On Parareal Algorithms for Semilinear Parabolic Stochastic PDEs

Abstract

Parareal algorithms are studied for semilinear parabolic stochastic partial differential equations. These algorithms proceed as two-level integrators, with fine and coarse schemes, and have been designed to achieve a "parallel in real time" implementation. In this work, the fine integrator is given by the exponential Euler scheme. Two choices for the coarse integrator are considered: the linear implicit Euler scheme, and the exponential Euler scheme.The influence on the performance of the parareal algorithm, of the choice of the coarse integrator, of the regularity of the noise, and of the number of parareal iterations, is investigated, with theoretical analysis results and with extensive numerical experiments.

Strong approximation for fractional wave equation forced by fractional Brownian motion with Hurst parameter $H\in(0,\frac{1}{2})$

“…Stochastic partial differential equations (SPDEs) can realistically simulate many phenomena in physical scientific and engineering applications; see [9,10,19]. The theoretical and numerical studies of SPDEs have received much attention [7,8,13,16,22]. While most works of stochastic model in fractional Brownian motion (FBM) are carried out for Hurst parameter H ∈ [ 1 2 , 1) ( [2,12,14,15,20]), a FBM with H ∈ (0, 1 2 ) might be more reasonable to model sequences with intermittency and anti-persistence, such as visual feedback effects in biology [6] and option prices in market practice [5,11,21].…”

Convergence of a spatial semi-discretization for a backward semilinear stochastic parabolic equation

“…In view of the fact that the numerical analysis of the infinite-dimensional stochastic differential equations (SDEs for short) differs considerably from that of the finite-dimensional SDEs, generally it is difficult to extend the numerical analysis of finite-dimensional BSDEs to the infinitedimensional BSDEs directly. We note that there is a huge list of papers in the literature on the numerical analysis of the infinite-dimensional SDEs; see [1,2,7,9,10,13,14,18,19,29,30,51,54] and the references therein. Despite this fact, the numerical analysis of the infinite-dimensional nonlinear BSDEs is expected to be a challenging problem, since the SDEs and the BSDEs are essentially different.…”