Convergence rates for nonequilibrium Langevin dynamics

Abstract

International audienceWe study the exponential convergence to the stationary state for nonequilibrium Langevin dynamics, by a perturbative approach based on hypocoercive techniques developed for equilibrium Langevin dynamics. The Hamiltonian and overdamped limits (corresponding respectively to frictions going to zero or infinity) are carefully investigated. In particular, the maximal magnitude of admissible perturbations are quantified as a function of the friction. Numerical results based on a Galerkin discretization of the generator of the dynamics confirm the theoretical lower bounds on the spectral gap

Using Perturbed Underdamped Langevin Dynamics to Efficiently Sample from Probability Distributions

“…Under the above choices (27), we show that the large perturbation limit lim μ→∞ σ 2 f exists and is finite and we provide an explicit expression for it (see Corollary 4). From this expression, we derive an algorithm for finding optimal choices for J 1 in the case of quadratic observables (see Algorithm 2).…”

“…In particular, in Proposition 3 we show that under certain conditions on the spectrum of J 1 , for any antisymmetric observable f ∈ L 2 (π) it holds that lim μ→∞ σ 2 f = 0. Numerical experiments and analysis show that departing significantly from (27) in fact possibly decreases the performance of the sampler. This is in stark contrast to (7), where it is not possible to increase the asymptotic variance by any perturbation.…”

“…In the case of Bayesian inverse problems and diffusion bridge sampling, the target measure π is given with respect to a Gaussian prior. We demonstrate the effectiveness of our approach in these applications, taking the prior Gaussian covariance as S in (27).…”

“…In fact, we conjecture that this statement is true for arbitrary observables f ∈ L 2 (π), but we have not been able to prove this. The dynamics (8) [used in conjunction with the conditions (27)] proves to be especially effective when the observable is antisymmetric (i.e. when it is invariant under the substitution q → −q) or when it has a significant antisymmetric part.…”

“…Using the framework of [15], we conjecture that the assumption on the boundedness of the Hessian of V can be substantially weakened and more quantitative decay estimates (in particular with respect to μ and ν) can be obtained. This approach has recently been successfully applied to equilibrium and nonequilibirum Langevin dynamics, see [27,53]. We leave this work track for future study.…”