Change search
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf
Disguised and new quasi-Newton methods for nonlinear eigenvalue problems
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, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, Centres, SeRC - Swedish e-Science Research Centre.
2018 (English)In: Numerical Algorithms, ISSN 1017-1398, E-ISSN 1572-9265, Vol. 79, no 1, p. 311-335Article in journal (Refereed) Published
Abstract [en]

In this paper, we take a quasi-Newton approach to nonlinear eigenvalue problems (NEPs) of the type M(λ)v = 0, where (Formula presented.) is a holomorphic function. We investigate which types of approximations of the Jacobian matrix lead to competitive algorithms, and provide convergence theory. The convergence analysis is based on theory for quasi-Newton methods and Keldysh’s theorem for NEPs. We derive new algorithms and also show that several well-established methods for NEPs can be interpreted as quasi-Newton methods, and thereby, we provide insight to their convergence behavior. In particular, we establish quasi-Newton interpretations of Neumaier’s residual inverse iteration and Ruhe’s method of successive linear problems.

Place, publisher, year, edition, pages
Springer, 2018. Vol. 79, no 1, p. 311-335
Keywords [en]
Nonlinear eigenvalue problems, Inverse iteration, Iterative methods, Quasi-Newton methods
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-218702DOI: 10.1007/s11075-017-0438-2ISI: 000442612400014Scopus ID: 2-s2.0-85034666004OAI: oai:DiVA.org:kth-218702DiVA, id: diva2:1161440
Note

QC 20171211

Available from: 2017-11-30 Created: 2017-11-30 Last updated: 2020-01-09Bibliographically approved
In thesis
1. Krylov methods for nonlinear eigenvalue problems and matrix equations
Open this publication in new window or tab >>Krylov methods for nonlinear eigenvalue problems and matrix equations
2020 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Nonlinear eigenvalue problems (NEPs) arise in many fields of science and engineering. Such problems are often defined by large matrices, which have specific structures, such as being sparse, low-rank, etc. Like the linear eigenvalue problem, the eigenvector appears in a linear form, whereas the eigenvalue appears in a nonlinear form. This feature allows for an extension of several methods, which are originally derived for the linear eigenvalue problem, to the nonlinear case. Among these methods, Krylov algorithms have been successfully extended in various ways. These methods are designed to take advantage of the matrix structures mentioned above. In this thesis, we present two Krylov-based methods for solving NEPs: the tensor infinite Arnoldi (TIAR), with its restarting variant, and infinite Lanczos (ILAN). We illustrate the flexibility of TIAR by adapting it for solving a NEP which comes from the study of waves propagating in periodic mediums. Despite the fact that Krylov methods are, in a sense, globally convergent, the convergence to the targeted eigenvalues, in certain cases, may be slow. When an accurate solution is required, the obtained approximations are refined with methods which have higher convergence order, e.g., Newton-like methods, which are also analyzed in this thesis. In the context of eigenvalue computation, the framework used to analyse Newton methods can be combined with the Keldysh theorem in order to better characterize the convergence factor. We also show that several well-established methods, such as residual inverse iteration and Ruhe’s method of successive linear problems, belong to the class of Newton-like methods. In this spirit, we derive a new quasi-Newton method, which is, in terms of convergence properties, equivalent to residual inverse iteration, but does not require the solution of a nonlinear system per iteration. The mentioned methods are implemented in NEP-PACK, which is a registered Julia package for NEPs that we develop. This package consists of: many state-of-the-art, but also well-established, methods for solving NEPs, a vast problem collection, and types and structures to efficiently represent and do computations with NEPs.Many problems in control theory, and many discretized partial differential equations, can be efficiently solved if formulated as matrix equations. Moreover, matrix equations arise in a very large variety of areas as intrinsic problems. In our framework, for certain applications, solving matrix equations is a part of the process of solving a NEP. In this thesis we derive a preconditioning technique which is applicable to linear systems which can be formulate as generalized Sylvester equation. More precisely, we assume that the matrix equation can be formulated as the sum of a Sylvester operator and another term which can be low-rank approximated. Such linear systems arise, e.g., when solving certain NEPs which come from wave propagation problems.We also derive an algorithm, which consists of applying a Krylov method directly to the the matrix equation rather then to the vectorized linear system, that exploits certain structures in the matrix coefficients.

