Randomized Spectral and Fourier-Wavelet Methods for multidimensional Gaussian random vector fields

We develop some new variations of the Randomized Spectral Method (RSM) and Fourier-Wavelet Method (FWM) for the simulation of homogeneous Gaussian random vector fields based on plane wave decompositions. We then compare the various random field simulation methods through the examination of their efficiency and the quality of the Eulerian and Lagrangian statistical characteristics of a three-dimensional isotropic incompressible random field as simulated by ensemble or by spatial averaging. We find that the RSM, which is easier to implement, can simulate random fields with accurate second order Eulerian and Lagrangian correlations more efficiently than the FWM when the statistics are collected through ensemble averages. When the correlation structure is inferred from spatial averages of a single realization of the random field (through appeal to an ergodic theorem), the FWM is more efficient in simulating an accurate Eulerian correlation function, and both the FWM and RSM are of comparable efficiency in simulating second order Lagrangian statistics accurately.