The quantification of probabilistic uncertainties in the outputs of physical, biological, and social systems governed by partial differential equations with random inputs require, in practice, the discretization of those equations. Stochastic finite element methods refer to an extensive class of algorithms for the approximate solution of partial differential equations having random input data, for which spatial discretization is effected by a finite element method. Fully discrete approximations require further discretization with respect to solution dependences on the random variables. For this purpose several approaches have been developed, including intrusive approaches such as stochastic Galerkin methods, for which the physical and probabilistic degrees of freedom are coupled, and non-intrusive approaches such as stochastic sampling and interpolatory-type stochastic collocation methods, for which the physical and probabilistic degrees of freedom are uncoupled. All these method classes are surveyed in this article, including some novel recent developments. Details about the construction of the various algorithms and about theoretical error estimates and complexity analyses of the algorithms are provided. Throughout, numerical examples are used to illustrate the theoretical results and to provide further insights into the methodologies.

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In this paper we propose a stable multistep scheme on time-space grids for solving backward stochastic differential equations. In our scheme, the integrands, which are conditional mathematical expectations derived from the original equations, are approximated by using Lagrange interpolating polynomials with values of the integrands at multiple time levels. They are then numerically evaluated using the Gauss-Hermite quadrature rules and polynomial interpolations on the spatial grids. Error estimates are rigorously proved for the semidiscrete version of the proposed scheme for backward stochastic differential equations with certain types of simplified generator functions. Finally, various numerical examples and comparisons with some other methods are presented to demonstrate high accuracy of the proposed multistep scheme.

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