Computing f(A)b via least squares polynomial approximations

Jie Chen, Mihai Anitescu, Yousef Saad

Research output: Contribution to journalArticle

26 Scopus citations

Abstract

Given a certain function f, various methods have been proposed in the past for addressing the important problem of computing the matrix-vector product f(A)b without explicitly computing the matrix f(A). Such methods were typically developed for a specific function f, a common case being that of the exponential. This paper discusses a procedure based on least squares polynomials that can, in principle, be applied to any (continuous) function f. The idea is to start by approximating the function by a spline of a desired accuracy. Then a particular definition of the function inner product is invoked that facilitates the computation of the least squares polynomial to this spline function. Since the function is approximated by a polynomial, the matrix A is referenced only through a matrix-vector multiplication. In addition, the choice of the inner product makes it possible to avoid numerical integration. As an important application, we consider the case when f(t) = √t and A is a sparse, symmetric positive-definite matrix, which arises in sampling from a Gaussian process distribution. The covariance matrix of the distribution is defined by using a covariance function that has a compact support, at a very large number of sites that are on a regular or irregular grid. We derive error bounds and show extensive numerical results to illustrate the effectiveness of the proposed technique.

Original languageEnglish (US)
Pages (from-to)195-222
Number of pages28
JournalSIAM Journal on Scientific Computing
Volume33
Issue number1
DOIs
StatePublished - Mar 10 2011

Keywords

  • Gaussian process
  • Least squares polynomials
  • Matrix function
  • Sampling

Fingerprint Dive into the research topics of 'Computing f(A)b via least squares polynomial approximations'. Together they form a unique fingerprint.

  • Cite this