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Sylvester-based preconditioning for the waveguide eigenvalue problem
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.ORCID iD: 0000-0001-6279-6145
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.ORCID iD: 0000-0001-9443-8772
2017 (English)In: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856Article in journal (Refereed) Published
Abstract [en]

We consider a nonlinear eigenvalue problem (NEP) arising from absorbing boundary conditions in the study of a partial differential equation (PDE) describing a waveguide. We propose a new computational approach for this large-scale NEP based on residual inverse iteration (Resinv) with preconditioned iterative solves. Similar to many preconditioned iterative methods for discretized PDEs, this approach requires the construction of an accurate and efficient preconditioner. For the waveguide eigenvalue problem, the associated linear system can be formulated as a generalized Sylvester equation AX+XB+A1XB1+A2XB2+K(ring operator)X=C, where (ring operator) denotes the Hadamard product. The equation is approximated by a low-rank correction of a Sylvester equation, which we use as a preconditioner. The action of the preconditioner is efficiently computed by using the matrix equation version of the Sherman-Morrison-Woodbury (SMW) formula. We show how the preconditioner can be integrated into Resinv. The results are illustrated by applying the method to large-scale problems.

Place, publisher, year, edition, pages
Elsevier, 2017.
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-218737DOI: 10.1016/j.laa.2017.06.027OAI: oai:DiVA.org:kth-218737DiVA: diva2:1161455
Funder
Swedish Research Council
Note

QC 20171212

Available from: 2017-11-30 Created: 2017-11-30 Last updated: 2017-12-12Bibliographically approved

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Ringh, EmilMele, GiampaoloKarlsson, JohanJarlebring, Elias
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Optimization and Systems TheoryNumerical Analysis, NA
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Linear Algebra and its Applications
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