| Various methods for sampling stationary, Gaussian stochastic processes are investigated and compared with an emphasis on applications to processes with power law energy spectra. These methods use a spectral representation of a stochastic process as a stochastic integral followed by an approximate evaluation of that integral.; Several approaches to performing this quadrature are considered here. They include a Riemann summation using left endpoints, the use of random wave numbers to sample a given part of the spectrum in proportion to the energy it contains, a combination of the two, and higher order methods. A quadrature based on a wavelet expansion, the Fourier-wavelet method of Elliott, Horntrop, and Majda, is also investigated and its behavior is compared to that of the simpler methods. All these methods are tested and evaluated in terms of their ability to simulate 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 simulates fields whose energy spectrum decays more rapidly and contains more energy near the origin than the Kolmogorov spectrum. The Fourier-wavelet method has roughly the same behavior independently of the exponent of the power law.; The Fourier-wavelet method returns simulations which are Gaussian over the range of values where the structure function is well-approximated. By contrast, Gaussianity may be lost at the smaller length scales when one uses methods with random wave numbers. |
