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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.

Abstracts-A new stochastic model (a diffusion approximation) for the relative dispersion of pair of particles in high-Reynolds-number incompressible turbulence is proposed. An attempt is made to uniquely define the coefficients of SDE governing the relative dispersion process under a closure assumption about the quasi-one dimensional model. An approach for constructing a diffusion approximation of the relative dispersion taking into account the intermittency is proposed.

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Stochastic Lagrangian Models of relative motion of two fluid particles in one-and three dimensions for locally isotropic incompressible turbulent flow are presented. A principle of consistency of statistics between the Eulerian and Lagrangian velocity fields for general random forcing models of a diffusion type is proposed. This enables us to analyze and improve some well known models. An analog of the well-mixed condition for the relative dispersion is proposed. An explicit form of an co-order consistency model is given. While there is a series of applications (both in theory and applied turbulence) of the relative motion of two particles, a relation to the problem of concentration fluctuations reflect our motivation by the problem of pollutant scattering in the turbulent atmosphere.