Publications

2025

• Mathematical analysis of an observer for solving inverse source wave problem
  
  • Delaunay Tiphaine
  • Imperiale Sébastien
  • Moireau Philippe
  Inverse Problems and Imaging, AIMS American Institute of Mathematical Sciences, 2025. The objective of this work is to propose a method using observers to estimate a source term of a wave equation from internal measurements in a subdomain. The first part of the work consists in proving an identifiability result from classical observability conditions for wave equations. We show that the source reconstruction is an ill-posed inverse problem of degree 1 or 2 depending on the measurements type. This inverse problem is solved using observers -- a sequential strategy -- that is proven to be equivalent to a minimization of a cost functional with Tikhonov regularization. (10.3934/ipi.2025043)
  DOI : 10.3934/ipi.2025043
• Aligning individual brains with Fused Unbalanced Gromov-Wasserstein
  
  • Thual Alexis
  • Tran Huy
  • Zemskova Tatiana
  • Courty Nicolas
  • Flamary Rémi
  • Dehaene Stanislas
  • Thirion Bertrand
  , 2022. Individual brains vary in both anatomy and functional organization, even within a given species. Inter-individual variability is a major impediment when trying to draw generalizable conclusions from neuroimaging data collected on groups of subjects. Current co-registration procedures rely on limited data, and thus lead to very coarse inter-subject alignments. In this work, we present a novel method for inter-subject alignment based on Optimal Transport, denoted as Fused Unbalanced Gromov Wasserstein (FUGW). The method aligns cortical surfaces based on the similarity of their functional signatures in response to a variety of stimulation settings, while penalizing large deformations of individual topographic organization. We demonstrate that FUGW is well-suited for whole-brain landmark-free alignment. The unbalanced feature allows to deal with the fact that functional areas vary in size across subjects. Our results show that FUGW alignment significantly increases between-subject correlation of activity for independent functional data, and leads to more precise mapping at the group level.
• A comparison of different approaches to compute surface tension contribution in incompressible two-phase flows
  
  • Orlando Giuseppe
  • Barbante Paolo Francesco
  • Bonaventura Luca
  , 2024. We perform a quantitative assessment of different strategies to compute the contribution due to surface tension in incompressible two-phase flows using a conservative level set (CLS) method. More specifically, we compare classical approaches, such as the direct computation of the curvature from the level set or the Laplace-Beltrami operator, with an evolution equation for the mean curvature recently proposed in literature. We consider the test case of a static bubble, for which an exact solution for the pressure jump across the interface is available, and the test case of an oscillating bubble, showing pros and cons of the different approaches.
• Modelling of relative velocity, velocity fluctuations and their interactions for two-fluid models by Stationary Action Principle
  
  • Haegeman Ward
  • Orlando Giuseppe
  • Kokh Samuel
  • Massot Marc
  , 2025. The objective of this contribution is the derivation of a two-fluid model including a relative velocity between the two phases and velocity fluctuations, describing pseudo-turbulent effects, as internal variables based on Stationary Action Principle. The variational derivation, used to obtain the model, relies on the variation of a single trajectory related to the mass-weighted average velocity under the barotropic assumption. The model is hyperbolic, satisfies a second principle of thermodynamics, and admits either linearly degenerate or genuinely nonlinear characteristic fields. Moreover, the variational approach yields a fully closed model and its non-conservative products are uniquely defined for weak solutions in 1D, i.e. jump conditions can be derived. In the laminar case, when velocity fluctuations are negligible, we recover previously derived multi-fluid models which have been analyzed in several contributions. As such, the present framework allows for an original extension of the existing models to include velocity fluctuations of each phase for pseudo-turbulent flows, their coupling with the relative velocity between phases, as well as dissipative effects compatible with the thermodynamics of irreversible processes. Eventually, we provide a discussion of the limitations of the proposed model, especially regarding the extension to the open problem of non-barotropic flows.
• Computing the Congestion Phases of Dynamical Systems with Priorities and Application to Emergency Departments
  
