Publications

2026

• A functional inequalities approach for the field-road diffusion model with (symmetric) nonlinear exchanges
  
  • Alfaro Matthieu
  • Chainais-Hillairet Claire
  • Nabet Flore
  Applied Mathematics Letters, Elsevier, 2026, 180, pp.109980. In this note, we consider the so-called field-road diffusion model in a bounded domain, consisting of two parabolic PDEs posed on sets of different dimensions and coupled through (symmetric) nonlinear exchange terms. We propose a new and rather direct functional inequalities approach to prove the exponential decay of a relative entropy, and thus the convergence of the solution towards the stationary state selected by the total mass of the initial datum. (10.1016/j.aml.2026.109980)
  DOI : 10.1016/j.aml.2026.109980
• Surrogate-Based Strategies for Accelerated Bayesian Calibration of Computer Codes With Complete Maximum a Posteriori Estimation of Model Error
  
  • Kahol Omar
  • Le Maître Olivier
  • Congedo Pietro M
  • Denimal Goy Enora
  Journal of Mechanical Design, American Society of Mechanical Engineers, 2026, 148 (9). The calibration of a computer code is a process that reduces the uncertainty of model parameters by matching the code’s predictions to experimental observations of a quantity of interest. A more faithful representation of the global uncertainty is achieved by including a model error term, a discrepancy between the physical system and the computer code. The recently proposed complete maximum a posteriori (CMP) method is able to infer both a posterior distribution of the model parameters and a model error term, improving upon traditional frameworks. On the other hand, the CMP method relies on an optimization step which increases the cost of complex calibration problems. This article proposes a surrogate-based strategy to reduce the computational cost of the CMP method. First, we build a surrogate model of the model error’s hyperparameters using Gaussian processes. Second, we propose an iterative algorithm that builds a training set in regions of the parameter space that are more likely, reducing the overall cost of the algorithm and improving the accuracy of the surrogate. The proposed strategy is applied to four different examples, including a design problem in solid mechanics and a complex test case in fluid dynamics. The results show that the proposed strategy is able to accelerate the CMP method without losing accuracy, making it suitable for real-world applications. In an industrial application, we demonstrate a speed-up of almost 100 compared to the original CMP method. (10.1115/1.4071071)
  DOI : 10.1115/1.4071071
• A robust computational framework for the mixture-energy-consistent six-equation two-phase model with instantaneous mechanical relaxation terms
  
  • Orlando Giuseppe
  • Haegeman Ward
  • Pelanti Marica
  • Massot Marc
  Computers and Fluids, Elsevier, 2026, 317, pp.107193. We present a robust computational framework for the numerical solution of a hyperbolic 6-equation single-velocity two-phase system. The system's main interest is that, when combined with instantaneous mechanical relaxation, it recovers the solution of the 5-equation model of Kapila. Several numerical methods based on this strategy have been developed over the years. However, neither the 5- nor 6-equation model admits a complete set of jump conditions because they involve non-conservative products. Different discretizations of these terms in the 6-equation model exist. The precise impact of these discretizations on the numerical solutions of the 5-equation model, in particular for shocks, is still an open question to which this work provides new insights. We consider the phasic total energies as prognostic variables to naturally enforce discrete conservation of total energy and compare the accuracy and robustness of different discretizations for the hyperbolic operator. Namely, we discuss the construction of an HLLC approximate Riemann solver in relation to jump conditions. We then compare an HLLC wave-propagation scheme which includes the non-conservative terms, with Rusanov and HLLC solvers for the conservative part in combination with suitable approaches for the non-conservative terms. We show that some approaches for the discretization of non-conservative terms fit within the framework of path-conservative schemes for hyperbolic problems. We then analyze the use of various numerical strategies on several relevant test cases, showing both the impact of the theoretical shortcomings of the models as well as the importance of the choice of a robust framework for the global numerical strategy. (10.1016/j.compfluid.2026.107193)
  DOI : 10.1016/j.compfluid.2026.107193
• On the simulation of extreme events with neural networks
  
