ADAPTIVE WICK-MALLIAVIN APPROXIMATION TO NONLINEAR SPDEs WITH DISCRETE RANDOM VARIABLES

作者：Zheng Mengdi^*; Rozovsky Boris; Karniadakis George E

来源：SIAM Journal on Scientific Computing, 2015, 37(4): A1872-A1890.

DOI：10.1137/140975930

• discrete stochastic process
• general polynomial chaos
• high dimensions
• Wiener-Askey scheme

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 >= 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.

• 出版日期2015