A Fourier–Wavelet Monte Carlo Method for Fractal Random Fields

Abstract

A new hierarchical method for the Monte Carlo simulation of random fields called the Fourier–wavelet method is developed and applied to isotropic Gaussian random fields with power law spectral density functions. This technique is based upon the orthogonal decomposition of the Fourier stochastic integral representation of the field using wavelets. The Meyer wavelet is used here because its rapid decay properties allow for a very compact representation of the field. The Fourier–wavelet method is shown to be straightforward to implement, given the nature of the necessary precomputations and the run-time calculations, and yields comparable results with scaling behavior over as many decades as the physical space multiwavelet methods developed recently by two of the authors. However, the Fourier–wavelet method developed here is more flexible and, in particular, applies to anisotropic spectra generated through solutions of differential equations. Simulation results using this new technique and the well-known nonhierarchical simulation technique, the randomization method, are given and compared for both a simple shear layer model problem as well as a two-dimensional isotropic Gaussian random field. The Fourier–wavelet method results are more accurate for statistical quantities depending on moments higher than order 2, in addition to showing a quite smooth decay to zero on the scales smaller than the scaling regime when compared with the randomization method results. The only situation in which the nonhierarchical randomization method is more computationally efficient occurs when no more than four decades of scaling behavior are needed and the statistical quantities of interest depend only on second moments.