Pathwise accuracy and ergodicity of metropolized integrators for SDEs

Bou-Rabee, Nawaf; Vanden‐Eijnden, Eric

doi:10.1002/cpa.20306

Cited by 62 publications

(122 citation statements)

References 38 publications

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“…Diffusion limit results similar to ours are proved in [4,5] for finite dimensional problems. In those papers, an accept-reject mechanism is appended to various standard integrators for the first and second order Langevin equations, and shown not to destroy the strong pathwise convergence of the underlying methods.…”

Section: Introductionsupporting

confidence: 79%

“…We will be particularly interested in choices of parameters in the algorithm which ensure that the output (suitability interpolated to continuous time) behaves like (2.8) whilst, as is natural for MCMC methods, exactly preserving the invariant measure. This perspective on discretization of the (physicists) Langevin equation in finite dimensions was introduced in [4,5].…”

Section: Function Space Algorithmmentioning

confidence: 99%

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A function space HMC algorithm with second order Langevin diffusion limit

Ottobre¹,

Pillai

Pinski³

et al. 2016

Bernoulli

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We describe a new MCMC method optimized for the sampling of probability measures on Hilbert space which have a density with respect to a Gaussian; such measures arise in the Bayesian approach to inverse problems, and in conditioned diffusions. Our algorithm is based on two key design principles: (i) algorithms which are well defined in infinite dimensions result in methods which do not suffer from the curse of dimensionality when they are applied to approximations of the infinite dimensional target measure on R N ; (ii) nonreversible algorithms can have better mixing properties compared to their reversible counterparts. The method we introduce is based on the hybrid Monte Carlo algorithm, tailored to incorporate these two design principles. The main result of this paper states that the new algorithm, appropriately rescaled, converges weakly to a second order Langevin diffusion on Hilbert space; as a consequence the algorithm explores the approximate target measures on R N in a number of steps which is independent of N. We also present the underlying theory for the limiting nonreversible diffusion on Hilbert space, including characterization of the invariant measure, and we describe numerical simulations demonstrating that the proposed method has favourable mixing properties as an MCMC algorithm.

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Section: Introductionsupporting

confidence: 79%

Section: Function Space Algorithmmentioning

confidence: 99%

A function space HMC algorithm with second order Langevin diffusion limit

Ottobre¹,

Pillai

Pinski³

et al. 2016

Bernoulli

View full text Add to dashboard Cite

show abstract

“…If either D(x) or r eq (x) are discontinuous with respect to x, then the drift term will have a singularity that is not resolved by standard time-integration schemes, and numerical simulations may not correctly approximate the equilibrium density. We propose using a variant of MALA as introduced by Roberts & Tweedie (1996) and analysed by Bou-Rabee & Vanden-Eijnden (2010). MALA is obtained by taking a convergent numerical method for the SDE (in this case, Euler-Maruyama) and introducing Metropolis step-rejections in order to have a discrete-time process with the correct equilibrium density.…”

Section: (C) Numerical Simulationmentioning

confidence: 99%

A paradox of state-dependent diffusion and how to resolve it

Tupper

Yang

2012

Proc. R. Soc. A.

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Consider a particle diffusing in a confined volume which is divided into two equal regions. In one region, the diffusion coefficient is twice the value of the diffusion coefficient in the other region. Will the particle spend equal proportions of time in the two regions in the long term? Statistical mechanics would suggest yes, since the number of accessible states in each region is presumably the same. However, another line of reasoning suggests that the particle should spend less time in the region with faster diffusion, since it will exit that region more quickly. We demonstrate with a simple microscopic model system that both predictions are consistent with the information given. Thus, specifying the diffusion rate as a function of position is not enough to characterize the behaviour of a system, even assuming the absence of external forces. We propose an alternative framework for modelling diffusive dynamics in which both the diffusion rate and equilibrium probability density for the position of the particle are specified by the modeller. We introduce a numerical method for simulating dynamics in our framework that samples from the equilibrium probability density exactly and is suitable for discontinuous diffusion coefficients.

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“…In many applications (see [25,16,5,19] and references therein), one is interested in simulating the invariant measure of a stochastic differential equation (SDE) by running a numerical scheme that approximates its time dynamics. In particular, one uses the numerical trajectories to construct an empirical measure either by averaging one single long trajectory or by averaging over many realisations to obtain a finite ensemble average (see for example [25]).…”

Section: Introductionmentioning

confidence: 99%

Long Time Accuracy of Lie--Trotter Splitting Methods for Langevin Dynamics

Abdulle¹,

Vilmart²,

Zygalakis³

2015

SIAM J. Numer. Anal.

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A new characterization of sufficient conditions for the Lie-Trotter splitting to capture the numerical invariant measure of nonlinear ergodic Langevin dynamics up to an arbitrary order is discussed. Our characterization relies on backward error analysis and needs weaker assumptions than assumed so far in the literature. In particular, neither high weak order of the splitting scheme nor symplecticity are necessary to achieve high order approximation of the invariant measure of the Langevin dynamics. Numerical experiments confirm our theoretical findings.

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Pathwise accuracy and ergodicity of metropolized integrators for SDEs

Cited by 62 publications

References 38 publications

A function space HMC algorithm with second order Langevin diffusion limit

A function space HMC algorithm with second order Langevin diffusion limit

A paradox of state-dependent diffusion and how to resolve it

Long Time Accuracy of Lie--Trotter Splitting Methods for Langevin Dynamics

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