Stochastic polynomial chaos based algorithm for solving PDEs with random coefficients

Abstract

A generalization of a polynomial chaos-based algorithm for solving PDEs with random input data is suggested. The input random field is assumed to be defined by its mean and correlation function. The method uses the Karhunen–Loève expansion, in its analytical form, for the input random field. Potentially, however, if desired, the Karhunen–Loève expansion can be also constructed by a randomized singular value decomposition of the correlation function recently suggested in our paper [Math. Comput. Simulation 82 (2011), 295–317]. The polynomial chaos expansion is then constructed by resolving a probabilistic collocation-based system of linear equations. The method is compared against a direct Monte Carlo method which solves repeatedly many times the PDE for a set of samples of the input random field. Along with the commonly used statistical characteristics like the mean and variance of the solution, we were able to calculate more sophisticated functionals like the instant velocity samples and the mean for Eulerian and Lagrangian velocity fields.