Simulation of stochastic processes using Karhunen-Loeve expansion

Abstract

This study focuses on extending the unified framework which has been established by Huang (2001) for the simulation of stationary and non-stationary, Gaussian and non-Gaussian processes using Karhunen-Loeve (K-L) expansion. The key steps involve solving for the eigensolutions from the homogeneous Fredholm integral equation based on the target covariance function and selecting uncorrelated standardized K-L random variables such that the expansion produces the desired marginal distribution. To improve the efficiency of obtaining eigensolutions, the wavelet-Galerkin approach is adopted to solve the Fredholm integral equation. The applicability of the K-L expansion as a simulation tool for Gaussian stochastic processes is examined numerically and compared with that of the wavelet expansion. The simulation methodology is extended to non-Gaussian processes with an iterative procedure that updates the appropriate K-L variables, which is originally proposed by Phoon et al (2000b). A modified version of the algorithm is developed in this study, where the simulated marginal distribution and covariance function can both match with the targets for strongly non-Gaussian processes. Finally, a new technique of generating non-Gaussian processes is discussed, which utilizes the fractile correlation instead of the usual product moment correlation.