j.jcp.2025.113889)
  DOI : 10.1016/j.jcp.2025.113889
• On reconstruction from imaginary part for radiation solutions in two dimensions
  
  • Nair Arjun
  • Novikov Roman
  , 2025. We consider a radiation solution ψ for the Helmholtz equation in an exterior region in R^2 . We show that ψ in the exterior region is uniquely determined by its imaginary part Im(ψ) on an interval of a line L lying in the exterior region. This result has holographic prototype in the recent work Nair, Novikov (2025, J. Geom. Anal. 35, 123). Some other curves for measurements instead of the lines L are also considered. Applications to the Gelfand-Krein-Levitan inverse problem (from boundary values of the spectral measure in R^2) and to passive imaging are also indicated.
• Avancées dans les Modèles Génératifs : Méthodologies et Applications à la Cardiologie
  
  • Bedin Lisa
  , 2025. This thesis investigates the application of generative models, particularly diffusion models, for the analysis and generation of electrocardiogram (ECG) signals. ECGs are essential diagnostic tools in cardiology, enabling rapid and non-invasive assessment of the heart's electrical activity. However, their interpretation is often complicated by noise, artifacts, and the need to reconstruct complete signals from partial observations.The research presented here focuses on developing advanced methods to address these challenges. We introduce BeatDiff, a lightweight diffusion model for generating 12-lead heartbeat morphologies, and Midpoint-Guidance Posterior Sampling (MGPS), a novel algorithm for high-dimensional inverse problems, which enhances the robustness and accuracy of ECG reconstructions. Additionally, we apply these methods to generate realistic ECGs with RhythmDiff, a diffusion model designed for complete and high-quality ECG signals.The contributions of this thesis include the development of efficient models for generating heartbeat morphologies, the introduction of new techniques for inverse problems, and the application of these methods to generate realistic ECGs. These advancements have the potential to improve cardiac diagnostics and develop more accessible and effective cardiac monitoring technologies, with significant implications for clinical practice and public health. (10.70675/58f8ffe6z5eb0z4b5fza7fez4aa8c78eed69)
  DOI : 10.70675/58f8ffe6z5eb0z4b5fza7fez4aa8c78eed69
• On discrete X-ray transform
  
  • Novikov Roman
  • Sharma Basant Lal
  , 2025. We consider a discrete version of X-ray transform going back, in particular, to Strichartz (1982). We suggest non-overdetermined reconstruction for this discrete transform. Extensions to weighted (attenuated) analogues are given. Connections to the continuous case are presented.
• Convergence and Wave Propagation for a System of Branching Rank-Based Interacting Brownian Particles
  
  • Demircigil Mete
  • Tomasevic Milica
  , 2025. In this work we study a branching particle system of diffusion processes on the real line interacting through their rank in the system. Namely, each particle follows an independent Brownian motion, but only K ≥ 1 particles on the far right are allowed to branch with constant rate, whilst the remaining particles have an additional positive drift of intensity χ > 0. This is the so called Go or Grow hypothesis, which serves as an elementary hypothesis to model cells in a capillary tube moving upwards a chemical gradient. Despite the discontinuous character of the coefficients for the movement of particles and their demographic events, we first obtain the limit behavior of the population as K → ∞ by weighting the individuals by 1/K. Then, on the microscopic level when K is fixed, we investigate numerically the speed of propagation of the particles and recover a threshold behavior according to the parameter χ consistent with the already known behavior of the limit. Finally, by studying numerically the ancestral lineages we categorize the traveling fronts as pushed or pulled according to the critical parameter χ.
• Non-convex functionals penalizing simultaneous oscillations along two independent directions: structure of the defect measure
  
  • Goldman Michael
  • Merlet Benoît
  , 2025. We continue the analysis of a family of energies penalizing oscillations in oblique directions: they apply to functions $u(x_1,x_2)$ with $x_l\in\mathbb{R}^{n_l}$ and vanish when $u(x)$ is of the form $u_1(x_1)$ or $u_2(x_2)$. We mainly study the rectifiability properties of the defect measure $\nabla_1\nabla_2u$ of functions with finite energy. The energies depend on a parameter $\theta\in(0,1]$ and the set of functions with finite energy grows with $\theta$. For $\theta<1$ we prove that the defect measure is $(n_1-1,n_2-1)$-tensor rectifiable in $\Omega_1\times\Omega_2$. We first get the result for $n_1=n_2=1$ and deduce the general case through slicing using White's rectifiability criterion. When $\theta=1$ the situation is less clear as measures of arbitrary dimensions from zero to $n_1+n_2-1$ are possible. We show however, in the case $n_1=n_2=1$ and for Lipschitz continuous functions, that the defect measures are $1\,$-rectifiable. This case bears strong analogies with the study of entropic solutions of the eikonal equation.
• Volatility models in practice: Rough, Path-dependent or Markovian?
  
