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A stochastically generated preconditioner for stable matrices

by F. M. Buchmann and W. P. Petersen

(Report number 2002-24)

Abstract
In this paper we simulate the Ornstein-Uhlenbeck process (OUP) to generate an approximate inverse of any real valued stable matrix. Matrix A being stable means that analytically the OUP will converge to a stationary Gaussian process in n dimensions with a covariance (2 A)-1. If the eigenvalues of A are widely separated in absolute value, however, the stiffness of the simulated linear stochastic differential equations must be considered. Hence, we consider a splitting scheme to permit large step sizes but keep convergence. Methods are described for both symmetric and non-symmetric matrices. Our preconditioner is also tested in the symmetric positive definite case by its effect on the convergence of conjugate gradient iterations.

Keywords:

@Techreport{BP02_310,
  author = {F. M. Buchmann and W. P. Petersen},
  title = {A stochastically generated preconditioner for stable matrices},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2002-24},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2002/2002-24.pdf },
  year = {2002}
}

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