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Krylov methods for nonlinear eigenvalue problems and matrix equations
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
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: urn:nbn:se:kth:diva-266368ISBN: 978-91-7873-392-7 (print)OAI: oai:DiVA.org:kth-266368DiVA, id: diva2:1384086
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
List of papers
1. The waveguide eigenvalue problem and the tensor infinite Arnoldi method
Open this publication in new window or tab >>The waveguide eigenvalue problem and the tensor infinite Arnoldi method
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.

Keywords
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:nbn:se:kth:diva-263719 (URN)10.1137/15M1044667 (DOI)000404763200024 ()2-s2.0-85020429543 (Scopus ID)
Funder
Swedish Research Council, 621-2013-4640
Note

QC 20191111

Available from: 2019-11-11 Created: 2019-11-11 Last updated: 2020-01-09Bibliographically approved
2. Sylvester-based preconditioning for the waveguide eigenvalue problem
Open this publication in new window or tab >>Sylvester-based preconditioning for the waveguide eigenvalue problem
2018 (English)In: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 542, no 1, p. 441-463Article 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, 2018
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-218737 (URN)10.1016/j.laa.2017.06.027 (DOI)000425197500022 ()2-s2.0-85021359703 (Scopus ID)
Funder
Swedish Research Council, 621-2013-4640
Note

QC 20171212

Available from: 2017-11-30 Created: 2017-11-30 Last updated: 2020-01-09Bibliographically approved
3. On restarting the tensor infinite Arnoldi method
Open this publication in new window or tab >>On restarting the tensor infinite Arnoldi method
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
Keywords
Krylov subspace method, Krylov–Schur method, Nonlinear eigenvalue problem, Restart, Tensor infinite Arnoldi
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-218704 (URN)10.1007/s10543-017-0671-z (DOI)000426857000007 ()2-s2.0-85023781566 (Scopus ID)
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: 2020-01-09Bibliographically approved
4. Disguised and new quasi-Newton methods for nonlinear eigenvalue problems
Open this publication in new window or tab >>Disguised and new quasi-Newton methods for nonlinear eigenvalue problems
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
Keywords
Nonlinear eigenvalue problems, Inverse iteration, Iterative methods, Quasi-Newton methods
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-218702 (URN)10.1007/s11075-017-0438-2 (DOI)000442612400014 ()2-s2.0-85034666004 (Scopus ID)
Note

QC 20171211

Available from: 2017-11-30 Created: 2017-11-30 Last updated: 2020-01-09Bibliographically approved
5. Krylov methods for low-rank commuting generalized Sylvester equations
Open this publication in new window or tab >>Krylov methods for low-rank commuting generalized Sylvester equations
2018 (English)In: Numerical Linear Algebra with Applications, ISSN 1070-5325, E-ISSN 1099-1506, Vol. 25, no 6, article id e2176Article in journal (Refereed) Published
Abstract [en]

We consider generalizations of the Sylvester matrix equation, consisting of the sum of a Sylvester operator and a linear operator pi with a particular structure. More precisely, the commutators of the matrix coefficients of the operator pi and the Sylvester operator coefficients are assumed to be matrices with low rank. We show (under certain additional conditions) low-rank approximability of this problem, that is, the solution to this matrix equation can be approximated with a low-rank matrix. Projection methods have successfully been used to solve other matrix equations with low-rank approximability. We propose a new projection method for this class of matrix equations. The choice of the subspace is a crucial ingredient for any projection method for matrix equations. Our method is based on an adaption and extension of the extended Krylov subspace method for Sylvester equations. A constructive choice of the starting vector/block is derived from the low-rank commutators. We illustrate the effectiveness of our method by solving large-scale matrix equations arising from applications in control theory and the discretization of PDEs. The advantages of our approach in comparison to other methods are also illustrated.

Place, publisher, year, edition, pages
Wiley, 2018
Keywords
generalized Sylvester equation, iterative solvers, Krylov subspace, low-rank commutation, matrix equation, projection methods
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-239463 (URN)10.1002/nla.2176 (DOI)000449497500008 ()2-s2.0-85055043992 (Scopus ID)
Conference
7th Workshop on Matrix Equations and Tensor Techniques (METT), FEB 13-14, 2017, Pisa, Italy
Funder
Swedish e‐Science Research CenterSwedish Research Council, 621-2013-4640
Note

QC 20181127

Available from: 2018-11-27 Created: 2018-11-27 Last updated: 2020-01-09Bibliographically approved
6. The infinite Lanczos method for symmetric nonlinear eigenvalue problems
Open this publication in new window or tab >>The infinite Lanczos method for symmetric nonlinear eigenvalue problems
(English)Manuscript (preprint) (Other academic)
Abstract [en]

A new iterative method for solving large scale symmetric nonlineareigenvalue problems is presented. We firstly derive an infinite dimensional symmetric linearization of the nonlinear eigenvalue problem, then we apply the indefinite Lanczos method to this specific linearization, resulting in a short-term recurrence. We show how, under specific assumption on the starting vector, this method can be carried out in finite arithmetic and how the exploitation of the problem structure leads to improvements in terms of computation time. The eigenpair approximations are extracted with the nonlinear Rayleigh–Ritz procedure combined with aspecific choice of the projection space. We illustrate how this extraction technique resolves the instability issues that may occur due to the loss of orthogonality in many standard Lanczos-type methods.

Keywords
nonlinear eigenvalue problem, symmetric, Lanczos
National Category
Computational Mathematics
Research subject
Applied and Computational Mathematics, Numerical Analysis
Identifiers
urn:nbn:se:kth:diva-263723 (URN)
Funder
Swedish Research Council, 621-2013-4640
Note

QCR 20191111

Available from: 2019-11-11 Created: 2019-11-11 Last updated: 2020-01-09Bibliographically approved

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