Implementation of Karhunen–Loeve expansion for simulation using a wavelet-Galerkin scheme

Abstract

The feasibility of implementing Karhunen–Loeve (K–L) expansion as a practical simulation tool hinges crucially on the ability to compute a large number of K–L terms accurately and cheaply. This study presents a simple wavelet-Galerkin approach to solve the Fredholm integral equation for K–L simulation. The proposed method has significant computational advantages over the conventional Galerkin method. Wavelet bases provide localized compact support, which lead to sparse representations of functions and integral operators. Existing efficient numerical scheme to obtain wavelet coefficients and inverse wavelet transforms can be taken advantage of solving the integral equation. The computational efficiency of the wavelet-Garlekin method is illustrated using two stationary covariance functions (exponential and squared exponential) and one non-stationary covariance function (Wiener–Levy). The ability of the wavelet-Galerkin approach to compute a large number of eigensolutions accurately and cheaply can be exploited to great advantage in implementing the K–L expansion for practical simulation.

Introduction

Stochastic simulation of practical problems has become increasingly attractive with the rapid advancement in computer technology. A unified procedure based on Karhunen–Loeve (K–L) expansion to simulate stationary and non-stationary, Gaussian and non-Gaussian processes have earlier been proposed by the authors , , . Basically, K–L expansion provides a second-moment characterization of a random process in terms of deterministic orthogonal functions and uncorrelated random variables as follows:

$$\varpi (x,\theta )=\varpi (x)+\sum _{k=1}^M\sqrt{{\lambda }_k}{\xi }_k(\theta )f_k(x)$$

where $\varpi (x)$ is the mean of the process, λ_k and f_k(x) are the eigenvalues and eigenfunctions of the covariance function C(x₁,x₂), ξ_k(θ) is a set of uncorrelated random variables, and M is the number of K–L terms. The deterministic eigenfunctions f_k(x) are obtained from the spectral decomposition of the covariance function. Hence, the first essential step is to solve for the eigenvalues and eigenfunctions from the homogeneous Fredholm integral equation of the second kind given by

$${\int }_{D}C(x_{1,}{x}_2)f_k(x_1)d{x}_1={\lambda }_k{f}_k(x_2)$$

The second important step is the selection of uncorrelated standardized K–L random variables such that the expansion produces the desired marginal distribution. Details are given elsewhere [3].

The efficiency of K–L expansion for simulating random processes hinges crucially on the availability of accurate eigenvalues and eigenfunctions of the covariance function. However, for most covariance functions, numerical methods such as the Galerkin, collocation or Rayleigh–Ritz methods are required, usually employing polynomial or trigonometric bases. Conventional Galerkin methods lead to dense matrices that are very costly to compute and invert. Representation of integral operators using conventional bases, such as polynomials and trigonometrics, also requires approximate integration quadratures that are tedious to evaluate. The difficulty of solving the Fredholm integral equation accurately for higher order eigenvalues and eigenfunctions is quite well known , , .

It is tempting to dismiss this difficulty as academic since high order eigenvalues are typically very small compared to the first few eigenvalues. For example, the 10th eigenvalue of an exponential covariance function is less than 1% of the first eigenvalue (Table 1). It is not unreasonable to assume that a short K–L expansion should suffice for practical applications. Studies involving application of K–L expansion in stochastic finite element method seem to justify such a commonsense point of view , , . A direct examination of the K–L expansion using simulation will however reveal that this is not true in general [1].

For processes with non-smooth covariance functions and long weakly correlated processes, ‘small’ high order eigenvalues cannot be neglected without having a very serious impact on the accuracy of the simulation. Many K–L terms are needed to even reproduce the target variance correctly for these classes of problems. This may partially explain why the K–L expansion has not been widely used for simulation despite its theoretical importance and its obvious advantage of providing an elegant unified framework for both stationary and non-stationary processes. The feasibility of implementing K–L expansion as a practical simulation tool therefore hinges crucially on the ability to compute a large number of K–L terms accurately and cheaply.

This study proposes the wavelet-Galerkin approach to solve Fredholm integral equation to overcome the shortcomings discussed earlier. The proposed method has significant computational advantages over the conventional Galerkin method , . Wavelet bases provide localized compact support, which lead to sparse representations of functions and integral operators , [10], [11], [12]. Existing efficient numerical schemes to obtain wavelet coefficients and inverse wavelet transforms can be taken advantage of to solve the integral equation [13], [14]. The wavelet transform has been used by various authors for purposes such as representation of random fields and simulation of random processes [15], [16], [17], but it has not been applied within the context of developing K–L expansion as a practical simulation tool. The validity and convergence characteristics of the wavelet-Garlekin method for solving the Fredholm integral equation will be illustrated by numerical examples. Results based on the conventional Galerkin method with polynomial bases and trigonometric bases will be presented for comparison. The ability of the wavelet-Galerkin approach to compute a large number of eigensolutions accurately and rapidly can be exploited to great advantage in implementing the K–L expansion for practical simulation.