  • Allamigeon Xavier
  • Capetillo Pascal
  • Gaubert Stéphane
  , 2025. Medical emergency departments are complex systems in which patients must be treated according to priority rules based on the severity of their condition. We develop a model of emergency departments using Petri nets with priorities, described by nonmonotone piecewise linear dynamical systems. The collection of stationary solutions of such systems forms a "phase diagram", in which each phase corresponds to a subset of bottleneck resources (like senior doctors, interns, nurses, consultation rooms, etc.). Since the number of phases is generally exponential in the number of resources, developing automated methods is essential to tackle realistic models. We develop a general method to compute congestion diagrams. A key ingredient is a polynomial time algorithm to test whether a given "policy" (configuration of bottleneck tasks) is achievable by a choice of resources. This is done by reduction to a feasibility problem for an unusual class of lexicographic polyhedra. Furthermore, we show that each policy uniquely determines the system's throughput. We apply our approach to a case study, analyzing a simplified model of an emergency department from Assistance Publique - Hôpitaux de Paris.
• Constrained optimization of a zoom lens with CMA-ES algorithm
  
  • Marty Tristan
  • Héron Sébastien
  • Semet Yann
  , 2025, 335, pp.02008. In the present paper we investigate how optimization algorithm can be tailored to improve the lens design process. We replaced gradient-based optimization methods by the Covariance Matrix Adaptation Evolution Strategy (CMA-ES). This stochastic algorithm is considered more robust and is well suited to avoid local optima often found in optical design. In addition, the algorithm is paired with an augmented Lagrangian method to incorporate constraints handling inside the computation framework. Performances are illustrated on a photographic zoom lens. (10.1051/epjconf/202533502008)
  DOI : 10.1051/epjconf/202533502008
• Quantifying the impact of different forms of stress on fungal growth: an inference method based on high-resolution pictures of the mycelial network
  
  • Kuwata Lena
  • Chassereau Thibault
  • Chapeland-Leclerc Florence
  • David Pascal
  • Herbert Eric
  • Ruprich-Robert Gwenaël
  • Tomašević Milica
  • Véber Amandine
  , 2025. In previous work [5], a complete methodology for monitoring the growth of a filamentous fungus was introduced, covering all aspects of this complex task going from the multi-scale imaging of the network of filaments to the automated extraction of the graph structure and its key statistics at regular time points. This methodology was applied to the fungus Podospora anserina grown in the lab under various conditions. In parallel, a stochastic growth-fragmentation model for the dynamics of such mycelial networks was introduced and studied in [23]. This simple model depends on three parameters only: the elongation speed v of a single filament, the branching rate b_1 of a filament at its open end, and the per unit length rate b_2 at which a budding event happens, resulting in a new filament branching off from an existing one. In this work, we develop a statistical inference method based on the large-time behaviour of the growth-fragmentation model shown in [23], and on the high-resolution pictures of the mycelial network obtained using the methodology described in [5], to reconstruct the parameters v, b_1 and b_2 from experimental data. We use this method to analyse the growth of P. anserina observed under standard conditions and when several forms of stress are applied, in order to quantify the effect of these stresses on the different mechanisms of fungal growth. By comparing with the parameter estimates obtained from the dynamical tracking of individual filaments, we show that reliable estimates of the individual elongation speed and branching rates can be computed from the easily accessible data consisting in a single panorama of the filament network pictured after several hours of growth and an empirical measure of the exponential growth rate of the number of branch points and free extremities.
• Discrete non-abelian X-ray transforms
  
  • Gupta Pranav
  • Novikov Roman
  , 2025. We define a discrete version of the non-abelian X-ray transform, going back in particular to Manakov, Zakharov (1981) and Strichartz (1982). We extend to this transform non-overdetermined reconstruction results obtained for the abelian case in the recent article by Novikov, Sharma (2025). In addition, we establish relations with the continuous non-abelian X-ray transform. In this respect, our results include an explicit and exact non-overdetermined layer-stripping reconstruction procedure for piecewise constant matrix-valued functions from their continuous non-abelian X-ray transform. To our knowledge, this result is new even for the classical X-ray transform.
• Order isomorphisms of sup-stable function spaces: Continuous, Lipschitz, c-convex, and beyond
  
  • Aubin-Frankowski Pierre-Cyril
  • Gaubert Stéphane
  Communications in Contemporary Mathematics, World Scientific Publishing, 2025. There have been many parallel streams of research studying order isomorphisms of some specific sets [Formula: see text] of functions from a set [Formula: see text] to [Formula: see text], such as the sets of convex or Lipschitz functions. We develop in this paper a unified approach inspired by [Formula: see text]-convex functions. Our results are obtained highlighting the role of inf and sup-irreducible elements of [Formula: see text] and the usefulness of characterizing them, to subsequently derive the structure of order isomorphisms, and in particular of those commuting with the addition of scalars. We show that in many cases all these isomorphisms [Formula: see text] are of the form [Formula: see text] for a translation [Formula: see text] and a bijective reparametrization [Formula: see text]. Given a reference anti-isomorphism, this characterization then allows to recover all the other anti-isomorphisms. We apply our theory to the sets of [Formula: see text]-convex functions on compact Hausdorff spaces, to the set of lower semicontinuous (convex) functions on a Hausdorff topological vector space and to 1-Lipschitz functions of complete metric spaces. The latter application is obtained using properties of the horoboundary of a metric space. (10.1142/S0219199725500762)
  DOI : 10.1142/S0219199725500762
• Design of experiments for efficient and conform Bayesian learning of seismic fragility curves
  
