Adaptive Wick--Malliavin Approximation to Nonlinear SPDEs with Discrete Random Variables

Abstract

We propose an adaptive Wick--Malliavin (WM) expansion in terms of the Malliavin derivative of order $Q$ to simplify the propagator of general polynomial chaos (gPC) of order $P$ (a system of deterministic equations for the coefficients of gPC) and to control the error growth with respect to time. Specifically, we demonstrate the effectiveness of the WM method by solving a stochastic reaction equation and a Burgers equation with several discrete random variables. Exponential convergence is shown numerically with respect to $Q$ when $Q \geq P-1$. We also analyze the computational complexity of the WM method and identify a significant speedup with respect to gPC, especially in high dimensions.