  • Allouche Michaël
  • Girard Stéphane
  • Gobet Emmanuel
  , 2026. This article aims at investigating the use of generative methods based on neural networks to simulate extreme events. Although very popular, these methods are mainly invoked in empirical works. Therefore, providing theoretical guidelines for using such models in extreme values context is of primal importance. To this end, we propose an overview of most recent generative methods dedicated to extremes, giving some theoretical and practical tips on their tail behaviour thanks to both extreme-value and copula tools.
• A sequential Bayesian approach to sparse Gaussian process quantile optimization: application to hydrofoil design
  
  • Nicolas Hugo
  • Le Maître Olivier
  , 2026. In previous work, we developed a sequential Bayesian approach to estimate conditional quantiles of a response variable--a problem framed as quantile regression [1]. The latent conditional quantile function was modeled via a sparse Gaussian process with inducing points. The asymmetric Laplace distribution was considered for the likelihood of the data. The inference of the posterior distribution over the inducing variables was recast as its Laplace approximation. The importance of inducing input locations to predictive accuracy was identified. As a result, we proposed a novel approach to optimally select their locations by leveraging the Gaussian process predictive variance. Moreover, we exploited uncertainties in the inducing variables to guide the acquisition of new observations of the response variable. Adaptive inducing point allocation and active learning were combined into a sequential algorithm, which further enabled automatic selection of the required number of inducing points. The present work adapts the above methodology, which focused on global predictive accuracy, to the problem of finding the optimum of the conditional quantile function--a problem framed as quantile optimization. New observations are selected using Thompson sampling by drawing samples from the posterior and choosing the corresponding optimal inputs. We leverage the decoupled strategy by [2] to efficiently sample from the sparse Gaussian quantile posterior within the Thompson sampling. This framework for quantile optimization is applied to the design of a hydrofoil under uncertainties. We focus on the hydrodynamic performance of the hydrofoil, with XFOIL as the flow solver, supplemented by analytical corrections to the hydrodynamic coefficients to account for free-surface proximity effects. Multiple sources of uncertainty are considered, including operational and manufacturing uncertainties. This application demonstrates the effectiveness of our algorithm and highlights its key advantages in high-dimensional uncertainty settings, as commonly encountered in engineering design [3].
• Affine Structure of the Brownian Signature
  
  • Jaber Eduardo Abi
  • Attal Elie
  • Sotnikov Dimitri
  , 2026. We establish an infinite-dimensional affine transform theory for the time-augmented Brownian signature. Our first main result shows that, for a suitable class of linear functions of the signature, the conditional Fourier-Laplace transform admits an entire signature expansion. We prove that the associated coefficients solve an infinite-dimensional linear differential equation on the extended tensor algebra. Our second main result shows that the logarithm admits a local signature expansion whose coefficients satisfy a Riccati equation on the extended tensor algebra, revealing a generalized affine structure of the Brownian signature in a genuinely path-dependent setting. In contrast to conventional affine processes, we show that this representation is intrinsically local: zeros of the Fourier-Laplace transform in the complex plane prevent any global expansion. To recover global representations, we introduce a new class of randomized Riccati equations with path-dependent terminal conditions through a recentering argument. Furthermore, we establish uniqueness of solutions to the linear and Riccati equations within a suitable class of solutions. Our results provide a theoretical framework for transform methods in non-Markovian settings, with applications to the computation of conditional distributions.
• Stochastic control with signatures via Riccati equations on the tensor algebra
  
  • Abi Jaber Eduardo
  • Attal Elie
  • Sotnikov Dimitri
  , 2026. We solve in semi-explicit form a class of non-Markovian stochastic optimal control problems with pathdependent rewards, using path signatures. We reformulate the control problem as the computation of Laplace transforms of signature functionals thanks to the Boué-Dupuis representation. Exploiting recent signature representations of such transforms on tensor algebras, we determine the value process and the optimal control through an infinite-dimensional system of Riccati equations on the extended tensor algebra. We establish an explicit feedback representation of the optimal control and the value process as an infinite linear combination of the time-extended signature of the controlled process, with time-dependent coefficients. The expansions being intrinsically local, we propose a dynamic recentering algorithm to ensure a global representation over the entire time horizon. We illustrate the approach on genuinely path-dependent, non-linear examples that go beyond the tractable linear-quadratic setting, including the tracking of linear functionals of the signature and signature lifts of Volterra control problems.
• Strong law of large numbers and L log L condition for supercritical branching processes
  