  • Abi Jaber Eduardo
  • Li Shaun Xiaoyuan
  Mathematical Finance, Wiley, 2025, 35 (4), pp.796–817. An extensive empirical study of the class of Volterra Bergomi models using SPX options data between 2011 and 2022 reveals the following fact-check on two fundamental claims echoed in the rough volatility literature: Do rough volatility models with Hurst index H ∈ (0, 1/2) really capture well SPX implied volatility surface with very few parameters? No, rough volatility models are inconsistent with the global shape of SPX smiles. They suffer from severe structural limitations imposed by the roughness component, with the Hurst parameter H ∈ (0, 1/2) controlling the smile in a poor way. In particular, the SPX at-the-money skew is incompatible with the power-law shape generated by rough volatility models. The skew of rough volatility models increases too fast on the short end, and decays too slow on the longer end where "negative" H is sometimes needed. Do rough volatility models really outperform consistently their classical Markovian counterparts? No, for short maturities they underperform their one-factor Markovian counterpart with the same number of parameters. For longer maturities, they do not systematically outperform the one-factor model and significantly underperform when compared to an under-parametrized two-factor Markovian model with only one additional calibratable parameter. On the positive side: our study identifies a (non-rough) path-dependent Bergomi model and an under-parametrized two-factor Markovian Bergomi model that consistently outperform their rough counterpart in capturing SPX smiles between one week and three years with only 3 to 4 calibratable parameters.
• Estimation of the lifetime distribution from fluctuations in Bellman-Harris processes
  
  • Olayé Jules
  • Bouzidi Hala
  • Aristov Andrey
  • Barizien Antoine
  • Ramos Salomé Gutiérrez
  • Baroud Charles
  • Bansaye Vincent
  Journal of Mathematical Biology, Springer, 2025, 90, pp.56. The growth of populations without interactions can often be modeled by branching processes where each individual evolves independently and with the same law. In Bellman–Harris processes, each individual lives a random time and is then replaced by a random number of offspring. We are interested in the estimation of the parameters of this model. Our motivation comes from the estimation of cell division time and we focus on Gamma distribution for lifetime and binary reproduction. The mean of the lifetime is closely related to the growth rate of the population. Going farther and describing lifetime variability from fixed time observations is a challenging task, due to the complexity of the fluctuations of non-Markovian branching processes. Using fine results on these fluctuations, we describe two time-asymptotic regimes and explain how to discriminate between them and estimate the parameters. Then, we consider simulations and biological data to validate and discuss our method. It allows to determine single-cell parameters from time-resolved measurements of populations without the need to track each individual or to know the details of the initial condition. The results can be extended to more general branching processes. (10.1007/s00285-025-02219-8)
  DOI : 10.1007/s00285-025-02219-8
• Stationary regimes of piecewise linear dynamical systems with priorities
  
  • Allamigeon Xavier
  • Capetillo Pascal
  • Gaubert Stéphane
  , 2025, pp.1-11. Dynamical systems governed by priority rules appear in the modeling of emergency organizations and road traffic. These systems can be modeled by piecewise linear time-delay dynamics, specifically using Petri nets with priority rules. A central question is to show the existence of stationary regimes (i.e., steady state solutions) -- taking the form of invariant half-lines -- from which essential performance indicators like the throughput and congestion phases can be derived. Our primary result proves the existence of stationary solutions under structural conditions involving the spectrum of the linear parts within the piecewise linear dynamics. This extends to a broader class of systems a fundamental theorem of Kohlberg (1980) dealing with nonexpansive dynamics. The proof of our result relies on topological degree theory and the notion of ``Blackwell optimality'' from the theory of Markov decision processes. Finally, we validate our findings by demonstrating that these structural conditions hold for a wide range of dynamics, especially those stemming from Petri nets with priority rules. This is illustrated on real-world examples from road traffic management and emergency call center operations. (10.1145/3716863.3718053)
  DOI : 10.1145/3716863.3718053
• Federated UCBVI: Communication-Efficient Federated Regret Minimization with Heterogeneous Agents
  
