Stochastic Galerkin Methods

Abstract

Chapter 11considered spectral expansions of square-integrable random variables, random vectors and random fields of the form U=∑k∈ℕ0uk$Ψ$k,{\$}{\$}{\backslash}displaystyle{\{}U ={\backslash}sum {\_}{\{}k{\backslash}in {\backslash}mathbb{\{}N{\}}{\_}{\{}0{\}}{\}}u{\_}{\{}k{\}}{\backslash}varPsi {\_}{\{}k{\}},{\}}{\$}{\$}where U∈L2($Θ$,$μ$;?){\$}{\$}U {\backslash}in L^{\{}2{\}}({\backslash}varTheta,{\backslash}mu;{\backslash}mathcal{\{}U{\}}){\$}{\$}, ?{\$}{\$}{\backslash}mathcal{\{}U{\}}{\$}{\$}is a Hilbert space in which the corresponding deterministic variables/vectors/fields lie, and {\{}$Ψ$k∣k∈ℕ0{\}}{\$}{\$}{\backslash}{\{}{\backslash}varPsi {\_}{\{}k{\}}{\backslash}mid k {\backslash}in {\backslash}mathbb{\{}N{\}}{\_}{\{}0{\}}{\backslash}{\}}{\$}{\$}is some orthogonal basis for L2($Θ$,$μ$;ℝ){\$}{\$}L^{\{}2{\}}({\backslash}varTheta,{\backslash}mu; {\backslash}mathbb{\{}R{\}}){\$}{\$}.