  • Van Biesbroeck Antoine
  • Feau Cyril
  • Garnier Josselin
  , 2025 (D7). Seismic fragility curves quantify the probability of failure of mechanical structures as a function of a seismic intensity measure (IM) that is derived from the ground motion. Although based on a strong assumption, the probit-lognormal model is very popular among practitioners for estimating such curves. Since their estimates may be compromised when data is scarce, this paper presents a novel adaptive design of experiments (DoE) strategy, within a Bayesian framework, to address this issue. This strategy first takes the reference prior theory as a support, in order to minimize the incorporation of subjectivity in the method. It then proposes a sequential selection of seismic signals that maximizes their impact on the posterior distribution. An application of the method on a case study taken from the nuclear industry is proposed. The results demonstrate that our approach significantly improves the accuracy and robustness of fragility curves estimations, particularly in data-limited scenarios.
• Left heart hemodynamics simulations with fluid-structure interaction and reduced valve modeling
  
  • Ruz Oscar
  • Diaz Jérôme
  • Vidrascu Marina
  • Moireau Philippe
  • Chapelle Dominique
  • Fernández Miguel Angel
  International Journal for Numerical Methods in Biomedical Engineering, John Wiley and Sons, 2025, 41 (9), pp.e70088. The combination of reduced models of cardiac valve dynamics with a one-way kinematic uncoupling of blood flow and electromechanics is a widespread approach for reducing the complexity of cardiac hemodynamics simulations. This comes however with a number of shortcomings: artificial pressure oscillations, missing isovolumetric phases and valve laws without precise continuous formulation. This paper is aimed at overcoming these three difficulties while still mitigating computational cost. A novel reduced model of valve dynamics is proposed in which unidirectional flow is enforced in a mathematically sound fashion. Artificial pressure oscillations are overcome by considering a fluid-structure interaction model, which couples bi-ventricular electromechanics and blood flow in the left cavities. The interface coupling is solved in a partitioned fashion via an unconditionally stable loosely coupled scheme. A priori energy estimates are derived for both the continuous coupled problem and its numerical approximation. The benefits and limitations of the proposed approaches are illustrated in a comprehensive numerical study. (10.1002/cnm.70088)
  DOI : 10.1002/cnm.70088
• Homogeneous multigrid for hybrid discretizations: application to HHO methods
  
  • Di Pietro Daniele A.
  • Dong Zhaonan
  • Kanschat Guido
  • Matalon Pierre
  • Rupp Andreas
  Numerical Methods for Partial Differential Equations, Wiley, 2025, 41 (5). We prove the uniform convergence of the geometric multigrid V- cycle for hybrid high-order (HHO) and other discontinuous skeletal methods. Our results generalize previously established results for HDG methods, and our multigrid method uses standard smoothers and local solvers that are bounded, convergent, and consistent. We use a weak version of elliptic regularity in our proofs. Numerical experiments confirm our theoretical results. (10.1002/num.70023)
  DOI : 10.1002/num.70023
• Long time behavior and Yaglom limit for real trait-structured Birth and Death Processes
  
  • Collet Pierre
  • Méléard Sylvie
  • San Jaime
  , 2025. In this article we study the long time behaviour of measure-valued birth and death processes in continuous time, where the dynamics between jumps are one-dimensional Markov processes including diffusion and jumps. We consider the three regimes, critical, subcritical and supercritical. Under suitable hypotheses on the Feynman-Kac semigroup, we prove a new recurrence for the moments and the extinction probability, their time asymptotics and the convergence in law for the measure-valued birth and death process conditioned to non extinction, leading to the existence of Q-process and Yaglom limit (in this infinite dimensional setting). We develop three classes of natural examples where our results apply.
• Convergence analysis of a high-order Multi-scale Finite Element Method (MsFEM) for Stokes flows in heterogeneous media
  
