Preprint ANL/MCS-P1888-0511 DIFFERENCE FILTER PRECONDITIONING FOR LARGE COVARIANCE MATRICES

In many statistical applications one must solve linear systems corresponding to large, dense, and possibly irregularly structured covariance matrices. These matrices are often illconditioned; for example, the condition number increases at least linearly with respect to the size of the matrix when observations of a random process are obtained from a fixed domain. This paper discusses a preconditioning technique based on a differencing approach such that the preconditioned covariance matrix has a bounded condition number independent of the size of the matrix for some important process classes. When used in large scale simulations of random processes, significant improvement is observed for solving these linear systems with an iterative method.