Simulation of non-stationary and non-Gaussian stochastic processes by the AFD-Type Sparse Representations

Abstract

Simulation of a stochastic process to simultaneously meet a given marginal distribution condition and be compatible with a given covariance function is a well-known problem called the two-match problem in the sequel. We propose the adaptive Fourier decomposition (AFD) type methods as constructive steps to solve the two-match problem. AFD-Type methods used in the physical domain (e.g., time or frequency) are replacements of the Karhunen–Loève expansion, or spectral decomposition, in general, while, when being used in the probability space are replacements of the polynomial chaos or other types of chaos methods. The AFD-Type methods in both circumstances play the role of an approximation. Being compared with the Karhunen–Loève expansion, the proposed AFD-Type methods do not need to compute out the eigenvalues and eigenfunctions of the kernel integral operator defined by the target covariance function, while compared with the polynomial chaos or other chaos methods the AFD-Type methods offer a great deal of flexibility and efficiency. The proposed methods can be applied to stationary and non-stationary, weakly, and strongly non-Gaussian stochastic processes. We provide several examples to show effectiveness and efficiency of the AFD-Type methods.