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Preconditioned eigensolvers for large-scale nonlinear Hermitian eigenproblems with variational characterizations. I. Extreme eigenvalues

Authors: Daniel B. Szyld and Fei Xue
Journal: Math. Comp. 85 (2016), 2887-2918
MSC (2010): Primary 65F15, 65F50, 15A18, 15A22
Published electronically: February 19, 2016
MathSciNet review: 3522974
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Abstract: Efficient computation of extreme eigenvalues of large-scale linear Hermitian eigenproblems can be achieved by preconditioned conjugate gradient (PCG) methods. In this paper, we study PCG methods for computing extreme eigenvalues of nonlinear Hermitian eigenproblems of the form $ T(\lambda )v=0$ that admit a nonlinear variational principle. We investigate some theoretical properties of a basic CG method, including its global and asymptotic convergence. We propose several variants of single-vector and block PCG methods with deflation for computing multiple eigenvalues, and compare them in arithmetic and memory cost. Variable indefinite preconditioning is shown to be effective to accelerate convergence when some desired eigenvalues are not close to the lowest or highest eigenvalue. The efficiency of variants of PCG is illustrated by numerical experiments. Overall, the locally optimal block preconditioned conjugate gradient (LOBPCG) is the most efficient method, as in the linear setting.

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Additional Information

Daniel B. Szyld
Affiliation: Department of Mathematics, Temple University (038-16), 1805 N. Broad Street, Philadelphia, Pennsylvania 19122-6094

Fei Xue
Affiliation: Department of Mathematics, University of Louisiana at Lafayette, P.O. Box 41010, Lafayette, Louisiana 70504-1010

Keywords: Nonlinear Hermitian eigenproblems, variational principle, preconditioned conjugate gradient, convergence analysis
Received by editor(s): August 27, 2014
Received by editor(s) in revised form: April 28, 2015, and June 9, 2015
Published electronically: February 19, 2016
Additional Notes: The first author was supported by NSF under grants DMS-1115520 and DMS-1418882.
The second author was supported by NSF under grants DMS-1115520 and DMS-1419100.
Article copyright: © Copyright 2016 American Mathematical Society

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