Ergodic SDEs on submanifolds and related numerical sampling schemes

Abstract

In many applications, it is often necessary to sample the mean value of certain quantity with respect to a probability measure µ on the level set of a smooth function ξ :A specially interesting case is the so-called conditional probability measure, which is useful in the study of free energy calculation and model reduction of diffusion processes. By Birkhoff's ergodic theorem, one approach to estimate the mean value is to compute the time average along an infinitely long trajectory of an ergodic diffusion process on the level set whose invariant measure is µ. Motivated by the previous work of Ciccotti, Lelièvre, and Vanden-Eijnden [11], as well as the work of Lelièvre, Rousset, and Stoltz [33], in this paper we construct a family of ergodic diffusion processes on the level set of ξ whose invariant measures coincide with the given one. For the conditional measure, in particular, we show that the corresponding SDEs of the constructed ergodic processes have relatively simple forms, and, moreover, we propose a consistent numerical scheme which samples the conditional measure asymptotically. The numerical scheme doesn't require computing the second derivatives of ξ and the error estimates of its long time sampling efficiency are obtained.

Coulomb Gases Under Constraint: Some Theoretical and Numerical Results

“…A recent strategy has been to introduce a reversibility check in addition to the standard acceptionrejection rule, which makes the HMC scheme under constraint reversible [52,35]. Note that [53] proposes an interesting alternative to the scheme used here, which is however not compatible with a Metropolis selection procedure in its current form. We thus present the algorithm as written in [35], with some simplifications and adaptations to our context, for which we introduce next the constrained Langevin dynamics.…”

Jarzynski’s Equality, Fluctuation Theorems, and Variance Reduction: Mathematical Analysis and Numerical Algorithms

“…First of all, we recall some notations as well as some results from the work [70,69] in order to introduce the problem under investigation.…”

“…the scalar product u, v a −1 = u T a −1 v, for u, v ∈ R n . It is shown in [69] that, starting from y(0) ∈ Σ z , the process…”

“…dy is the normalization constant. Letting τ → 0 and applying [69,Proposition 3], we obtain Σ z(0) E y,0 e −βW (t) ϕ(y(t)) µ z(0) (dy)…”

Order Conditions for Sampling the Invariant Measure of Ergodic Stochastic Differential Equations on Manifolds

“…We quote in particular the Markov chain Monte Carlo (MCMC) methods [10,27,45] and the hybrid Monte Carlo methods [46,65], where the need for a reverse projection check is shown to be a key step. We also mention the integrators in [47,66] that are based on an Euler discretization and present new approaches for projecting on the manifold. The alternative approach of using Metropolis-Hastings rejection procedure allows to fully remove the bias on the invariant measure.…”