  • Balazi Loïc
  • Allaire Grégoire
  • Omnes Pascal
  , 2025. An enriched non-conforming Multi-scale Finite Element Method (MsFEM) to solve viscous incompressible flow problems in genuine heterogeneous or porous media was proposed in [Q. Feng, G. Allaire, and P. Omnes, Multiscale Model. Simul., 20(1):462-492, 2022]. The main feature of this MsFEM is the consideration of high-order sets of weighting functions: for the velocity, they are polynomials of order n on the faces and of order n-1 in the volume of the elements; for the pressure they are polynomials of order n in the element volume. In the previously cited reference, only the case n = 1 was numerically tested. The present paper proposes the first implementation for the case n = 2 in two and three dimensions. Furthermore, a discrete analysis of this MsFEM applied to the Stokes problem in heterogeneous media is performed. In particular, a first error estimate is obtained, proving the convergence of this MsFEM for the Stokes problem in periodic perforated media. In addition, it has been shown in the previously cited reference, that the continuous local problems involved in this MsFEM are well-posed. Here, their discrete counterparts are also proved to be well-posed, for any n in two dimensions and for n equal to 1 and 2 in three dimensions, with a judicious choice of non-conforming pairs of finite elements.
• Fourier-Laplace transforms in polynomial Ornstein-Uhlenbeck volatility models
  
  • Abi Jaber Eduardo
  • Li Shaun Xiaoyuan
  • Lin Xuyang
  Finance and Stochastics, Springer Verlag (Germany), 2025. We consider the Fourier-Laplace transforms of a {broad} class of polynomial Ornstein-Uhlenbeck (OU) volatility models, including the well-known Stein-Stein, Schöbel-Zhu, one-factor Bergomi, and the recently introduced Quintic OU models motivated by the SPX-VIX joint calibration problem. We show the connection between the joint {Fourier-Laplace} functional of the log-price and the integrated variance, and the solution of an infinite dimensional Riccati equation. Next, under some non-vanishing conditions of the Fourier-Laplace transforms, we establish an existence result for such Riccati equation and we provide a discretized approximation of the joint characteristic functional that is exponentially entire. On the practical side, we develop a numerical scheme to solve the stiff infinite dimensional Riccati equations and demonstrate the efficiency and accuracy of the scheme for pricing SPX options and volatility swaps using Fourier and Laplace inversions, with specific examples of the Quintic OU and the one-factor Bergomi models and their calibration to real market data.
• Hedging with memory: shallow and deep learning with signatures
  
  • Abi Jaber Eduardo
  • Gérard Louis-Amand
  , 2025. <div><p>We investigate the use of path signatures in a machine learning context for hedging exotic derivatives under non-Markovian stochastic volatility models. In a deep learning setting, we use signatures as features in feedforward neural networks and show that they outperform LSTMs in most cases, with orders of magnitude less training compute. In a shallow learning setting, we compare two regression approaches: the first directly learns the hedging strategy from the expected signature of the price process; the second models the dynamics of volatility using a signature volatility model, calibrated on the expected signature of the volatility. Solving the hedging problem in the calibrated signature volatility model yields more accurate and stable results across different payoffs and volatility dynamics.</p></div>
• A Gyromoment Approach for Electron Dynamics in Low-Temperature E × B Plasmas of Hall Thrusters
  
  • Tazakkati Zoubaïr
  • Laguna Alejandro Alvarez
  • Massot Josselin
  • Massot Marc
  • Pichard Teddy
  , 2025. <div><p>We study the electron dynamics in the acceleration region of a Hall thruster (HT). The strong crossed electric and magnetic fields induce both a fast electron cyclotron gyration around the magnetic field lines and a E × B drift. Starting from a Boltzmann-Poisson system, we perform a dimensional analysis to identify the dominant physical effects within this zone. This yields a non-dimensional kinetic equation tailored to the regime of interest, featuring multiple small parameters and a clear scale separation. The fast cyclotron gyration are filtered out through a Hilbert expansion combined with a gyroaveraging operator, yielding a reduced gyrokinetic model. Then a gyrofluid model is derived using a moment method with an entropy-based closure. Owing to the symmetries introduced by the gyroaveraging process, the number of required moments is reduced, and the closure corresponds to an anisotropic Gaussian requiring only four moments: the density, parallel momentum, and two directional temperatures. A numerical strategy using common tools from the literature is provided to handle the remaining small scales. Numerical experiments exhibit promising results for our applications.</p></div>
• Uniform attachment with freezing
  
