Coarse-Grid Sampling Interpolatory Methods for Approximating Gaussian Random Fields

Abstract

Random fields can be approximated using grid-based discretizations of their covariance functions followed by, e.g., an eigendecomposition (i.e., a Karhunen--Loeve expansion) or a Cholesky factorization of the resulting covariance matrix. In this paper, we consider Gaussian random fields and we analyze the efficiency gains obtained by using low-rank approximations based on constructing a coarse grid covariance matrix, followed by either an eigendecomposition or a Cholesky factorization of that matrix, followed by interpolation from the coarse grid onto the fine grid. The result is coarser sampling and smaller decomposition or factorization problems than that for full-rank approximations. With one-dimensional experiments we examine the relative merits, with respect to accuracy achieved for the same computational complexity, of the different low-rank approaches. We find that interpolation from the coarse grid combined with the Cholesky factorization of the coarse grid covariance matrix yields the most effici...