 
Summary: Preprint ANL/MCSP18880511
DIFFERENCE FILTER PRECONDITIONING FOR LARGE
COVARIANCE MATRICES
MICHAEL L. STEIN, JIE CHEN, AND MIHAI ANITESCU
Abstract. In many statistical applications one must solve linear systems corresponding to
large, dense, and possibly irregularly structured covariance matrices. These matrices are often ill
conditioned; 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.
Key words. Condition number, preconditioner, stochastic process, random field, spectral anal
ysis, fixeddomain asymptotics
AMS subject classifications. 65F35, 60G25, 62M15
1. Introduction. A problem that arises in many statistical applications is the
solution of linear systems of equations for large positive definite covariance matrices
(see, e.g., [15]). An underlying challenge for solving such linear systems is that co
variance matrices are often dense and illconditioned. Specifically, if one considers
taking an increasing number of observations of some random process in a fixed and
