laeuchli-estimatingthetrace-2016.pdf (3.43 MB)
Download file

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

Download (3.43 MB)
journal contribution
posted on 2016-12-01, 00:00 authored by L Wu, Jesse LaeuchliJesse Laeuchli, V Kalantzis, A Stathopoulos, E Gallopoulos
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.

History

Journal

Journal of computational physics

Volume

326

Pagination

828 - 844

Publisher

Elsevier

Location

Amsterdam, The Netherlands

ISSN

0021-9991

eISSN

1090-2716

Language

eng

Publication classification

C1.1 Refereed article in a scholarly journal

Copyright notice

2016, Elsevier Inc.