Comparative analysis of multiscale Gaussian random field simulation algorithms

Abstract

We analyze and compare the efficiency and accuracy of two simulation methods for homogeneous random fields with multiscale resolution. We consider in particular the Fourier-wavelet method and three variants of the Randomization method: (A) without any stratified sampling of wavenumber space, (B) with stratified sampling of wavenumbers with equal energy subdivision, (C) with stratified sampling with a logarithmically uniform subdivision. We focus primarily on fractal Gaussian random fields with Kolmogorov-type spectra. Previous work has shown that variants (A) and (B) of the Randomization method are only able to generate a self-similar structure function over three to four decades with reasonable computational effort. By contrast, variant (C), along with the Fourier-wavelet method, is able to reproduce accurate self-similar scaling of the structure function over a number of decades increasing linearly with computational effort (for our examples we will show that nine decades can be reproduced). We provide some conceptual and numerical comparison of the various cost contributions to each random field simulation method.We find that when evaluating ensemble averaged quantities like the correlation and structure functions, as well as some multi-point statistical characteristics, the Randomization method can provide good accuracy with less cost than the Fourierwavelet method. The cost of the Randomization method relative to the Fourierwavelet method, however, appears to increase with the complexity of the random field statistics which are to be calculated accurately. Moreover, the Fourier-wavelet method has better ergodic properties, and hence becomes more efficient for the computation of spatial (rather than ensemble) averages which may be important in simulating the solutions to partial differential equations with random field coefficients.

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“…The k n are a set of wavenumbers and, finally, the paramaters V n are k-space volume co-efficients. The method of fixing the N k k's, V 's, A's and B's for a realization of F is 'variant C' of the Randomization Method described in Kramer, Kurbanmuradov, & Sabelfeld [45] which we have generalized to three dimensions. The Randomization Method in general partitions the space of wavenumbers into N k regions and from each we select a random wavevector using the power-spectrum, E(k), as a probability distribution.…”

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“…The accuracy of the method is influenced by the sampling of the wave numbers (components of k j ). Improvements are achieved if one divides the range of wave numbers into a finite number of bins, of uniform or logarithmic distributed dimensions, and one samples randomly wave numbers in each bin [2]. Since U (x) = log(K(x)) is normally distributed, the mean and the variance of the hydraulic conductivity can be finally computed from the relations E(K) = e …”