  • Bellin Étienne
  • Blanc-Renaudie Arthur
  • Kammerer Emmanuel
  • Kortchemski Igor
  The Annals of Applied Probability, Institute of Mathematical Statistics (IMS), 2025, 35 (4). In the classical model of random recursive trees, trees are recursively built by attaching new vertices to old ones. What happens if vertices are allowed to freeze, in the sense that new vertices cannot be attached to already frozen ones? We are interested in the impact of freezing on the height of such trees. (10.1214/25-AAP2190)
  DOI : 10.1214/25-AAP2190
• Scaling limit of graph classes through split decomposition
  
  • Bassino Frédérique
  • Bouvel Mathilde
  • Féray Valentin
  • Gerin Lucas
  • Pierrot Adeline
  The Australasian Journal of Combinatorics, Combinatorial Mathematics Society of Australasia (Inc.), 2025, 92 (3), pp.266-319. We prove that Aldous' Brownian CRT is the scaling limit, with respect to the Gromov--Prokhorov topology, of uniform random graphs in each of the three following families of graphs: distance-hereditary graphs, $2$-connected distance-hereditary graphs and $3$-leaf power graphs. Our approach is based on the split decomposition and on analytic combinatorics. (10.48550/arXiv.2207.12253)
  DOI : 10.48550/arXiv.2207.12253
• Deciphering the Replication-Division Coordination in E. coli: A Unified Mathematical framework for Systematic Model Comparison
  
  • Perrin Alexandre
  • Doumic Marie
  • El Karoui Meriem
  • Méléard Sylvie
  , 2025. While significant efforts have been made to model bacterial cell division, few models have incorporated DNA replication into the control of this process. To date, models that attempt to capture the coordination between replication and division cycles are based on fundamentally different assumptions, and yet conflicting results have emerged. As a result, key questions regarding how replication affects cell size at division remain unclear. To address this, we develop in a first part, a robust mathematical framework to study models of coordination of replication and division cycles proposed in the literature. Through theoretical analysis, we highlight necessary and sufficient conditions to apply to replication-agnostic and division-agnostic models to ensure physiologically-coherent behaviors. Then, in a second part, we lead a comprehensive statistical analysis to assess the ability of the models to reproduce the division volume distribution conditioned on cell covariates. This in-depth analysis highlighted remarkable performances of a novel model, yielding promising results for future refinements toward a universal model of replication-division coordination in E. coli. (10.1101/2025.07.25.666816)
  DOI : 10.1101/2025.07.25.666816
• SKADA-Bench: Benchmarking Unsupervised Domain Adaptation Methods with Realistic Validation On Diverse Modalities
  
  • Lalou Yanis
  • Gnassounou Theo
  • Collas Antoine
  • de Mathelin Antoine
  • Kachaiev Oleksii
  • Odonnat Ambroise
  • Moreau Thomas
  • Gramfort Alexandre
  • Flamary Rémi
  Transactions on Machine Learning Research Journal, [Amherst Massachusetts]: OpenReview.net, 2022, 2025, pp.1-48. Unsupervised Domain Adaptation (DA) consists of adapting a model trained on a labeled source domain to perform well on an unlabeled target domain with some data distribution shift. While many methods have been proposed in the literature, fair and realistic evaluation remains an open question, particularly due to methodological difficulties in selecting hyperparameters in the unsupervised setting. With SKADA-bench, we propose a framework to evaluate DA methods on diverse modalities, beyond computer vision task that have been largely explored in the literature. We present a complete and fair evaluation of existing shallow algorithms, including reweighting, mapping, and subspace alignment. Realistic hyperparameter selection is performed with nested cross-validation and various unsupervised model selection scores, on both simulated datasets with controlled shifts and real-world datasets across diverse modalities, such as images, text, biomedical, and tabular data. Our benchmark highlights the importance of realistic validation and provides practical guidance for real-life applications, with key insights into the choice and impact of model selection approaches. SKADA-bench is open-source, reproducible, and can be easily extended with novel DA methods, datasets, and model selection criteria without requiring re-evaluating competitors. SKADA-bench is available on Github: https://github.com/scikit-adaptation/skada-bench.
• Federated Majorize-Minimization: Beyond Parameter Aggregation
  
