Spectral Expansions

Abstract

In this chapter, we discuss fundamental and practical aspects of spectral expansions of random model data, and of model solutions. We focus on a specific class of random process in L ² and seek Fourier-like expansions that are convergent with respect to the norm associated with the corresponding inner product. We focus our attention the Karhunen-Lov̀e (KL) representation of the stochastic process u. This classical approach essentially amounts to a biorthogonal decomposition based on the eigenfunctions obtained through analysis of its correlation function. Basic results pertaining to these decompositions are first outlined; implementation of KL decompositions is then outlined through specific examples. Polynomial Chaos (PC) representations of random variables are then outlined, starting with the classical concepts of Homogeneous Chaos, and the associated one-dimensional and multi-dimensional Hermite bases expansions. These concepts are later extended to generalized PC expansions based on different families of orthogonal polynomials, as well as situations involving dependent random variables. Finally these expansion are extended to random vectors and stochastic processes, and an elementary road map is provided for the applications of spectral representations.