doi:10.1016/S0266-8920(02)00013-9 
Copyright © 2002 Elsevier Science Ltd All rights reserved.
Implementation of Karhunen–Loeve expansion for simulation using a wavelet-Galerkin scheme
K.K Phoon
, 
, S.P Huang and S.T Quek
Department of Civil Engineering, National University of Singapore, Block E1A #07-03, 1 Engineering Drive 2, Singapore, Singapore 117576
Available online 28 May 2002.
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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.
Keywords: Karhunen–Loeve expansion; simulation; Fredholm integral equation; Wavelet-Galerkin; Discrete wavelet transform; Mallat's tree algorithm; Harr wavelets
Fig. 1. Eighth- to 10th-order eigenfunctions for exponential covariance function (Example 1)—analytical solution (short dash); wavelet solution using N=64 (solid); degree 10 polynomial solution (dash-dot) and trigonometrics with five harmonics (long dash).
Fig. 2. Convergence of eigenvalues for squared exponential covariance function (Example 2).
Fig. 3. Comparsion of relative error in eigenvalue produced by different Galerkin approaches for (a) exponential covariance function (Example 1), (b) squared exponential covariance function (Example 2), and (c) Wiener–Levy process covariance function (Example 3).
Fig. 4. Relative error in eigenvalue and CPU time versus the: (a) level of resolution of wavelet solution, (b) number of harmonics in trigonometric solution, and (c) degree of polynomial (Example 1—exponential covariance).
Fig. 5. Relative error in eigenvalue and CPU time versus the: (a) level of resolution of wavelet solution, (b) number of harmonics in trigonometric solution, and (c) degree of polynomial (Example 2—squared exponential covariance).
Fig. 6. Computed exponential covariance function (Example 1) using M=5, 20 and 100 terms in K–L expansion (eigensolutions from wavelet-Galerkin approach using N=128).
Fig. 7. Normalized variance versus number of terms in K–L at different dyadic points for non-stationary Wiener–Levy process (eigensolutions from wavelet-Galerkin approach with N=64).
Table 1.
Comparison of eigenvalues for exponential covariance function
Table 2.
Comparison of eigenvalues for squared exponential covariance function
Table 3.
Comparison of eigenvalues for non-stationary Wiener–Levy process
Table 4.
Variance of K–L expansion truncated at M terms for various normalized length of the exponential process a/b (eigensolutions from wavelet-Galerkin approach with N=128)
Target variance=1.
Table 5.
Variance of K–L expansion truncated at M terms for various normalized length of the squared exponential process a/b (eigensolutions from wavelet-Galerkin approach with N=128)
Target variance=1.
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