Abstract [sv]

Icke-linjära egenvärdesproblem, förkortat NEP från engelskans nonlinear eigenvalue problem, uppstår inom många områden inom vetenskap och teknik. Sådana problem definieras ofta av stora matriser med specifika strukturer, såsom gleshet, låg rang osv. Liksom det i linjära egenvärdesproblemet är beroendet på egenvektorn linjärt, medan beroendet på egenvärdet är icke-linjärt. Denna egenskap möjliggör en utvidgning av flera metoder, som ursprungligen härleds för det linjära egenvärdesproblemet, till det icke-linjära fallet. Bland dessa metoder har Krylov-metoder framgångsrikt vidareutvecklats på olika sätt. Dessa metoder är utformade för att dra fördel av de ovan nämnda matrisstrukturerna. I den här avhandlingen presenterar vi två Krylov-baserade metoder för att lösa NEP: tensor Infinite Arnoldi (TIAR), med en variant som möjliggör omstart, och Infinite Lanczos (ILAN). Vi illustrerar flexibiliteten i TIAR genom att anpassa den till att lösa en NEP som kommer från studien av vågor som sprider sig i periodiska medier.Även om Krylov-metoder på sätt och vis är globalt konvergenta, kan konvergensen till de önskade egenvärdena i vissa fall vara långsam. När en noggrann lösning erfordras kan de erhållna approximationerna förfinas med metoder som har högre konvergensordning, t.ex. Newton-liknande metoder, som också analyseras i denna avhandling. I detta sammanhang kan ramverket som används för att analysera Newton-metoder kombineras med Keldysh sats för att bättre karakterisera konvergensfaktorn. Vi visar också att flera väletablerade metoder, såsom residual inversiteration och Ruhes metod för successiva linjära problem, tillhör klassen Newton-liknande metoder. I denna anda erhåller vi en ny quasi-Newton metod som motsvarar residual inversiteration, när det gäller konvergensegenskaper, men inte kräver lösning av en olinjär ekvation per iteration.De nämnda metoderna är implementerade i NEP-PACK, som är ett registrerat Julia-paket för NEP som vi har utvecklat. Detta paket består av många nyutvecklade samt väletablerade metoder för att lösa NEP, en stor problemsamling samt typer och strukturer för att effektivt representera och göra beräkningar med NEP.Många problem inom styrteori, samt många diskretiserade partiella differentialekvationer, kan lösas effektivt om de formuleras som matrisekvationer. Dessutom uppstår matrisekvationer som delproblem i ett mycket stort antal områden. I vårt ramverk, är lösning av matrisekvationer en del av processen för att lösa en NEP i vissa tillämpningar. I denna avhandling härleder vi en förkonditioneringsteknik som är tillämplig på vissa linjära system vilka kan formuleras som en generaliserad Sylvesterekvation. Mer exakt antar vi att matrisekvationen kan skrivas som summan av en Sylvesteroperator och en annan term som kan approximeras med en operator med låg rang. Sådana linjära system uppstår, t.ex., vid lösning av vissa NEP som kommer från vågutbredningsproblem. Vi presenterar också en algoritm, som består av att tillämpa en Krylov-metod direkt på matrisekvationen snarare än på det vektoriserade linjära systemet, vilken utnyttjar vissa strukturer i matriskoefficienterna.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2020. p. 57
Series
TRITA-SCI-FOU ; 2019:59
National Category
Computational Mathematics
Research subject
Applied and Computational Mathematics, Numerical Analysis
Identifiers
urn:nbn:se:kth:diva-266368 (URN)978-91-7873-392-7 (ISBN)
Public defence
2020-02-11, F3, Lindstedtsvägen 26, Sing-Sing, floor 2, KTH Campus, Stockholm, 10:00 (English)
Opponent
Supervisors
Available from: 2020-01-09 Created: 2020-01-09 Last updated: 2020-01-13Bibliographically approved

Open Access in DiVA

fulltext(1186 kB)32 downloads
File information
File name FULLTEXT01.pdfFile size 1186 kBChecksum SHA-512
bbc150b5fcc282b83eee4e9effd7b12f504a19c57c9470c7e9a3f506f0bf54a48aaf2e42fab6fba624a764296c82a6d662285586866de4b9571ea799476254b3
Type fulltextMimetype application/pdf

Other links

Publisher's full textScopus

Search in DiVA

By author/editor
Jarlebring, EliasKoskela, AnttiMele, Giampaolo
By organisation
Numerical Analysis, NASeRC - Swedish e-Science Research Centre
In the same journal
Numerical Algorithms
Computational Mathematics

Search outside of DiVA

GoogleGoogle Scholar
Total: 32 downloads
The number of downloads is the sum of all downloads of full texts. It may include eg previous versions that are now no longer available

doi
urn-nbn

Altmetric score

doi
urn-nbn
Total: 72 hits
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf