White Noise Analysis for Stochastic Partial Differential Equations

Stochastic partial differential equations arise when modelling uncertain phenomena. Here the emphasis is on uncertain systems where the randomness is spatial. In contrast to traditional slow computational approaches like Monte Carlo simulation, the methods described here can be orders of magnitude more efficient. These more recent methods are based on some kind stochastic Galerkin approximations, approximating the unknown quantities as functions of independent random variables, hence the name "white noise analysis". We outline the steps leading to the fully discrete equations, commenting on one possible numerical solution method. Key to many of the developments is tensor product structure of the solution, which must be exploited both theoretically and numerically. For two examples with polynomial non-linearities the computations are shown to be quite explicit and can be performed largely analytically.