  • Bansaye Vincent
  • Berah Tresnia
  • Cloez Bertrand
  , 2025. We consider branching processes for structured populations: each individual is characterized by a type or trait which belongs to a general measurable state space. We focus on the supercritical recurrent case, where the population may survive and grow and the trait distribution converges. The branching process is then expected to be driven by the positive triplet of first eigenvalue problem of the first moment semigroup. Under the assumption of convergence of the renormalized semigroup in weighted total variation norm, we prove strong convergence of the normalized empirical measure and non-degeneracy of the limiting martingale. Convergence is obtained under an Llog L condition which provides a Kesten-Stigum result in infinite dimension and relaxes the uniform convergence assumption of the renormalized first moment semigroup required in the work of Asmussen and Hering in 1976. The techniques of proofs combine families of martingales and contraction of semigroups and the truncation procedure of Asmussen and Hering. We also obtain L^1 convergence of the renormalized empirical measure and contribute to unifying different results in the literature. These results greatly extend the class of examples where a law of large numbers applies, as we illustrate it with absorbed branching diffusion, the house of cards model and some growth-fragmentation processes.
• On the range of competing random walks
  
  • Baccara Maxence
  , 2026. We consider $N$ independent random walks $X^1,\dots,X^N$ in the lattice $\mathbb{Z}^d$ and prove limit theorems for the competitive range $\mathcal{R}_n^k$ of the $k$-th random walk $X^k$, which corresponds to the number of distinct sites that it has discovered before any of the other $X^\ell$, $\ell\ne k$, up to time $n$. This is a natural object to study foraging mechanisms in population ecology, in which context it is also natural to ask how the effect of competition for the access to resources affects the number of resources consumed by each individual. We work with random walks in the domain of attraction of a $\beta$-stable law and focus on the regime $d/\beta\in[1,3/2)$, in which classical results for the range show that the fluctuations are described by the renormalized self-intersection local time of the limiting process. We establish a central limit theorem in which a competition term emerges, thus answering the two previous questions we asked. We end the paper with a brief discussion on the remaining regimes $d/\beta\ge3/2$, in which the fluctuations are Gaussian and are not affected by the competition, and $d/\beta<1$ in which no strong law of large numbers holds and we expect the effect of the competition to strongly affect the first-order asymptotics.
• Sequential sparse Gaussian process quantile regression
  
  • Nicolas Hugo
  • Le Maître Olivier
  , 2026. Quantile regression aims to estimate the conditional quantiles of a response variable from observed data. In a Bayesian setting, Gaussian process quantile regression provides uncertainty quantification but faces significant computational challenges due to the nonconjugacy of the asymmetric Laplace likelihood and the cost of posterior inference. We develop a sparse Gaussian process framework in which the quantile function is represented through a reduced set of inducing variables and posterior inference is performed using a Laplace approximation. A decomposition of the predictive uncertainty into conditional-prior and posterior-induced variance components is then exploited to drive two complementary adaptive mechanisms: inducing-input infilling and data acquisition. These mechanisms are combined within a sequential algorithm that allocates computational effort toward the dominant source of predictive uncertainty and adaptively controls model complexity. Numerical experiments on benchmark problems demonstrate the accuracy of the Laplace approximation, the benefits of variance-based inducing-input placement, and the effectiveness of the proposed sequential enrichment strategy compared with predefined data-acquisition strategies.
• Threatening excursions in large population quasi-stationary birth and death systems. On a question of Antonio Galves.
  
  • Collet Pierre
  • Martínez Servet
  • Méléard Sylvie
  , 2026. We consider time continuous multispecies birth and death processes in a regime of large populations. The jump rates depend on a large scaling parameter K modeling the charge capacity. When K tends to infinity, the process is close (in finite time) to a dynamical system containing a non zero global attracting equilibrium and zero as unstable equilibrium. For each fixed K, extinction in finite time occurs almost surely and a quasi-stationary distribution occurs naturally in the study of the statistics over times scales which are large but smaller than the extinction time scale. Before this catastrophic event the process makes many unsuccessful large deviations attempts with time scales corresponding to how far it deviates from the quasi-equilibrium. The paper concerns the statistical description of these typical trajectories starting from the quasi-stationary distribution until extinction. An unusual mixing property yields large time scale behavior for the process starting from a fixed state. We give a precise statistical description of the successive exit times of the process rescaled by K from a neighborhood of the equilibrium of the dynamical system in a clumping time scale and prove their asymptotic Poisson distribution. We also give a precise description of the asymptotic distribution of the successive records until extinction.
• Stochastic-structural modelling of particle-laden turbulent flows based on wavelet reconstruction
  
