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  • 1.
    Jarlebring, Elias
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, Centres, SeRC - Swedish e-Science Research Centre.
    Koskela, Antti
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Mele, Giampaolo
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, Centres, SeRC - Swedish e-Science Research Centre.
    Disguised and new quasi-Newton methods for nonlinear eigenvalue problems2018In: Numerical Algorithms, ISSN 1017-1398, E-ISSN 1572-9265, Vol. 79, no 1, p. 311-335Article in journal (Refereed)
    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.

  • 2.
    Jarlebring, Elias
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.). KTH, Centres, SeRC - Swedish e-Science Research Centre.
    Mele, Giampaolo
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, Centres, SeRC - Swedish e-Science Research Centre.
    Palitta, Davide
    Univ Bologna, Dipartimento Matemat, Piazza Porta S Donato,5, I-40127 Bologna, Italy..
    Ringh, Emil
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.). KTH, Centres, SeRC - Swedish e-Science Research Centre.
    Krylov methods for low-rank commuting generalized Sylvester equations2018In: Numerical Linear Algebra with Applications, ISSN 1070-5325, E-ISSN 1099-1506, Vol. 25, no 6, article id e2176Article in journal (Refereed)
    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.

  • 3.
    Jarlebring, Elias
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, Centres, SeRC - Swedish e-Science Research Centre.
    Mele, Giampaolo
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, Centres, SeRC - Swedish e-Science Research Centre.
    Runborg, Olof
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, Centres, SeRC - Swedish e-Science Research Centre.
    The waveguide eigenvalue problem and the tensor infinite Arnoldi method2017In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 39, no 3, p. A1062-A1088Article in journal (Refereed)
    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.

  • 4.
    Mele, Giampaolo
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, Centres, SeRC - Swedish e-Science Research Centre.
    Krylov methods for nonlinear eigenvalue problems and matrix equations2020Doctoral 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.

  • 5.
    Mele, Giampaolo
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, Centres, SeRC - Swedish e-Science Research Centre.
    The infinite Lanczos method for symmetric nonlinear eigenvalue problemsManuscript (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.

  • 6.
    Mele, Giampaolo
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, Centres, SeRC - Swedish e-Science Research Centre.
    Jarlebring, Elias
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, Centres, SeRC - Swedish e-Science Research Centre.
    On restarting the tensor infinite Arnoldi method2018In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 58, no 1, p. 133-162Article in journal (Refereed)
    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. 

  • 7.
    Ringh, Emil
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Mele, Giampaolo
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Karlsson, Johan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Jarlebring, Elias
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Sylvester-based preconditioning for the waveguide eigenvalue problem2018In: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 542, no 1, p. 441-463Article in journal (Refereed)
    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.

    The full text will be freely available from 2020-05-03 09:14
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