Two Mathematical Tools to Analyze Metastable Stochastic Processes

Lelièvre, Tony

doi:10.1007/978-3-642-33134-3_83

Cited by 24 publications

(28 citation statements)

References 46 publications

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“…Proposition 8 Let ψ J t satisfy the Fokker-Planck equation (25), with an initial probability law with density ψ J 0 and such that R N e α x 2 ψ J 0 dx < +∞ for some positive α . Assume moreover that J = S −1/2J S −1/2 , that (J , Q) are chosen in P opt and that t 0 is given by (54).…”

Section: General Initial Densitiesmentioning

confidence: 99%

“…However, even for nonlinear problems it is always the case that the addition of a nonreversible perturbation can accelerate the convergence to equilibrium, as mentioned in the introduction. This is particularly the case for systems with metastable states and/or multiscale structure [25]; for such systems, a "clever" choice of the nonreversible perturbation can lead to a very significant increase in the rate of convergence to equilibrium. A systematic methodology for obtaining the optimal nonreversible perturbation for general reversible diffusions (i.e.…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…In view of (5), the algorithm is efficient if λ is large, which is typically not the case if X t is a metastable process (see [25]). A natural question is therefore how to design a Markovian dynamics which converges to equilibrium distribution ψ ∞ (much) faster than (2).…”

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confidence: 99%

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Optimal Non-reversible Linear Drift for the Convergence to Equilibrium of a Diffusion

2013

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We consider non-reversible perturbations of reversible diffusions that do not alter the invariant distribution and we ask whether there exists an optimal perturbation such that the rate of convergence to equilibrium is maximized. We solve this problem for the case of linear drift by proving the existence of such optimal perturbations and by providing an easily implementable algorithm for constructing them. We discuss in particular the role of the prefactor in the exponential convergence estimate. Our rigorous results are illustrated by numerical experiments.Keywords Non-reversible diffusion · Convergence to equilibrium · Wick calculus 1 Introduction MotivationThe problem of convergence to equilibrium for diffusion processes has attracted considerable attention in recent years. In addition to the relevance of this problem for the convergence to equilibrium of some systems in statistical T. Lelièvre at Université Paris-Est, Cermics, and INRIA, MicMac project team, Ecole des ponts, 6-8 avenue Blaise Pascal, 77455 Marne la Vallée Cedex 2, France. E-mail: lelievre@cermics.enpc.fr · F. Nier at IRMAR, Campus de Beaulieu, Université de Rennes 1, 35042 Rennes, France. E-mail: francis.nier@univ-rennes1.fr · G.A. Pavliotis at Imperial College London, Department of Mathematics, South Kensington Campus, London SW7 2AZ, England. E-mail: g.pavliotis@imperial.ac.uk 2 physics, see for example [29], such questions are also important in statistics, for example in the analysis of Markov Chain Monte Carlo (MCMC) algorithms [9]. Roughly speaking, one measure of efficiency of an MCMC algorithm is its rate of convergence to equilibrium, and increasing this rate is thus the aim of many numerical techniques (see for example [5]).Let us recall the basic approach for a reversible diffusion. Suppose that we are interested in sampling from a probability distribution functionwhere V : R N → R is a given smooth potential such that R N e −V dx < ∞ . A natural dynamics to use is the reversible dynamicswhere W t denotes a standard N -dimensional Brownian motion. Let us denote by ψ t the probability density function of the process X t at time t. It satisfies the Fokker-Planck equationUnder appropriate assumptions on the potential V (e.g. that 1 2 |∇V (x)| 2 − ∆V (x) → +∞ as |x| → +∞ , see [41, A.19]), the density ψ ∞ satisfies a Poincaré inequality: there exists λ > 0 such that for all probability density functions φ,The optimal parameter λ in (4) is the opposite of the smallest (in absolute value) non-zero eigenvalue of the Fokker-Planck operator ∇ · (∇V · +∇·), which is self-adjoint in L 2 (R N , ψ −1 ∞ dx) (see (7) below). Thus, λ is also called the spectral gap of the Fokker-Planck operator.It is then standard to show that (4) is equivalent to the following inequality, which shows exponential convergence to the equilibrium for (2): for all initial conditions ψ 0 ∈ L 2 (R N , ψ −1 ∞ dx), for all times t ≥ 0,where · L 2 (ψ −1 ∞ ) denotes the norm in L 2 (R N , ψ −1 ∞ ), namely f 2∞ (x) dx . This equivalence is a simple consequence of th...

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Section: General Initial Densitiesmentioning

confidence: 99%

Section: Numerical Experimentsmentioning

confidence: 99%

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Optimal Non-reversible Linear Drift for the Convergence to Equilibrium of a Diffusion

2013

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“…This is related to the fact that the timescale at the microscopic level is much smaller than the timescale at the macroscopic level. In [66], we propose a summary of the mathematical approaches to quantify metastability, and which appear to be useful to analyze the numerical methods used in molecular dynamics.…”

Section: Computational Statistical Physicsmentioning

confidence: 99%

Computational Statistical Physics

Santra¹,

Ray²

2011

Texts and Readings in Physical Sciences

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“…It is often a detailed theoretical and numerical analysis of metastability that allows for a proper upscaling of microscopic models -defined at the atomic or molecular level -to macroscopic descriptions catching experimental observations in materials science, biology and chemistry. We refer to Kramers' influential [14], to [10] and to [2,16,7] for recent brief reviews on some of the mathematical techniques used in the analysis of metastability.…”

Section: Introductionmentioning

confidence: 99%

Full Metastable Asymptotic of the Fisher Information

Gesù¹,

Mariani²

2017

SIAM J. Math. Anal.

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Abstract. We establish an expansion by Γ-convergence of the Fisher information relative to the reference measure e −βV dx, where V is a generic multiwell potential and β → ∞. The expansion reveals a hierarchy of multiple scales reflecting the metastable behavior of the underlying overdamped Langevin dynamics: distinct scales emerge and become relevant depending on whether one considers probability measures concentrated on local minima of V , probability measures concentrated on critical points of V , or generic probability measures on R d . We thus fully describe the asymptotic behavior of minima of the Fisher information over regular sets of probabilities. The analysis mostly relies on spectral properties of diffusion operators and the related semiclassical Witten Laplacian and covers also the case of a compact smooth manifold as underlying space.

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Two Mathematical Tools to Analyze Metastable Stochastic Processes

Cited by 24 publications

References 46 publications

Optimal Non-reversible Linear Drift for the Convergence to Equilibrium of a Diffusion

Optimal Non-reversible Linear Drift for the Convergence to Equilibrium of a Diffusion

Computational Statistical Physics

Full Metastable Asymptotic of the Fisher Information

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