Simulation of non-Gaussian stochastic processes and fields with applications to structural engineering problems

Abstract

Three classes of non-Gaussian functions are defined and Monte Carlo algorithms are developed for generating samples of the random functions in these classes. These classes consist of translation random functions, diffusion processes and memoryless transformations of these processes, and spectral representation based non-Gaussian processes. A translation random function X(t) Rd, t Rd, is defined by $$ \begin{array}{*{20}c} {X_i \left( t \right) = F_i ^{ - 1} \circ \Phi \left( {G_i \left( t \right)} \right) = h_i \left( {G_i \left( t \right)} \right),} & {i = 1, \ldots ,d,} \\ \end{array} $$ where F is the distribution of the standard Gaussian variable N(0, 1) and G(t), t Rd, is an Rd- valued stationary Gaussian function with coordinates Gi(t),i=1,...,d, of mean 0, variance 1, and covariance functions \( \rho _{ij} \left( \tau \right) = E\left[ {G_i \left( {t + \tau } \right)G_j \left( t \right)} \right],\tau \in \mathbb{R}^{d'} \). Hence, X is specified by its marginal distribution and second-moment properties. Diffusion processes can be viewed as outputs of dynamic systems to Gaussian white noise. For example, X is said to be a diffusion process if it is defined by the stochastic differential equation $$ d\begin{array}{*{20}c} {X\left( t \right) = a\left( {X\left( t \right)} \right)dt + a\left( {X\left( t \right)} \right)d\underline ( t),} & {t \geqslant 0,} \\ \end{array} $$ where a and b denote the drift and diffusion of X and B is a Brownian motion. Spectral representation based non-Gaussian real-valued processes are defined by $$ \begin{array}{*{20}c} {X\left( t \right) = \int_0^\infty {\left[ {\cos \left( {\nu t} \right)dM_1 \left( \nu \right) + \sin \left( {\nu t} \right)dM_2 \left( \nu \right)} \right]} ,} & {t \geqslant 0,} \\ \end{array} $$ where M1 and M2 are square integrable martingale. Monte Carlo simulation algorithms are developed for all non-Gaussian functions considered in the paper. The algorithms are simple, efficient, and can be based onMATLAB functions. Numerical examples are used to illustrate the implementation of some of the Monte Carlo simulation algorithms presented in the paper. It is shown that translation functions are versatile and their construction involves relatively relatively simple concepts.