Weak convergence and invariant measure of a full discretization for non-globally Lipschitz parabolic SPDE

Abstract

Approximating the invariant measure and the expectation of the functionals for parabolic stochastic partial differential equations (SPDEs) with non-globally Lipschitz coefficients is an active research area and is far from being well understood. In this article, we study such problem in terms of a full discretization based on the spectral Galerkin method and the temporal implicit Euler scheme. By deriving the a priori estimates and regularity estimates of the numerical solution via a variational approach and Malliavin calculus, we establish the sharp weak convergence rate of the full discretization. When the SPDE admits a unique V -uniformly ergodic invariant measure, we prove that the invariant measure can be approximated by the full discretization. The key ingredients lie on the time-independent weak convergence analysis and time-independent regularity estimates of the corresponding Kolmogorov equation. Finally, numerical experiments confirm the theoretical findings.

Weak convergence rates for an explicit full-discretization of stochastic Allen-Cahn equation with additive noise

“…Since no expression for the exact solution is available, we identify the 'exact' solution by using sufficiently small step-size. Particularly, we take M exact = 2 20 ; N exact = 2 10 to compute the 'exact' solution for the spatial discretization and take N exact = 1000, M exact = 2 15 for the temporal discretization, respectively. In the left picture of Figure 1, we depict the weak errors due to the spatial discretization, against space step-sizes 1 N , N = 2 i , i = 1, 2, .…”

“…In this paper, however, we develop a different and more direct approach for the weak error analysis, which does not rely on the use of the regularized Kolmogorov equation. Furthermore, we highlight that the proposed fully discrete scheme with explicit time-stepping is more computationally efficient than the nonlinearity-implicit time-stepping in [15], which is the first paper to analyze weak error of a fully discrete scheme for the stochastic Allen-Cahn equations.…”

“…By contrast, the study of weak approximations of SPDEs with non-globally Lipschitz coefficients is still at an early stage. Existing publications include [10], [14] and [15], where the authors analyzed the weak error of temporal semi-discretization splitting schemes, spatial finite element semi-discretization and implicit full-discretization of the stochastic Allen-Cahn equations, respectively. All of these three works employed a splitting strategy to construct a nice auxiliary continuous-time process with appropriate spatio-temporal regularity properties and deduce the weak convergence rate from the regularity of the regularized Kolmogorov equation.…”

Numerical analysis of a full discretization for stochastic Cahn--Hilliard equation driven by additive noise

“…In contrast to the Lipschitz case, convergence and convergence rate of numerical approximation for SPDEs with non-globally Lipschitz continuous nonlinearity, become more involved recently (see e.g. [1,3,4,5,6,7,8,9,10,11,13,14,15,16,17,18,20,22,23,25,28,29,30,31]) and are far from well-understood. Up to now, there have been some works focusing on the strong convergence and strong convergence rates of numerical schemes for the stochastic Cahn-Hilliard equation (see e.g.…”

Finite-Volume approximation of the invariant measure of a viscous stochastic scalar conservation law

“…Those results hold under a global Lipschitz assumption on the nonlinearity. In the more recent works [12,13], non-Lipschitz nonlinearities are considered, but they still need to satisfy a one-sided Lipschitz condition.…”