  • Morhain Clément
  • Letournel Roxane
  • Massot Marc
  • Vié Aymeric
  Computers and Fluids, Elsevier, 2026. Reduced-order modelling and simulation of turbulent particle-laden flows is required in numerous configurations, where the resolution of the whole spectrum of turbulent scales through DNS is out of reach. Whereas structural or stochastic models have been derived in order to provide a synthetic turbulent model for the non-resolved scales of the fluid flow field, reproducing particle dynamics is challenging because it requires capturing both spatial and temporal correlations. We present a reduced-order framework that combines wavelet-based structural modelling with stochastic evolution. Using compactly supported divergence-free wavelets within a multiresolution analysis, the method provides direct control over spatial structures and correlations of synthetic multiscale incompressible velocity fields. In contrast to Fourier modes, the wavelet basis functions are localized in space and spectrally non-sharp in Fourier space, and spread over a range of wavenumbers, which requires a dedicated procedure to enforce a prescribed turbulent energy spectrum. The stochastic evolution of wavelet coefficients further ensures consistent temporal correlations. The proposed framework is evaluated in homogeneous isotropic turbulence under a fully reduced setting, where all turbulent scales must be provided by the model. When coupled to a disperse phase in the one-way coupled framework, results show that it reproduces particle preferential concentration across a wide range of Stokes numbers as well as the pair-dispersion regimes, achieving similar agreement with DNS data as for classical Fourier-based Kinematic Simulation. This establishes a physically consistent turbulence model, which combines structural fidelity with stochastic dynamics, providing an alternative framework for synthetic turbulence modelling and investigating particle–turbulence interactions resolution.
• A two-scale two-phase flow model for the separate-to-disperse phase transition in atomizing flows
  
  • Haegeman Ward
  • Orlando Giuseppe
  • Kokh Samuel
  • Massot Marc
  , 2026. An original two-scale, isothermal compressible two-phase flow model with surface tension is presented. The model allows for a unified description of the separate interface and disperse phase regimes. The inter-scale mass transfer terms, activated when the local curvatures exceed a physical and grid-independent length threshold, allow for the transition from the former regime, to the latter, through atomization. This mass transfer process is obtained through a pressure relaxation towards a modified Laplace law such that local curvatures do not exceed the prescribed threshold. It leads to a local and dissipative regularization of the large-scale interface, while retaining a sub-scale representation of the small-scale flow features. The backbone of the model is derived through the use of Hamilton's Stationary Action Principle. The source terms are derived such that the inter-scale mass transfer is dissipative for the extended thermodynamics, which includes the surface energies at both scales. The methodology that is developed allows for the derivation of a thermodynamically consistent model which admits a supplementary conservation law for the entropy and real characteristics.
• Fourier-Laplace Transforms of the Brownian Signature via Riccati Equations on the Tensor Algebra
  
  • Abi Jaber Eduardo
  • Attal Elie
  • Sotnikov Dimitri
  , 2026. We establish an infinite-dimensional affine transform theory for the time-augmented Brownian signature. Our first main result shows that, for a suitable class of linear functions of the signature, the conditional Fourier-Laplace transform admits an entire signature expansion. We prove that the associated coefficients solve an infinite-dimensional linear differential equation on the extended tensor algebra. Our second main result shows that the logarithm admits a local signature expansion whose coefficients satisfy a Riccati equation on the extended tensor algebra, revealing a generalized affine structure of the Brownian signature in a genuinely path-dependent setting. In contrast to conventional affine processes, we show that this representation is intrinsically local: zeros of the Fourier-Laplace transform in the complex plane prevent any global expansion. To recover global representations, we introduce a new class of randomized Riccati equations with path-dependent terminal conditions through a recentering argument. Furthermore, we establish uniqueness of solutions to the linear and Riccati equations within a suitable class of solutions. Our results provide a theoretical framework for transform methods in non-Markovian settings, with applications to the computation of conditional distributions.
• Rémy's diffusion on Brownian trees
  
