Matrix Probing and its Conditioning
Name
Chiu-2012-MATRIX PROBING AND I.pdf
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Author(s)
Chiu, Jiawei
Demanet, Laurent
Date Issued
February 2012
Journal
SIAM Journal on Numerical Analysis
Publisher
Society for Industrial and Applied Mathematics
Citation
Chiu, Jiawei, and Laurent Demanet. “Matrix Probing and Its Conditioning.” SIAM Journal on Numerical Analysis 50.1 (2012): 171–193. Web.
Version
Final published version
Abstract
When a matrix A with n columns is known to be well-approximated by a linear combination
of basis matrices B1, . . . , Bp, we can apply A to a random vector and solve a linear system to recover this linear combination. The same technique can be used to obtain an approximation to A[superscript −1].
A basic question is whether this linear system is well-conditioned. This is important for two reasons:
a well-conditioned system means (1) we can invert it and (2) the error in the reconstruction can be controlled. In this paper, we show that if the Gram matrix of the Bj ’s is sufficiently well-conditioned and each Bj has a high numerical rank, then n [infinity symbol] p log[superscript 2] n will ensure that the linear system is well-conditioned with high probability. Our main application is probing linear operators with smooth pseudodifferential symbols such as the wave equation Hessian in seismic imaging [L. Demanet et al., Appl. Comput. Harmonic Anal., 32 (2012), pp. 155–168]. We also demonstrate numerically that matrix probing can produce good preconditioners for inverting elliptic operators in variable media.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
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DOI of Published Version
https://doi.org/10.1137/110825972
