Recent Developments in Spectral Stochastic Methods for the Numerical Solution of Stochastic Partial Differential Equations

Uncertainty quantification appears today as a crucial point in numerous branches of science and engineering. In the last two decades, a growing interest has been devoted to a new family of methods, called spectral stochastic methods, for the propagation of uncertainties through physical models governed by stochastic partial differential equations. These approaches rely on a fruitful marriage of probability theory and approximation theory in functional analysis. This paper provides a review of some recent developments in computational stochastic methods, with a particular emphasis on spectral stochastic approaches. After a review of different choices for the functional representation of random variables, we provide an overview of various numerical methods for the computation of these functional representations: projection, collocation, Galerkin approaches…. A detailed presentation of Galerkin-type spectral stochastic approaches and related computational issues is provided. Recent developments on model reduction techniques in the context of spectral stochastic methods are also discussed. The aim of these techniques is to circumvent several drawbacks of spectral stochastic approaches (computing time, memory requirements, intrusive character) and to allow their use for large scale applications. We particularly focus on model reduction techniques based on spectral decomposition techniques and their generalizations.