Randomized algorithms for generalized Hermitian eigenvalue problems with application to computing Karhunen-Loève expansion

Summary We describe randomized algorithms for computing the dominant eigenmodes of the generalized Hermitian eigenvalue problem Ax = λBx, with A Hermitian and B Hermitian and positive definite. The algorithms we describe only require forming operations Ax,Bx and B−1x and avoid forming square roots o...

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Bibliographic Details
Published in:Numerical linear algebra with applications Vol. 23; no. 2; pp. 314 - 339
Main Authors: Saibaba, Arvind K., Lee, Jonghyun, Kitanidis, Peter K.
Format: Journal Article
Language:English
Published: Oxford Blackwell Publishing Ltd 01.03.2016
Wiley Subscription Services, Inc
Subjects:
Algorithms
Approximation
Computation
Eigenvalues
Error analysis
Expansion
Forming
generalized Hermitian eigenvalue problems
Karhunen-Loeve expansion
Karhunen-Loève expansion
Mathematical analysis
Matrix
Randomized algorithms
ISSN:1070-5325, 1099-1506
Online Access:Get full text
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Summary:Summary We describe randomized algorithms for computing the dominant eigenmodes of the generalized Hermitian eigenvalue problem Ax = λBx, with A Hermitian and B Hermitian and positive definite. The algorithms we describe only require forming operations Ax,Bx and B−1x and avoid forming square roots of B (or operations of the form, B1/2x or B−1/2x). We provide a convergence analysis and a posteriori error bounds and derive some new results that provide insight into the accuracy of the eigenvalue calculations. The error analysis shows that the randomized algorithm is most accurate when the generalized singular values of B−1A decay rapidly. A randomized algorithm for the generalized singular value decomposition is also provided. Finally, we demonstrate the performance of our algorithm on computing an approximation to the Karhunen–Loève expansion, which involves a computationally intensive generalized Hermitian eigenvalue problem with rapidly decaying eigenvalues. Copyright © 2015 John Wiley & Sons, Ltd.
Bibliography:ark:/67375/WNG-S8B8V1P6-P
istex:1A8236D73465AED8BFFD5300C06E6620807240FC
ArticleID:NLA2026
National Science Foundation - No. NSF EEC-1028968
ObjectType-Article-1
SourceType-Scholarly Journals-1
ObjectType-Feature-2
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ISSN:1070-5325
1099-1506
DOI:10.1002/nla.2026