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# Documents  Lelièvre, Tony | enregistrements trouvés : 19

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## An introduction to molecular dynamics Stoltz, Gabriel | CIRM H

Post-edited

Research talks;Partial Differential Equations;Probability and Statistics

The aim of this two-hour lecture is to present the mathematical underpinnings of some common numerical approaches to compute average properties as predicted by statistical physics. The first part provides an overview of the most important concepts of statistical physics (in particular thermodynamic ensembles). The aim of the second part is to provide an introduction to the practical computation of averages with respect to the Boltzmann-Gibbs measure using appropriate stochastic dynamics of Langevin type. Rigorous ergodicity results as well as elements on the estimation of numerical errors are provided. The last part is devoted to the computation of transport coefficients such as the mobility or autodiffusion in fluids, relying either on integrated equilibrium correlations à la Green-Kubo, or on the linear response of nonequilibrium dynamics in their steady-states. The aim of this two-hour lecture is to present the mathematical underpinnings of some common numerical approaches to compute average properties as predicted by statistical physics. The first part provides an overview of the most important concepts of statistical physics (in particular thermodynamic ensembles). The aim of the second part is to provide an introduction to the practical computation of averages with respect to the Boltzmann-Gibbs ...

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## Large-time behavior in (hypo)coercive ODE-systems and kinetic models Arnold, Anton | CIRM H

Post-edited

Research talks

In this talk we discuss the convergence to equilibrium in conservative-dissipative ODE-systems, kinetic relaxation models (of BGK-type), and Fokker-Planck equation. This will include symmetric, non-symmetric and hypocoercive evolution equations. A main focus will be on deriving sharp decay rates.
We shall start with hypocoercivity in ODE systems, with the ”hypocoercivity index” characterizing its structural complexity.
BGK equations are kinetic transport equations with a relaxation operator that drives the phase space distribution towards the spatially local equilibrium, a Gaussian with the same macroscopic parameters. Due to the absence of dissipation w.r.t. the spatial direction, convergence to the global equilibrium is only possible thanks to the transport term that mixes various positions. Hence, such models are hypocoercive.
We shall prove exponential convergence towards the equilibrium with explicit rates for several linear, space periodic BGK-models in dimension 1 and 2. Their BGK-operators differ by the number of conserved macroscopic quantities (like mass, momentum, energy), and hence their hypocoercivity index. Our discussion includes also discrete velocity models, and the local exponential stability of a nonlinear BGK-model.
The third part of the talk is concerned with the entropy method for (non)symmetric Fokker-Planck equations, which is a powerful tool to analyze the rate of convergence to the equilibrium (in relative entropy and hence in L1). The essence of the method is to first derive a differential inequality between the first and second time derivative of the relative entropy, and then between the entropy dissipation and the entropy. For hypocoercive Fokker-Planck equations, i.e. degenerate parabolic equations (with drift terms that are linear in the spatial variable) we modify the classical entropy method by introducing an auxiliary functional (of entropy dissipation type) to prove exponential decay of the solution towards the steady state in relative entropy. The obtained rate is indeed sharp (both for the logarithmic and quadratic entropy). Finally, we extend the method to the kinetic Fokker-Planck equation (with nonquadratic potential).
In this talk we discuss the convergence to equilibrium in conservative-dissipative ODE-systems, kinetic relaxation models (of BGK-type), and Fokker-Planck equation. This will include symmetric, non-symmetric and hypocoercive evolution equations. A main focus will be on deriving sharp decay rates.
We shall start with hypocoercivity in ODE systems, with the ”hypocoercivity index” characterizing its structural complexity.
BGK equations are kinetic ...

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## Adaptive inexact Newton methods and their application to multi-phase flows Vohralík, Martin | CIRM H

Multi angle

Research talks;Partial Differential Equations

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## Adaptive low-rank approximations for stochastic and parametric equations: a subspace point of view Nouy, Anthony | CIRM H

Multi angle

Research talks;Partial Differential Equations;Mathematics in Science and Technology;Probability and Statistics

Tensor methods have emerged as an indispensable tool for the numerical solution of high-dimensional problems in computational science, and in particular problems arising in stochastic and parametric analyses. In many practical situations, the approximation of functions of multiple parameters (or random variables) is made computationally tractable by using low-rank tensor formats. Here, we present some results on rank-structured approximations and we discuss the connection between best approximation problems in tree-based low-rank formats and the problem of finding optimal low-dimensional subspaces for the projection of a tensor. Then, we present constructive algorithms that adopt a subspace point of view for the computation of sub-optimal low-rank approximations with respect to a given norm. These algorithms are based on the construction of sequences of suboptimal but nested subspaces.

