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The waveguide eigenvalue problem and the tensor infinite Arnoldi method
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, Centres, SeRC - Swedish e-Science Research Centre.ORCID iD: 0000-0001-9443-8772
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, Centres, SeRC - Swedish e-Science Research Centre.ORCID iD: 0000-0002-6990-445X
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, Centres, SeRC - Swedish e-Science Research Centre.ORCID iD: 0000-0002-6321-8619
2017 (English)In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 39, no 3, p. A1062-A1088Article in journal (Refereed) Published
Abstract [en]

We present a new computational approach for a class of large-scale nonlinear eigenvalue problems (NEPs) that are nonlinear in the eigenvalue. The contribution of this paper is two fold. We derive a new iterative algorithm for NEPs, the tensor infinite Arnoldi method (TIAR), which is applicable to a general class of NEPs, and we show how to specialize the algorithm to a specific NEP: the waveguide eigenvalue problem. The waveguide eigenvalue problem arises from a finite-element discretization of a partial differential equation used in the study waves propagating in a periodic medium. The algorithm is successfully applied to accurately solve benchmark problems as well as complicated waveguides. We study the complexity of the specialized algorithm with respect to the number of iterations "m" and the size of the problem "n", both from a theoretical perspective and in practice. For the waveguide eigenvalue problem, we establish that the computationally dominating part of the algorithm has complexity O(nm^2+sqrt(n)m^3). Hence, the asymptotic complexity of TIAR applied to the waveguide eigenvalue problem, for n→ ∞, is the same as for Arnoldi’s method for standard eigenvalue problems.

Place, publisher, year, edition, pages
2017. Vol. 39, no 3, p. A1062-A1088
Keywords [en]
nonlinear eigenvalue problems, iterative methods, Krylov methods, Helmholtz equation, Arnoldi’s method
National Category
Computational Mathematics
Research subject
Applied and Computational Mathematics, Numerical Analysis
Identifiers
URN: urn:nbn:se:kth:diva-263719DOI: 10.1137/15M1044667ISI: 000404763200024Scopus ID: 2-s2.0-85020429543OAI: oai:DiVA.org:kth-263719DiVA, id: diva2:1369184
Funder
Swedish Research Council, 621-2013-4640
Note

QC 20191111

Available from: 2019-11-11 Created: 2019-11-11 Last updated: 2019-11-11Bibliographically approved

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Jarlebring, EliasMele, GiampaoloRunborg, Olof
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