Relative efficiency of Gaussian stochastic process sampling procedures

Abstract

Various methods for sampling stationary, Gaussian stochastic processes are investigated and compared with an emphasis on applications to processes with power law energy spectra. Several approaches are considered, including a Riemann summation using left endpoints, the use of random wave numbers to sample a the spectrum in proportion to the energy it contains, and a combination of the two. The Fourier-wavelet method of Elliott et al. is investigated and compared with other methods, all of which are evaluated in terms of their ability to sample the stochastic process over a large number of decades for a given computational cost. The Fourier-wavelet method has accuracy which increases linearly with the computational complexity, while the accuracy of the other methods grows logarithmically. For the Kolmogorov spectrum, a hybrid quadrature method is as efficient as the Fourier-wavelet method, if no more than eight decades of accuracy are required. The effectiveness of this hybrid method wanes when one samples fields whose energy spectrum decays more rapidly near the origin. The Fourier-wavelet method has roughly the same behavior independently of the exponent of the power law. The Fourier-wavelet method returns samples which are Gaussian over the range of values where the structure function is well approximated. By contrast, (multi-point) Gaussianity may be lost at the smaller length scales when one uses methods with random wave numbers.