Estimating the trace of the matrix inverse by interpolating from the diagonal of an approximate inverse

JOURNAL OF COMPUTATIONAL PHYSICS

卷:326

Article

Wu, Lingfei¹  Laeuchli, Jesse¹  Kalantzis, Vassilis²  Stathopoulos, Andreas¹  Gallopoulos, Efstratios³ 

[1] Coll William & Mary, Dept Comp Sci, Williamsburg, VA 23187 USA

[2] Univ Minnesota, Dept Comp Sci, Minneapolis, MN 55455 USA

[3] Univ Patras, Dept Comp Engn & Informat, Patras, Greece

关键词: Matrix trace;    Monte Carlo method;    Variance reduction;    Preconditioner;    Fitting;    Interpolation;   

DOI  :  10.1016/j.jcp.2016.09.001

来源: Elsevier

【 摘 要 】

A number of applications require the computation of the trace of a matrix that is implicitly available through a function. A common example of a function is the inverse of a large, sparse matrix, which is the focus of this paper. When the evaluation of the function is expensive, the task is computationally challenging because the standard approach is based on a Monte Carlo method which converges slowly. We present a different approach that exploits the pattern correlation, if present, between the diagonal of the inverse of the matrix and the diagonal of some approximate inverse that can be computed inexpensively. We leverage various sampling and fitting techniques to fit the diagonal of the approximation to the diagonal of the inverse. Depending on the quality of the approximate inverse, our method may serve as a standalone kernel for providing a fast trace estimate with a small number of samples. Furthermore, the method can be used as a variance reduction method for Monte Carlo in some cases. This is decided dynamically by our algorithm. An extensive set of experiments with various technique combinations on several matrices from some real applications demonstrate the potential of our method. (C) 2016 Published by Elsevier Inc.

【 授权许可】

Free