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

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.