  • Labbi Safwan
  • Tiapkin Daniil
  • Mancini Lorenzo
  • Mangold Paul
  • Moulines Eric
  , 2025, vol. 258 of PMLR, pp.1315-1323. In this paper, we present the Federated Upper Confidence Bound Value Iteration algorithm ($\texttt{Fed-UCBVI}$), a novel extension of the $\texttt{UCBVI}$ algorithm (Azar et al., 2017) tailored for the federated learning framework. We prove that the regret of $\texttt{Fed-UCBVI}$ scales as $\tilde{\mathcal{O}}(\sqrt{H^3 |\mathcal{S}| |\mathcal{A}| T / M})$, with a small additional term due to heterogeneity, where $|\mathcal{S}|$ is the number of states, $|\mathcal{A}|$ is the number of actions, $H$ is the episode length, $M$ is the number of agents, and $T$ is the number of episodes. Notably, in the single-agent setting, this upper bound matches the minimax lower bound up to polylogarithmic factors, while in the multi-agent scenario, $\texttt{Fed-UCBVI}$ has linear speed-up. To conduct our analysis, we introduce a new measure of heterogeneity, which may hold independent theoretical interest. Furthermore, we show that, unlike existing federated reinforcement learning approaches, $\texttt{Fed-UCBVI}$'s communication complexity only marginally increases with the number of agents. (10.48550/arXiv.2410.22908)
  DOI : 10.48550/arXiv.2410.22908
• Narrowing the Gap between Adversarial and Stochastic MDPs via Policy Optimization
  
  • Tiapkin Daniil
  • Chzhen Evgenii
  • Stoltz Gilles
  , 2025. We consider the problem of learning in adversarial Markov decision processes [MDPs] with an oblivious adversary in a full-information setting. The agent interacts with an environment during $T$ episodes, each of which consists of $H$ stages, and each episode is evaluated with respect to a reward function that will be revealed only at the end of the episode. We propose an algorithm, called APO-MVP, that achieves a regret bound of order $\tilde{\mathcal{O}}(\mathrm{poly}(H)\sqrt{SAT})$, where $S$ and $A$ are sizes of the state and action spaces, respectively. This result improves upon the best-known regret bound by a factor of $\sqrt{S}$, bridging the gap between adversarial and stochastic MDPs, and matching the minimax lower bound $\Omega(\sqrt{H^3SAT})$ as far as the dependencies in $S,A,T$ are concerned. The proposed algorithm and analysis completely avoid the typical tool given by occupancy measures; instead, it performs policy optimization based only on dynamic programming and on a black-box online linear optimization strategy run over estimated advantage functions, making it easy to implement. The analysis leverages two recent techniques: policy optimization based on online linear optimization strategies (Jonckheere et al., 2023) and a refined martingale analysis of the impact on values of estimating transitions kernels (Zhang et al., 2023).
• Learning signals defined on graphs with optimal transport and Gaussian process regression
  
  • Carpintero Perez Raphaël
  • da Veiga Sébastien
  • Garnier Josselin
  • Staber Brian
  , 2025. In computational physics, machine learning has now emerged as a powerful complementary tool to explore efficiently candidate designs in engineering studies. Outputs in such supervised problems are signals defined on meshes, and a natural question is the extension of general scalar output regression models to such complex outputs. Changes between input geometries in terms of both size and adjacency structure in particular make this transition non-trivial. In this work, we propose an innovative strategy for Gaussian process regression where inputs are large and sparse graphs with continuous node attributes and outputs are signals defined on the nodes of the associated inputs. The methodology relies on the combination of regularized optimal transport, dimension reduction techniques, and the use of Gaussian processes indexed by graphs. In addition to enabling signal prediction, the main point of our proposal is to come with confidence intervals on node values, which is crucial for uncertainty quantification and active learning. Numerical experiments highlight the efficiency of the method to solve real problems in fluid dynamics and solid mechanics.
• Personalized Convolutional Dictionary Learning of Physiological Time Series
  
  • Roques Axel
  • Gruffaz Samuel
  • Kim Kyurae
  • Durmus Alain O
  • Oudre Laurent
  , 2025. Human physiological signals tend to exhibit both global and local structures: the former are shared across a population, while the latter reflect inter-individual variability. For instance, kinetic measurements of the gait cycle during locomotion present common characteristics, although idiosyncrasies may be observed due to biomechanical disposition or pathology. To better represent datasets with local-global structure, this work extends Convolutional Dictionary Learning (CDL), a popular method for learning interpretable representations, or dictionaries, of time-series data. In particular, we propose Personalized CDL (PerCDL), in which a local dictionary models local information as a personalized spatiotemporal transformation of a global dictionary. The transformation is learnable and can combine operations such as time warping and rotation. Formal computational and statistical guarantees for PerCDL are provided and its effectiveness on synthetic and real human locomotion data is demonstrated.
• The evolution of partner preferences: Evidence using matrimonial ads from Canada, France, India and the United States
  
  • Lippmann Quentin
  • Surana Khushboo
  Journal of Economic Behavior and Organization, Elsevier, 2025, 233, pp.106950. Using the text from matrimonial ads, we assemble a novel data set to describe the evolution of partner preferences over time and space. Analyzing ads published in Canada, France and India between 1950 and 1995, we show that stated preferences for economic criteria have fallen sharply in favor of personality traits in the two Western countries while they remain the most prevalent in India. Using ads covering various regions from the US and Canada in 1995, we show that personality traits are consistently more demanded than economic criteria. We provide evidence that these results are unlikely to be driven by the composition effects over time, role of parents or changing social norms. We show that the changes over time are particularly strong for women and accompany narrowing gender gaps in labor force participation in Western countries. We discuss the implications for understanding the evolution of assortative mating over time. (10.1016/j.jebo.2025.106950)
  DOI : 10.1016/j.jebo.2025.106950
• Stochastic Dynamics of Incoherent Branched Flow
  
