Representation of Gaussian fields in series with independent coefficients

Abstract

The numerical discretization of problems with stochastic data or stochastic parameters generally involves the introduction of coordinates that describe the stochastic behavior, such as coefficients in a series expansion or values at discrete points. The series expansion of a Gaussian field with respect to any orthonormal basis of its Cameron–Martin space has independent standard normal coefficients. A standard choice for numerical simulations is the Karhunen–Loeve series, which is based on eigenfunctions of the covariance operator. We suggest an alternative, the hierarchic discrete spectral expansion, which can be constructed directly from the covariance kernel. The resulting basis functions are often well localized, and the convergence of the series expansion seems to be comparable to that of the Karhunen–Loeve series. We provide explicit formulas for particular cases, and general numerical methods for computing exact representations of such bases. Finally, we relate our approach to numerical discretizations based on replacing a random field by its values on a finite set. Introduction The numerical discretization of problems with stochastic data or stochastic parameters requires that the random inputs are approximated by finite quantities. This is generally done in one of two ways. Either the random data is expanded in a series, which can be truncated for numerical computations, or it is replaced by a finite dimensional random variable, describing for example the value of a random field on a discrete set of points. A standard approach, falling strictly into the first category, is to expand a random field into its Karhunen–Loeve series. For Gaussian fields, the coefficients in this series are independent standard normal random variables. The independence of these coefficients is crucial to many numerical methods. For example, in Monte Carlo simulation, coefficient sequences can be generated by independent draws of pseudorandom numbers. The construction of polynomial chaos bases as tensor products of orthonormal bases with respect to the distributions of the coefficients also requires that these are independent. Similarly, in collocation and quasi Monte Carlo methods, constructions of collocation points make use of the product structure of the joint distribution of the coefficients. Nevertheless, the Karhunen–Loeve series is often ill-suited for numerical computations, as it requires eigenfunctions of the covariance operator. These are usually not known exactly, and are expensive to approximate numerically. Furthermore, the eigenfunctions generally have global supports. We suggest an alternative to the Karhunen–Loeve series for general continuous Gaussian fields on bounded domains, which we call the hierarchic discrete spectral expansion. Assuming that the covariance kernel is given, the basis functions in our series expansion can be constructed exactly. As these form an orthonormal basis of Date: September 29, 2010. Research supported in part by the Swiss National Science Foundation grant No. 200021120290/1. The author wishes to thank Hans-Jorg Starkloff for his insightful suggestions. 1