Analysis of continuous spectral method for sampling stationary Gaussian random fields

Problems of uncertainty quantification usually involve large number realiza-tions of a stationary spatial Gaussian random field over a regular grid points. This paper analyzes the convergence of the continuous spectral method for generating a stationary Gaussian random field. The continuous spectral method is the classical approach which discretizes the spectral representation integral to construct an approximation of the field within the Fast Fourier Transform algorithm. The method can be used as an alternative of circulant embedding approach when the discrete covariance matrix is not valid. We demonstrate that the method is computationally attractive when the spectral is a smooth function and decreases rapidly to zero at infinity. In such case, The spectral method is a very versatile approach for generating Gaussian stochastic fields. A simulation results are realized using pseudo-random data based on Monte-Carlo simulations to illustrate the theoretical bound of the method regarding the regularity of the random field and its spectral density.