  • Garnier Josselin
  • Picozzi Antonio
  • Torres Theo
  Physical Review Letters, American Physical Society, 2025, 134, pp.223803. Waves propagating through weakly disordered smooth linear media undergo a universal phenomenon called branched flow. Branched flow has been observed and studied experimentally in various systems by considering coherent waves. Recent experiments have reported the observation of optical branched flow by using an incoherent light source, thus revealing the key role of coherent phase-sensitive effects in the development of incoherent branched flow. By considering the paraxial wave equation as a generic representative model, we elaborate a stochastic theory of both coherent and incoherent branched flow. We derive closed-form equations that determine the evolution of the intensity correlation function, as well as the value and the propagation distance of the maximum of the scintillation index, which characterize the dynamical formation of incoherent branched flow. We report accurate numerical simulations that are found in quantitative agreement with the theory without free parameters. Our theory highlights the important impact of coherence and interference on branched flow, thereby providing a framework for exploring branched flow in nonlinear media, in relation with the formation of freak waves in oceans. (10.48550/arXiv.2502.07028)
  DOI : 10.48550/arXiv.2502.07028
• Improving the evaluation of samplers on multi-modal targets
  
  • Grenioux Louis
  • Noble Maxence
  • Gabrié Marylou
  , 2025. Addressing multi-modality constitutes one of the major challenges of sampling. In this reflection paper, we advocate for a more systematic evaluation of samplers towards two sources of difficulty that are mode separation and dimension. For this, we propose a synthetic experimental setting that we illustrate on a selection of samplers, focusing on the challenging criterion of recovery of the mode relative importance. These evaluations are crucial to diagnose the potential of samplers to handle multi-modality and therefore to drive progress in the field.
• A Γ-convergence result for 2D type-I superconductors
  
  • Cosenza Alessandro
  • Goldman Michael
  • Zilio Alessandro
  Interfaces and Free Boundaries : Mathematical Analysis, Computation and Applications, European Mathematical Society, 2025. We consider a 2D non-standard Modica-Mortola type functional. This functional arises from the Ginzburg-Landau theory of type-I superconductors in the case of an infinitely long sample and in the regime of comparable penetration and coherence lengthes. We prove that the functional Γ-converges to the perimeter functional. This result is a first step in understanding how to extend the results of Conti, Goldman, Otto, Serfaty (2018) to the regime of non vanishing Ginzburg-Landau parameter κ.
• Simulating integrated Volterra square-root processes and Volterra Heston models via Inverse Gaussian
  
  • Abi Jaber Eduardo
  • Attal Elie
  , 2025. <div><p>We introduce a novel simulation scheme, iVi (integrated Volterra implicit), for integrated Volterra square-root processes and Volterra Heston models based on the Inverse Gaussian distribution. The scheme is designed to handle $L^1$ kernels with singularities by relying solely on integrated kernel quantities, and it preserves the non-decreasing property of the integrated process. We establish weak convergence of the iVi scheme by reformulating it as a stochastic Volterra equation with a measure kernel and proving a stability result for this class of equations. Numerical results demonstrate that convergence is achieved with very few time steps. Remarkably, for the rough fractional kernel, unlike existing schemes, convergence seems to improve as the Hurst index H decreases and approaches -1/2.</p></div>
• Watermark anything with localized messages
  
  • Sander Tom
  • Fernandez Pierre
  • Durmus Alain
  • Furon Teddy
  • Douze Matthijs
  , 2025. Image watermarking methods are not tailored to handle small watermarked areas. This restricts applications in real-world scenarios where parts of the image may come from different sources or have been edited. We introduce a deep-learning model for localized image watermarking, dubbed the Watermark Anything Model (WAM). The WAM embedder imperceptibly modifies the input image, while the extractor segments the received image into watermarked and non-watermarked areas and recovers one or several hidden messages from the areas found to be watermarked. The models are jointly trained at low resolution and without perceptual constraints, then post-trained for imperceptibility and multiple watermarks. Experiments show that WAM is competitive with state-of-the art methods in terms of imperceptibility and robustness, especially against inpainting and splicing, even on high-resolution images. Moreover, it offers new capabilities: WAM can locate watermarked areas in spliced images and extract distinct 32-bit messages with less than 1 bit error from multiple small regions -no larger than 10% of the image surface -even for small 256 × 256 images. Training and inference code and model weights are available at github.com/facebookresearch/watermark-anything.