  • Curien Nicolas
  • Marzouk Cyril
  , 2026. Rémy's algorithm is a famous recursive construction of uniform random binary trees of growing size by a local grafting operation. In this work we construct a continuous version, a new local diffusion on the space of real trees of growing Brownian Continuum Random Trees (CRT's). It appears as the scaling limit of a variant of Rémy's algorithm due to Bacher, Bodini, and Jacquot. Once the trees are rescaled to have constant mass, this diffusion uncovers an ergodic dynamics on trees with the Brownian CRT as unique invariant law.
• Wiring the Fly Brain into a Hierarchical World Model A self-supervised JEPA world model of a complete connectome Team Piaget
  
  • Thil Lucas
  • Nowak Assis Daniel
  • Berthier Louis
  • Jlidi Adam
  , 2026. World models predict the future of an environment in an abstract latent space, the kind of prediction a brain is thought to perform. We ask whether such a model can learn the dynamics of a real brain rather than a metaphor for one. We take the FlyWire connectome of the adult Drosophila, the only animal whose brain is fully mapped (∼139,000 neurons, ∼50M synapses), turn it into a spiking brain model embodied in a physics simulation of the fly's body, and let it act in a closed sensorimotor loop. On the resulting whole-brain spike trains we train a hierarchy of region-level world models (Joint-Embedding Predictive Architectures, JEPAs), one per brain region, coupled along the connectome's own sensory → integration → motor flow. To our knowledge this is the first JEPA world model of a complete brain. We report three findings: the motor (descending) pathway behaves as a genuine world model, forecasting its own activity and linearly decoding real motor commands; the extreme sparsity of spike codes induces a characteristic latent collapse, diagnosable by participation ratio, which a connectome-derived encoder helps counteract; and coupling regions along the connectome's feedforward structure measurably improves prediction, with the gain growing over longer horizons. The brain's wiring does not merely inspire the architecture, it improves the model of the world.
• Nonequilibrium and multiscale fluids: Models and approximations
  
  • Pichard Teddy
  , 2026.
• Optimized high-order IMEX-RK schemes for degenerate diffusion-reaction problems with application to travelling waves phenomena
  
  • Antonietti Paola F.
  • Corti Mattia
  • Orlando Giuseppe
  , 2026. We study a class of IMplicit-EXplicit Runge--Kutta (IMEX-RK) schemes for the numerical approximation of reaction and diffusion-reaction problems arising in a variety of biological and physical applications. Such models may admit travelling wave solutions, with the Fisher--Kolmogorov equation representing a prototypical example. Motivated by this feature, the proposed time integration schemes are designed to accurately capture sharp propagating fronts. We also investigate a less standard use of IMEX-RK methods that circumvents a splitting of reaction terms into linear and nonlinear components, while still requiring the solution of linear systems at each stage. This semi-implicit formulation, referred to as SI-IMEX-RK, enables a targeted treatment of stiffness by isolating its relevant contributions. The time discretization is coupled with a high-order polygonal discontinuous Galerkin method for space discretization, resulting in a flexible and robust framework for the treatment of multiscale dynamics in complex geometries. A comprehensive validation strategy is presented to assess the accuracy and stability properties of the proposed schemes across a hierarchy of increasingly challenging test problems.
• A 3D-shell model of left atrial electromechanics
  
  • Ruz Oscar
  • Brito-Pacheco Carlos
  • Vidrascu Marina
  • Chapelle Dominique
  • Fernández Miguel Angel
  , 2026. The thin-walled nature of the atrial myocardium can lead to artificial stiffening when full 3D electromechanical models are discretized using standard finite elements. In this work, we propose an electromechanical model of the left atrium based on a 3D-shell formulation that overcomes these limitations. The model incorporates both passive and active components of atrial tissue mechanics, while atrioventricular interaction is described by the coupling with a 0D electromechanical model of the left ventricle. The proposed approach is assessed under physiological and pathological conditions and systematically compared with the standard full 3D formulation. The results demonstrate the superior robustness and computational efficiency of the proposed 3D-shell electromechanical model.
• Mean field games with incomplete information
  