Keywords: high dimensional problems - tensor numerical methods - projection-based model order reduction - low-rank tensor formats - greedy algorithms - proper generalized decomposition - uncertainty quantification - parametric equations
Tensor methods have emerged as an indispensable tool for the numerical solution of high-dimensional problems in computational science, and in particular problems arising in stochastic and parametric analyses. In many practical situations, the approximation of functions of multiple parameters (or random variables) is made computationally tractable by using low-rank tensor formats. Here, we present some results on rank-structured approximations ...

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## Brittleness and robustness of Bayesian inference for complex systems Sullivan, Tim | CIRM H

Multi angle

Research talks;Partial Differential Equations;Mathematics in Science and Technology;Probability and Statistics

The flexibility of the Bayesian approach to uncertainty, and its notable practical successes, have made it an increasingly popular tool for uncertainty quantification. The scope of application has widened from the finite sample spaces considered by Bayes and Laplace to very high-dimensional systems, or even infinite-dimensional ones such as PDEs. It is natural to ask about the accuracy of Bayesian procedures from several perspectives: e.g., the frequentist questions of well-specification and consistency, or the numerical analysis questions of stability and well-posedness with respect to perturbations of the prior, the likelihood, or the data. This talk will outline positive and negative results (both classical ones from the literature and new ones due to the authors) on the accuracy of Bayesian inference. There will be a particular emphasis on the consequences for high- and infinite-dimensional complex systems. In particular, for such systems, subtle details of geometry and topology play a critical role in determining the accuracy or instability of Bayesian procedures. Joint with with Houman Owhadi and Clint Scovel (Caltech). The flexibility of the Bayesian approach to uncertainty, and its notable practical successes, have made it an increasingly popular tool for uncertainty quantification. The scope of application has widened from the finite sample spaces considered by Bayes and Laplace to very high-dimensional systems, or even infinite-dimensional ones such as PDEs. It is natural to ask about the accuracy of Bayesian procedures from several perspectives: e.g., the ...

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## From cluster algorithms to PDMP algorithms: a Monte Carlo story of symmetry exploitation Michel, Manon | CIRM H

Multi angle

Research talks

During this talk, I will present how the development of non-reversible algorithms by piecewise deterministic Markov processes (PDMP) was first motivated by the impressive successes of cluster algorithms for the simulation of lattice spin systems. I will especially stress how the spin involution symmetry crucial to the cluster schemes was replaced by the exploitation of more general symmetry, in particular thanks to the factorization of the energy function. During this talk, I will present how the development of non-reversible algorithms by piecewise deterministic Markov processes (PDMP) was first motivated by the impressive successes of cluster algorithms for the simulation of lattice spin systems. I will especially stress how the spin involution symmetry crucial to the cluster schemes was replaced by the exploitation of more general symmetry, in particular thanks to the factorization of the ...

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## Large deviations for Poisson driven processes in epidemiology Kratz, Peter | CIRM H

Multi angle

Research talks;Mathematics in Science and Technology;Probability and Statistics

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## Mathématiques du hasard et de l'évolution Méléard, Sylvie | CIRM H

Multi angle

Outreach;Mathematics Education and Popularization of Mathematics;Mathematics in Science and Technology

