Convergence study of the truncated Karhunen–Loeve expansion for simulation of stochastic processes

Abstract

AbstractA random process can be represented as a series expansion involving a complete set of deterministic functions with corresponding random coefficients. Karhunen–Loeve (K–L) series expansion is based on the eigen‐decomposition of the covariance function. Its applicability as a simulation tool for both stationary and non‐stationary Gaussian random processes is examined numerically in this paper. The study is based on five common covariance models. The convergence and accuracy of the K–L expansion are investigated by comparing the second‐order statistics of the simulated random process with that of the target process. It is shown that the factors affecting convergence are: (a) ratio of the length of the process over correlation parameter, (b) form of the covariance function, and (c) method of solving for the eigen‐solutions of the covariance function (namely, analytical or numerical). Comparison with the established and commonly used spectral representation method is made. K–L expansion has an edge over the spectral method for highly correlated processes. For long stationary processes, the spectral method is generally more efficient as the K–L expansion method requires substantial computational effort to solve the integral equation. The main advantage of the K–L expansion method is that it can be easily generalized to simulate non‐stationary processes with little additional effort. Copyright © 2001 John Wiley & Sons, Ltd.