  • Bertucci Charles
  , 2026. This paper is concerned with mean field games in which the players do not know the repartition of the other players. First a case in which the players do not gain information is studied. Results of existence and uniqueness are proved and discussed. Then, a case in which the players observe the payments is investigated. A master equation is derived and partial results of uniqueness are given for this more involved case.
• Entropic Mirror Monte Carlo
  
  • Cherradi Anas
  • Janati Yazid
  • Durmus Alain
  • Le Corff Sylvain
  • Petetin Yohan
  • Stoehr Julien
  , 2026. Importance sampling is a Monte Carlo method which designs estimators of expectations under a target distribution using weighted samples from a proposal distribution. When the target distribution is complex, such as multimodal distributions in highdimensional spaces, the efficiency of importance sampling critically depends on the choice of the proposal distribution. In this paper, we propose a novel adaptive scheme for the construction of efficient proposal distributions. Our algorithm promotes efficient exploration of the target distribution by combining global sampling mechanisms with a delayed weighting procedure. The proposed weighting mechanism plays a key role by enabling rapid resampling in regions where the proposal distribution is poorly adapted to the target. Our sampling algorithm is shown to be geometrically convergent under mild assumptions and is illustrated through various numerical experiments.
• Numerical analysis of an optimal control approach to solve a tsunami inverse problem
  
  • Bourgeois Laurent
  • Moireau Philippe
  • Terrine Raphaël
  , 2026. This paper concerns the reconstruction of an abrupt bottom displacement of the ocean from the measurement of the induced perturbation of the free surface, which is a severely ill-posed inverse problem. This problem is solved by using an optimal control approach, the physics being governed by a time evolution system based on a simple oceanography model. We firstly recast the problem in an abstract framework, secondly propose an implicit Euler scheme for the time discretization combined with a Finite Element method for the space discretization. The main result is an error estimate between the solution to the discrete control optimal problem and the solution to the continuous optimal problem, which is obtained by considering the discrete and continuous weak mixed formulations that characterize the optimality for these two problems. Some numerical experiments illustrate the efficiency of our approach and the consistency of our error estimate.
• An inverse tsunami problem in the time domain: a well-posedness analysis of the forward problem and an inversion strategy based on a mixed formulation of the Tikhonov regularization
  
  • Bourgeois Laurent
  • Moireau Philippe
  • Terrine Raphaël
  , 2026. This contribution concerns an inverse problem related to a tsunami in the ocean, the tsunami being caused by a submarine earthquake. Considering the very beginning of the phenomenon, a simple linear model incorporating both gravity and acoustic waves is proposed. The main objective is to develop a strategy to solve the inverse problem of retrieving the bottom displacement from the induced free surface perturbation. Such strategy is based on a mixed formulation of the Tikhonov regularization in the space/time domain, the regularization parameter being determined by using the Morozov principle by means of duality in optimization. Some numerical experiments in 2D, which rely on a tensorized finite element method, show that our strategy is effective. A secondary objective is to prove existence and uniqueness of both strong and variational solutions to the forward problem.
• On the convergence of dynamic implementations of Hamiltonian Monte Carlo and No U-Turn Samplers
  
  • Durmus Alain
  • Gruffaz Samuel
  • Kailas Miika
  • Saksman Eero
  • Vihola Matti
  The Annals of Applied Probability, Institute of Mathematical Statistics (IMS), 2026, 36 (3). There is substantial empirical evidence about the success of dynamic implementations of Hamiltonian Monte Carlo (HMC), such as the No U-Turn Sampler (NUTS), in many challenging inference problems but theoretical results about their behavior are scarce. The aim of this paper is to fill this gap. More precisely, we consider a general class of MCMC algorithms we call dynamic HMC. We show that this general framework encompasses NUTS as a particular case, implying the invariance of the target distribution as a by-product. Second, we establish conditions under which NUTS is irreducible and aperiodic and as a corrolary ergodic. Under conditions similar to the ones existing for HMC, we also show that NUTS is geometrically ergodic. Finally, we improve existing convergence results for HMC showing that this method is ergodic without any boundedness condition on the stepsize and the number of leapfrog steps, in the case where the target is a perturbation of a Gaussian distribution. (10.1214/25-AAP2269)
  DOI : 10.1214/25-AAP2269
• 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
  , 2026. 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.