Lorsque l'on évoque Darwin et la théorie de l'évolution, on ne pense pas aux mathématiques. Pourtant dès que l'on s'intéresse aux mécanismes de la sélection naturelle, au hasard de la reproduction et au rôle des mutations, il est indispensable de les utiliser.
Après une introduction historique aux idées de Darwin sur l'évolution des espèces, nous expliquons l'impact de sa théorie et de ses réflexions sur la communauté scientifique et l'influence qu'il a eue sur la modélisation mathématique des dynamiques de population ou de la génétique des populations. Nous développons quelques exemples d'objets mathématiques, tels les processus de branchement, qui permettent de prédire le futur d'une population (son extinction, sa diversité…) ou au contraire d'en connaître le passé biologique (l'ancêtre commun d'un groupe d'individus par exemple). L'introduction du hasard dans la modélisation des questions liées à la biodiversité et à l'évolution est fondamentale. Elle permet de prendre en compte les variabilités individuelles et de mieux comprendre l'impact des facteurs écologiques et génétiques sur l'évolution des espèces.
Ces idées seront illustrées par des exemples issus de travaux récents développés entre mathématiciens et biologistes.
Lorsque l'on évoque Darwin et la théorie de l'évolution, on ne pense pas aux mathématiques. Pourtant dès que l'on s'intéresse aux mécanismes de la sélection naturelle, au hasard de la reproduction et au rôle des mutations, il est indispensable de les utiliser.
Après une introduction historique aux idées de Darwin sur l'évolution des espèces, nous expliquons l'impact de sa théorie et de ses réflexions sur la communauté scientifique et l'influence ...

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## Modélisation mathématique en lien avec le stockage des déchets radioactifs Ern, Alexandre | CIRM H

Multi angle

Outreach;Mathematics Education and Popularization of Mathematics;Mathematics in Science and Technology

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## Models and approximations of some geophysical events Nkonga, Boniface | CIRM H

Multi angle

Research talks;Mathematical Physics;Mathematics in Science and Technology

shallow water - slides - incompressible fluid

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## Multiscale model reduction for flows in heterogeneous porous media Calo, Victor | CIRM H

Multi angle

Research talks;Analysis and its Applications;Computer Science;Control Theory and Optimization;Partial Differential Equations;Mathematical Physics;Mathematics in Science and Technology

We combine discrete empirical interpolation techniques, global mode decomposition methods, and local multiscale methods, such as the Generalized Multiscale Finite Element Method (GMsFEM), to reduce the computational complexity associated with nonlinear flows in highly-heterogeneous porous media. To solve the nonlinear governing equations, we employ the GMsFEM to represent the solution on a coarse grid with multiscale basis functions and apply proper orthogonal decomposition on a coarse grid. Computing the GMsFEM solution involves calculating the residual and the Jacobian on the fine grid. As such, we use local and global empirical interpolation concepts to circumvent performing these computations on the fine grid. The resulting reduced-order approach enables a significant reduction in the flow problem size while accurately capturing the behavior of fully-resolved solutions. We consider several numerical examples of nonlinear multiscale partial differential equations that are numerically integrated using fully-implicit time marching schemes to demonstrate the capability of the proposed model reduction approach to speed up simulations of nonlinear flows in high-contrast porous media.

Keywords: generalized multiscale finite element method - nonlinear PDEs - heterogeneous porous media - discrete empirical interpolation - proper orthogonal decomposition
We combine discrete empirical interpolation techniques, global mode decomposition methods, and local multiscale methods, such as the Generalized Multiscale Finite Element Method (GMsFEM), to reduce the computational complexity associated with nonlinear flows in highly-heterogeneous porous media. To solve the nonlinear governing equations, we employ the GMsFEM to represent the solution on a coarse grid with multiscale basis functions and apply ...

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## Multiscale modeling for soft matter - Perspectives and challenges Kremer, Kurt | CIRM H

Multi angle

Research talks

Material properties of soft matter are governed by a delicate interplay of energetic and entropic contributions. In other words, generic universal aspects are as relevant as local chemistry specific properties. Thus many different time and length scales are intimately coupled, which often makes a clear separation of scales difficult. This introductory lecture will review recent advances in multiscale modeling of soft matter. This includes different approaches of sequential and concurrent coupling. Furthermore problems of representability and transferability will be addressed as well as the question of scaling of time upon coarse graining. Finally some new developments related to data driven methods will be shortly mentioned. Material properties of soft matter are governed by a delicate interplay of energetic and entropic contributions. In other words, generic universal aspects are as relevant as local chemistry specific properties. Thus many different time and length scales are intimately coupled, which often makes a clear separation of scales difficult. This introductory lecture will review recent advances in multiscale modeling of soft matter. This includes ...

