Stochastic galerkin and collocation methods for quantifying uncertainty in differential equations: a review

Abstract

The article reviews the mathematical theory of stochastic Galerkin and stochastic collocation methods, focusing on their strengths and limitations. The aim is to construct a first stop, widely accessible document that directs a reader to more detailed descriptions of stochastic Galerkin and stochastic collocation methods that are suitable for their application of interest. References point to rigorous convergence proofs and accuracy estimates, computational considerations and numerical examples. A supplementary document gives a quick look-up guide to the strengths and weaknesses of stochastic Galerkin and collocation methods. References M. K. Deb, I. M. Babuska, and J. T Oden. Solution of stochastic partial differential equations using Galerkin finite element techniques. Comput. Methods Appl. Mech. Engrg., 190:6359--6372, 2001. doi:10.1016/S0045-7825(01)00237-7 B. J. Debusschere, H. N. Najm, P. P. Pebay, O. M. Knio, R. G. Ghanem, and O. P. Le Maitre. 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