High Order Integrator for Sampling the Invariant Distribution of a Class of Parabolic Stochastic PDEs with Additive Space-Time Noise

Abstract

We introduce a time-integrator to sample with high order of accuracy the invariant distribution for a class of semilinear SPDEs driven by an additive space-time noise. Combined with a postprocessor, the new method is a modification with negligible overhead of the standard linearized implicit Euler-Maruyama method. We first provide an analysis of the integrator when applied for SDEs (finite dimension), where we prove that the method has order 2 for the approximation of the invariant distribution, instead of 1. We then perform a stability analysis of the integrator in the semilinear SPDE context, and we prove in a linear case that a higher order of convergence is achieved. Numerical experiments, including the semilinear heat equation driven by space-time white noise, confirm the theoretical findings and illustrate the efficiency of the approach.

Influence of the regularity of the test functions for weak convergence in numerical discretization of SPDEs

“…< ∞ almost surely, thanks to (9). Finally, letting ǫ → 0, for all N ∈ N and ∆t ∈ (0, 1) Then, thanks to the regularity results (7), (8) and (9) for all N ∈ N and ∆t ∈ (0, 1),…”

“…Deriving weak convergence rates is fundamental in infinite dimension, see for instance [25]. Moreover, it is the appropriate notion for the approximation of invariant distribution (in the asymptotic regime T → ∞), see [5], [8], [9]. The extension of the results of this article in this regime is straightforward.…”

“…Similar arguments will be used again below. 6 To prove the first statement in (9), let ∆t > 0, k ∈ N, and α ∈ [0, 1], then…”

Exotic aromatic B-series for the study of long time integrators for a class of ergodic SDEs

“…The approach combines the usual Talay-Tubaro methodology [46] and recent developments of the theory of backward error analysis and modified differential equations in the stochastic context [48,2,21,28,29], a major tool in the area of deterministic geometric numerical integration [23]. In [47] for finite dimensions and in [10] in the context of parabolic stochastic partial differential equations, this approach is combined with the idea of processing from Butcher [13], to design efficient postprocessed integrators with high order for the invariant measure at a negligible overcost compared to standard low order schemes. The postprocessor methodology is extended in [1] for a class of explicit stabilized schemes of order two for the invariant measure and with optimally large stability domains.…”

Parareal Exponential $\theta$-Scheme for Longtime Simulation of Stochastic Schrödinger Equations with Weak Damping

“…In the numerical approximation for both deterministic and stochastic evolution equations, several methods have been developed to improve the convergence order of classical schemes, such as (partitioned) Runge-Kutta methods, schemes via modified equations, predictorcorrector schemes and so on (see [4,14,17,18] and references therein). For high order numerical approximations of stochastic partial differential equations (SPDEs), the computing cost can be prohibitively large due to the high dimension in space, especially for longtime simulations.…”