A generalized spectral decomposition technique to solve a class of linear stochastic partial differential equations

Abstract

We propose a new robust technique for solving a class of linear stochastic partial differential equations. The solution is approximated by a series of terms, each of which being the product of a scalar stochastic function by a deterministic function. None of these functions are fixed a priori but determined by solving a problem which can be interpreted as an “extended” eigenvalue problem. This technique generalizes the classical spectral decomposition, namely the Karhunen–Loève expansion. Ad hoc iterative techniques to build the approximation, inspired by the power method for classical eigenproblems, then transform the problem into the resolution of a few uncoupled deterministic problems and stochastic equations. This method drastically reduces the calculation costs and memory requirements of classical resolution techniques used in the context of Galerkin stochastic finite element methods. Finally, this technique is particularly suitable to non-linear and evolution problems since it enables the construction of a relevant reduced basis of deterministic functions which can be efficiently reused for subsequent resolutions.