Extensions of multiscale Gaussian random field simulation algorithms
- Kramer, Peter
- Kurbanmuradov, Orazgeldy
- Sabelfeld, Karl
2010 Mathematics Subject Classification
- 65C05 65C20 65T60
2008 Physics and Astronomy Classification Scheme
- 02.50.Ng, 02.60.Cb
- Randomization method, Fourier-wavelet representation, multiscale random fields
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) stratified sampling with a logarithmically uniform subdivision. We focus on fractal Gaussian random fields with Kolmogorov-type spectra. As noted in previous work [3,6], 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), suggested by [34,22], along with the Fourier-wavelet method developed by , 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 (overhead, cost per realization, cost per evaluation). 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 considerably less cost than the Fourier-wavelet method. The Fourier-wavelet method, however, 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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