Approximation of non-stationary fractional Gaussian random fields

Abstract

Numerical approximations of fractional and multifractional Brownian fields are studied by measuring the numerical convergence order. In order to construct these nonstationary fields a study of Gaussian fields, fractal analysis and self-similarity is conducted. The random fields are defined through their covariance function. Simulations are constructed through the Cholesky method, which builds on the Cholesky decomposition of the covariance matrix in order to accurately simulate the nonstationary field. The strong error in L2(; L2(T;R)) is measured for the fractional Brownian motion defined by the fixed Hurst parameter H. It is shown numerically that the convergence rate _ satisfies _ > H for H 2 (0, 0.6). Furthermore the convergence rates are measured for multifractional Brownian motions defined by Hurst functions h : T ! (0, 1) of varying form.