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## Numerical studies of space filling designs: optimization algorithm and subprojection properties Iooss, Bertrand | CIRM H

Multi angle

Research talks;Probability and Statistics

discrepancy, optimal design, Latin Hypercube Sampling, computer experiment

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## Pourquoi l'eau tombe t-elle d'un verre qu'on retourne? Soulard, Olivier | CIRM H

Multi angle

Research talks;Mathematics Education and Popularization of Mathematics;Mathematical Physics

The purpose of this presentation is to describe the basic phenomenology of the Rayleigh-Taylor instability, from its early linear phase to its late turbulent and self-similar regime. Simple experiments are performed to illustrate this phenomenology.
fluid mechanics - Rayleigh-Taylor instability - turbulence

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## Rare event simulation for molecular dynamics Guyader, Arnaud | CIRM H

Multi angle

Research talks

This talk is devoted to the presentation of algorithms for simulating rare events in a molecular dynamics context, e.g., the simulation of reactive paths. We will consider $\mathbb{R}^d$ as the space of configurations for a given system, where the probability of a specific configuration is given by a Gibbs measure depending on a temperature parameter. The dynamics of the system is given by an overdamped Langevin (or gradient) equation. The problem is to find how the system can evolve from a local minimum of the potential to another, following the above dynamics. After a brief overview of classical Monte Carlo methods, we will expose recent results on adaptive multilevel splitting techniques. This talk is devoted to the presentation of algorithms for simulating rare events in a molecular dynamics context, e.g., the simulation of reactive paths. We will consider $\mathbb{R}^d$ as the space of configurations for a given system, where the probability of a specific configuration is given by a Gibbs measure depending on a temperature parameter. The dynamics of the system is given by an overdamped Langevin (or gradient) equation. The ...

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## Semiclassical methods for Langevin dynamics at low temperature Laurent, Michel | CIRM H

Multi angle

Research talks

Semiclassical methods have shown to be very efficient to get quantitative description of metastability of Langevin dynamics. In this talk we try to explain the main ideas of this approach in both reversible and non-reversible cases.

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## Variance reduction approaches for stochastic homogenization Legoll, Frédéric | CIRM H

Multi angle

Research talks;Partial Differential Equations;Probability and Statistics

The simulation of random heterogeneous materials is often very expensive. For instance, in a homogenization setting, the homogenized coefficient is defined from the so-called corrector function, that solves a partial differential equation set on the entire space. This is in contrast with the periodic case, where he corrector function solves an equation set on a single periodic cell. As a consequence, in the stochastic setting, the numerical approximation of the corrector function (and therefore of the homogenized coefficient) is a challenging computational task.
In practice, the corrector problem is solved on a truncated domain, and the exact homogenized coefficient is recovered only in the limit of infinitely large domains. As a consequence of this truncation, the approximated homogenized coefficient turns out to be stochastic, even though the exact homogenized coefficient is deterministic. One then has to resort to Monte-Carlo methods, in order to compute the expectation of the (approximated, apparent) homogenized coefficient within a good accuracy. Variance reduction questions thus naturally come into play, in order to increase the accuracy (e.g. reduce the size of the confidence interval) for a fixed computational cost. In this talk, we will present some variance reduction approaches to address this question.
The simulation of random heterogeneous materials is often very expensive. For instance, in a homogenization setting, the homogenized coefficient is defined from the so-called corrector function, that solves a partial differential equation set on the entire space. This is in contrast with the periodic case, where he corrector function solves an equation set on a single periodic cell. As a consequence, in the stochastic setting, the numerical ...

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## Free energy computations:a mathematical perspective Lelièvre, Tony ; Rousset, Mathias ; Stoltz, Gabriel | Imperial College Press 2010

Ouvrage

- xii; 458 p.
ISBN 978-1-84816-247-1

Localisation : Ouvrage RdC (LELI)

physique # analyse numérique # energie libre # thermodynamique statistique # méthode adaptative

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## Mathematical methods for the magnetohydrodynamics of liquid metals Gerbeau, Jean-Frédéric ; Le Bris, Claude ; Lelièvre, Tony | Oxford University Press 2006

Ouvrage

- 305 p.
ISBN 978-0-19-856665-6

Numerical mathematics and scientific computation

Localisation : Ouvrage RdC (GERB)

magnétohydrodynamique # mécanique des fluides # approximation numérique # application industrielle

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Ressources Electroniques (Depuis le CIRM)

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