Accelerating Gaussian Diffusions

Hwang, Chii‐Ruey; Hwang-Ma, Shu-Yin; Sheu, Shuenn‐Jyi

doi:10.1214/aoap/1177005371

Cited by 97 publications

(116 citation statements)

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“…As noticed in [23,24], one way to accelerate the convergence to equilibrium is to depart from reversible dynamics (see also [10] for related discussions for Markov Chains). Let us recall that the dynamics (2) is reversible in the sense that if X 0 is distributed according to ψ ∞ (x) dx, then (X t ) 0≤t≤T and (X T −t ) 0≤t≤T have the same law.…”

Section: Non-reversible Diffusionmentioning

confidence: 99%

“…This problem was studied in [23] for a linear drift (namely V is quadratic and b is linear) and in [24] for the general case. It was shown in these works that the addition of a drift function b satisfying (9) helps to speed up convergence to equilibrium.…”

Section: Bibliographymentioning

confidence: 99%

“…In particular, the spectral gap of the generator L is determined by the eigenvalues of B, and this yields a simple way to design the optimal matrix J opt . On the other hand, the control of the constant C(S, J) requires a more elaborate analysis, using Wick (in the sense of Wick ordered) calculus, see Section 5.3 below.Compared to the related previous paper [23], our contributions are threefold: (i) we propose an algorithm to build the optimal matrix J opt , (ii) we discuss how to get estimates on the constant C(S, J) and (iii) we consider the longtime behavior of the partial differential equation (12) and not only of ordinary differential equations related to (17). In particular, our analysis covers also non Gaussian initial conditions for the SDE (17).…”

mentioning

confidence: 99%

“…Compared to the related previous paper [23], our contributions are threefold: (i) we propose an algorithm to build the optimal matrix J opt , (ii) we discuss how to get estimates on the constant C(S, J) and (iii) we consider the longtime behavior of the partial differential equation (12) and not only of ordinary differential equations related to (17). In particular, our analysis covers also non Gaussian initial conditions for the SDE (17).…”

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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: Non-reversible Diffusionmentioning

confidence: 99%

Section: Bibliographymentioning

confidence: 99%

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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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“…The asymptotic behaviour of this process was considered for Gaussian diffusions in [20], where the rate of convergence of the covariance to equilibrium was quantified in terms of the choice of γ . This work was extended to the case of non-Gaussian target densities, and consequently for nonlinear SDEs of the form (7) in [21].…”

Section: Introductionmentioning

confidence: 99%

Using Perturbed Underdamped Langevin Dynamics to Efficiently Sample from Probability Distributions

2017

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In this paper we introduce and analyse Langevin samplers that consist of perturbations of the standard underdamped Langevin dynamics. The perturbed dynamics is such that its invariant measure is the same as that of the unperturbed dynamics. We show that appropriate choices of the perturbations can lead to samplers that have improved properties, at least in terms of reducing the asymptotic variance. We present a detailed analysis of the new Langevin sampler for Gaussian target distributions. Our theoretical results are supported by numerical experiments with non-Gaussian target measures.

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Accelerating MCMC algorithms

Robert

Elvira

Tawn

et al. 2018

WIREs Computational Stats

112

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Markov chain Monte Carlo algorithms are used to simulate from complex statistical distributions by way of a local exploration of these distributions. This local feature avoids heavy requests on understanding the nature of the target, but it also potentially induces a lengthy exploration of this target, with a requirement on the number of simulations that grows with the dimension of the problem and with the complexity of the data behind it. Several techniques are available toward accelerating the convergence of these Monte Carlo algorithms, either at the exploration level (as in tempering, Hamiltonian Monte Carlo and partly deterministic methods) or at the exploitation level (with Rao–Blackwellization and scalable methods).This article is categorized under: Statistical and Graphical Methods of Data Analysis > Markov Chain Monte Carlo (MCMC)Algorithms and Computational Methods > AlgorithmsStatistical and Graphical Methods of Data Analysis > Monte Carlo Methods

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Accelerating Gaussian Diffusions

Cited by 97 publications

References 14 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

Using Perturbed Underdamped Langevin Dynamics to Efficiently Sample from Probability Distributions

Accelerating MCMC algorithms

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