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On restarting the tensor infinite Arnoldi method
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, Centres, SeRC - Swedish e-Science Research Centre.
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
2018 (English)In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 58, no 1, p. 133-162Article in journal (Refereed) Published
Abstract [en]

An efficient and robust restart strategy is important for any Krylov-based method for eigenvalue problems. The tensor infinite Arnoldi method (TIAR) is a Krylov-based method for solving nonlinear eigenvalue problems (NEPs). This method can be interpreted as an Arnoldi method applied to a linear and infinite dimensional eigenvalue problem where the Krylov basis consists of polynomials. We propose new restart techniques for TIAR and analyze efficiency and robustness. More precisely, we consider an extension of TIAR which corresponds to generating the Krylov space using not only polynomials, but also structured functions, which are sums of exponentials and polynomials, while maintaining a memory efficient tensor representation. We propose two restarting strategies, both derived from the specific structure of the infinite dimensional Arnoldi factorization. One restarting strategy, which we call semi-explicit TIAR restart, provides the possibility to carry out locking in a compact way. The other strategy, which we call implicit TIAR restart, is based on the Krylov–Schur restart method for the linear eigenvalue problem and preserves its robustness. Both restarting strategies involve approximations of the tensor structured factorization in order to reduce the complexity and the required memory resources. We bound the error introduced by some of the approximations in the infinite dimensional Arnoldi factorization showing that those approximations do not substantially influence the robustness of the restart approach. We illustrate the effectiveness of the approaches by applying them to solve large scale NEPs that arise from a delay differential equation and a wave propagation problem. The advantages in comparison to other restart methods are also illustrated. 

Place, publisher, year, edition, pages
Springer, 2018. Vol. 58, no 1, p. 133-162
Keywords [en]
Krylov subspace method, Krylov–Schur method, Nonlinear eigenvalue problem, Restart, Tensor infinite Arnoldi
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-218704DOI: 10.1007/s10543-017-0671-zISI: 000426857000007Scopus ID: 2-s2.0-85023781566OAI: oai:DiVA.org:kth-218704DiVA, id: diva2:1161443
Funder
Swedish Research Council, 621-2013-4640Swedish e‐Science Research Center
Note

QC 20171211

Available from: 2017-11-30 Created: 2017-11-30 Last updated: 2018-04-27Bibliographically approved

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Mele, GiampaoloJarlebring, Elias
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