A Krylov Subspace Method for Covariance Approximation and Simulation of Random Processes and Fields

Abstract

This paper proposes a new iterative algorithm for simultaneously computing an approximation to the covariance matrix of a random vector and drawing a sample from that approximation. The algorithm is especially suited to cases for which the elements of the random vector are samples of a stochastic process or random field. The proposed algorithm has close connections to the conjugate gradient method for solving linear systems of equations. A comparison is made between our algorithm's structure and complexity and other methods for simulation and covariance matrix approximation, including those based on FFTs and Lanczos methods. The convergence of our iterative algorithm is analyzed both analytically and empirically, and a preconditioning technique for accelerating convergence is explored. The numerical examples include a fractional Brownian motion and a random field with the spherical covariance used in geostatistics.