A Galerkin solution for stochastic algebraic equations

Abstract

Galerkin solutions are developed for linear stochastic algebraic equations, that is, linear algebraic equations with random coefficients. Two classes ofGalerkin solutions, referred to as optimal and sub-optimal Galerkin solutions, are constructed. It is shown that optimal Galerkin solutions for a linear stochastic algebraic equation are given by conditional expectations of the exact solution of this equation with respect to s-fields that are coarser than the σ-field relative to which the exact solution is measurable. The σ-fields needed to defined optimal Galerkin solutions can be constructed from, for example, σ-fields generated by measurable partitions of the sample space. Galerkin solutions that are not optimal are called sub-optimal. Optimal and sub-optimal Galerkin solutions are unbiased and biased approximations of the exact solution.