  • Dieuleveut Aymeric
  • Fort Gersende
  • Hegazy Mahmoud
  • Wai Hoi-To
  , 2025. This paper proposes a unified approach for designing stochastic optimization algorithms that robustly scale to the federated learning setting. Our work studies a class of Majorize-Minimization (MM) problems, which possesses a linearly parameterized family of majorizing surrogate functions. This framework encompasses (proximal) gradient-based algorithms for (regularized) smooth objectives, the Expectation Maximization algorithm, and many problems seen as variational surrogate MM. We show that our framework motivates a unifying algorithm called Stochastic Approximation Stochastic Surrogate MM (SA-SSMM), which includes previous stochastic MM procedures as special instances. We then extend SA-SSMM to the federated setting, while taking into consideration common bottlenecks such as data heterogeneity, partial participation, and communication constraints; this yields FedMM. The originality of FedMM is to learn locally and then aggregate information characterizing the surrogate majorizing function, contrary to classical algorithms which learn and aggregate the original parameter. Finally, to showcase the flexibility of this methodology beyond our theoretical setting, we use it to design an algorithm for computing optimal transport maps in the federated setting.
• Solving bihomogeneous polynomial systems with a zero-dimensional projection
  
  • Bender Matías R
  • Busé Laurent
  • Checa Carles
  • Tsigaridas Elias
  , 2025, pp.206-214. We study bihomogeneous systems defining, non-zero dimensional, biprojective varieties for which the projection onto the first group of variables results in a finite set of points. To compute (with) the 0-dimensional projection and the corresponding quotient ring, we introduce linear maps that greatly extend the classical multiplication maps for zero-dimensional systems, but are not those associated to the elimination ideal; we also call them multiplication maps. We construct them using linear algebra on the restriction of the ideal to a carefully chosen bidegree or, if available, from an arbitrary Gröbner bases. The multiplication maps allow us to compute the elimination ideal of the projection, by generalizing FGLM algorithm to bihomogenous, non-zero dimensional, varieties. We also study their properties, like their minimal polynomials and the multiplicities of their eigenvalues, and show that we can use the eigenvalues to compute numerical approximations of the zero-dimensional projection. Finally, we establish a single exponential complexity bound for computing multiplication maps and Gröbner bases, that we express in terms of the bidegrees of the generators of the corresponding bihomogeneous ideal. (10.1145/3747199.3747563)
  DOI : 10.1145/3747199.3747563
• An optimal transport based embedding to quantify the distance between playing styles in collective sports
  
  • Baouan Ali
  • Rosenbaum Mathieu
  • Pulido Sergio
  Journal of Quantitative Analysis in Sports, De Gruyter, 2025. This study presents a quantitative framework to compare teams in collective sports with respect to their style of play. The style of play is characterized by the team's spatial distribution over a collection of frames. As a first step, we introduce an optimal transport-based embedding to map frames into Euclidean space, allowing for the efficient computation of a distance. Then, building on this frame-level analysis, we leverage quantization to establish a similarity metric between teams based on a collection of frames from their games. For illustration, we present an analysis of a collection of games from the 2021-2022 Ligue 1 season. We are able to retrieve relevant clusters of game situations and calculate the similarity matrix between teams in terms of style of play. Additionally, we demonstrate the strength of the embedding as a preprocessing tool for relevant prediction tasks. Likewise, we apply our framework to analyze the dynamics in the first half of the NBA season in 2015-2016. (10.1515/jqas-2025-0007)
  DOI : 10.1515/jqas-2025-0007
• Any nonincreasing convergence curves are simultaneously possible for GMRES and weighted GMRES, as well as for left and right preconditioned GMRES
  
  • Matalon Pierre
  • Spillane Nicole
  , 2025. The convergence of the GMRES linear solver is notoriously hard to predict. A particularly enlightening result by [Greenbaum, Pták, Strakoš, 1996] is that, given any convergence curve, one can build a linear system for which GMRES realizes that convergence curve. What is even more extraordinary is that the eigenvalues of the problem matrix can be chosen arbitrarily. We build upon this idea to derive novel results about weighted GMRES. We prove that for any linear system and any prescribed convergence curve, there exists a weight matrix M for which weighted GMRES (i.e. GMRES in the inner product induced by M ) realizes that convergence curve, and we characterize the form of M . Additionally, we exhibit a necessary and sufficient condition on M for the simultaneous prescription of two convergence curves, one realized by GMRES in the Euclidean inner product, and the other in the inner product induced by M . These results are then applied to infer some properties of preconditioned GMRES when the preconditioner is applied either on the left or on the right. For instance, we show that any two convergence curves are simultaneously possible for left and right preconditioned GMRES.