(1.1)
A new hierarchical method for the Monte Carlo simulation of random fields called the Fourier-wavelet method is developed and where 0 Ͻ H Ͻ 1 is the Hurst exponent and ͗и͘ denotes applied to isotropic Gaussian random fields with power law spectral the expected value.
density functions. This technique is based upon the orthogonal
Here we develop a new Monte Carlo method based upon decomposition of the Fourier stochastic integral representation of a wavelet expansion of the Fourier space representation of the field using wavelets. The Meyer wavelet is used here because the fractal random fields in (1.1). This method is capable its rapid decay properties allow for a very compact representation of the field. The Fourier-wavelet method is shown to be straightfor-of generating a velocity field with the Kolmogoroff specward to implement, given the nature of the necessary precomputatrum (H ϭ in (1.1)) over many (10 to 15) decades of tions and the run-time calculations, and yields comparable results scaling behavior comparable to the physical space multiwith scaling behavior over as many decades as the physical space wavelet algorithm developed by two of the authors in remultiwavelet methods developed recently by two of the authors.
cent work [5][6][7]. However, since the Fourier-wavelet However, the Fourier-wavelet method developed here is more flexible and, in particular, applies to anisotropic spectra generated method developed below is a spectral method, it is much through solutions of differential equations. Simulation results using more flexible for generating random fields with anisotropic this new technique and the well-known nonhierarchical simulation spectra and, in particular, spectra generated through the technique, the randomization method, are given and compared for solutions of partial differential equations.
both a simple shear layer model problem as well as a two-dimen-A wide variety of computational approaches have been sional isotropic Gaussian random field. The Fourier-wavelet method results are more accurate for statistical quantities de-used to generate fractal random fields. Some of these algopending on moments higher than order 2, in addition to showing rithms involve hierarchical methods [4-7] while others utia quite smooth decay to zero on the scales smaller than the scaling lize nonhierarchical methods [8][9][10][11][12] with varying degrees regime when compared with the randomization method results. The of success. Hierarchical methods involve the decomposionly situation in which the nonhierarchical randomization method tion of a random field into distinct characteristic scales.
is more computationally efficient occurs when no more than four decades of scaling behavior are needed and the statistical quantities Because of the relatively compact nature of this type of of interest depend only on second moments. ᮊ 1997 Academic Press representation, hierarchical methods are particularly attractive for simulating random fields with large regimes of scaling behavior. The nonhierarchical methods, such as the