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Wim Michiels - One of the best experts on this subject based on the ideXlab platform.
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An Explicit Formula for the Splitting of Multiple Eigenvalues for Nonlinear Eigenvalue Problems and Connections with the Linearization for the Delay Eigenvalue Problem
SIAM Journal on Matrix Analysis and Applications, 2017Co-Authors: Wim Michiels, Islam Boussaada, Silviu-iulian NiculescuAbstract:We contribute to the perturbation theory of Nonlinear Eigenvalue problems in three ways. First, we extend the formula for the sensitivity of a simple Eigenvalue with respect to a variation of a parameter to the case of multiple nonsemisimple Eigenvalues, thereby providing an explicit expression for the leading coefficients of the Puiseux series of the emanating branches of Eigenvalues. Second, for a broad class of delay Eigenvalue problems, the connection between the finite-dimensional Nonlinear Eigenvalue problem and an associated infinite-dimensional linear Eigenvalue problem is emphasized in the developed perturbation theory. Finally, in contrast to existing work on analyzing multiple Eigenvalues of delay systems, we develop all theory in a matrix framework, i.e., without reduction of a problem to the analysis of a scalar characteristic quasi-polynomial.
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Computation of pseudospectral abscissa for large-scale Nonlinear Eigenvalue problems
IMA Journal of Numerical Analysis, 2017Co-Authors: Karl Meerbergen, Wim Michiels, Emre Mengi, Roel Van BeeumenAbstract:Author(s): Meerbergen, K; Michiels, W; Van Beeumen, R; Mengi, E | Abstract: © The authors 2017. We present an algorithm to compute the pseudospectral abscissa for a Nonlinear Eigenvalue problem. The algorithm relies on global under-estimator and over-estimator functions for the Eigenvalue and singular value functions involved. These global models follow from Eigenvalue perturbation theory. The algorithm has three particular features. First, it converges to the globally rightmost point of the pseudospectrum, and it is immune to nonsmoothness. The global convergence assertion is under the assumption that a global lower bound is available for the second derivative of a singular value function depending on one parameter. It may not be easy to deduce such a lower bound analytically, but assigning large negative values works robustly in practice. Second, it is applicable to large-scale problems since the dominant cost per iteration stems from computing the smallest singular value and associated singular vectors, for which efficient iterative solvers can be used. Furthermore, a significant increase in computational efficiency can be obtained by subspace acceleration, that is, by restricting the domains of the linear maps associated with the matrices involved to small but suitable subspaces, and solving the resulting reduced problems. Occasional restarts of these subspaces further enhance the efficiency for large-scale problems. Finally, in contrast to existing iterative approaches based on constructing low-rank perturbations and rightmost Eigenvalue computations, the algorithm relies on computing only singular values of complex matrices. Hence, the algorithm does not require solutions of Nonlinear Eigenvalue problems, thereby further increasing efficiency and reliability. This work is accompanied by a robust implementation of the algorithm that is publicly available.
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compact rational krylov methods for Nonlinear Eigenvalue problems
SIAM Journal on Matrix Analysis and Applications, 2015Co-Authors: Roel Van Beeumen, Karl Meerbergen, Wim MichielsAbstract:We propose a new uniform framework of compact rational Krylov (CORK) methods for solving large-scale Nonlinear Eigenvalue problems $A(\lambda) x = 0$. For many years, linearizations were used for solving polynomial and rational Eigenvalue problems. On the other hand, for the general Nonlinear case, $A(\lambda)$ can first be approximated by a (rational) matrix polynomial and then a convenient linearization is used. However, the major disadvantage of linearization-based methods is the growing memory and orthogonalization costs with the iteration count, i.e., in general they are proportional to the degree of the polynomial. Therefore, the CORK family of rational Krylov methods exploits the structure of the linearization pencils by using a generalization of the compact Arnoldi decomposition. In this way, the extra memory and orthogonalization costs due to the linearization of the original Eigenvalue problem are negligible for large-scale problems. Furthermore, we prove that each CORK step breaks down into an ...
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nleigs a class of fully rational krylov methods for Nonlinear Eigenvalue problems
SIAM Journal on Scientific Computing, 2014Co-Authors: Stefan Güttel, Karl Meerbergen, Roel Van Beeumen, Wim MichielsAbstract:A new rational Krylov method for the efficient solution of Nonlinear Eigenvalue problems is proposed. This iterative method, called fully rational Krylov method for Nonlinear Eigenvalue problems (abbreviated as NLEIGS), is based on linear rational interpolation and generalizes the Newton rational Krylov method proposed in [R. Van Beeumen, K. Meerbergen, and W. Michiels, SIAM J. Sci. Comput., 35 (2013), pp. A327-A350]. NLEIGS utilizes a dynamically constructed rational interpolant of the Nonlinear operator and a new companion-type linearization for obtaining a generalized Eigenvalue problem with special structure. This structure is particularly suited for the rational Krylov method. A new approach for the computation of rational divided differences using matrix functions is presented. It is shown that NLEIGS has a computational cost comparable to the Newton rational Krylov method but converges more reliably, in particular, if the Nonlinear operator has singularities nearby the target set. Moreover, NLEIGS implements an automatic scaling procedure which makes it work robustly independent of the location and shape of the target set, and it also features low-rank approximation techniques for increased computational efficiency. Small- and large-scale numerical examples are included.
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Fast algorithms for computing the distance to instability of Nonlinear Eigenvalue problems, with application to time-delay systems
International Journal of Dynamics and Control, 2014Co-Authors: Dries Verhees, Karl Meerbergen, Roel Beeumen, Nicola Guglielmi, Wim MichielsAbstract:A continuous dynamical system is stable if all Eigenvalues lie strictly in the left half of the complex plane. However, this is not a robust measure because stability is no longer guaranteed when the system parameters are slightly perturbed. Therefore, the pseudospectrum of a matrix and its pseudospectral abscissa are studied. Mostly, one is often interested in computing the distance to instability, because it is a robust measure for stability against perturbations. As a first contribution, this paper presents two iterative methods for computing the distance to instability, considering complex perturbations. The first one is based on locating a zero of the pseudospectral abscissa function. This method is particularly suitable for large sparse matrices as it is based on repeated Eigenvalue computations, where the original matrix is perturbed with a rank one matrix. The second method is based on a recently proposed global optimization technique. The advantages of both methods can be combined in a hybrid algorithm. As a second contribution we show that the methods apply to a broad class of Nonlinear Eigenvalue problems, in particular Eigenvalue problems inferred from linear delay-differential equations, and, therefore, they are useful for a wide range of problems. In the numerical examples the standard Eigenvalue problem, the quadratic Eigenvalue problem and the delay Eigenvalue problem are addressed.
Heinrich Voss - One of the best experts on this subject based on the ideXlab platform.
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Nonlinear Eigenvalue problems a challenge for modern Eigenvalue methods
Gamm-mitteilungen, 2004Co-Authors: Volker Mehrmann, Heinrich VossAbstract:We discuss the state of the art in numerical solution methods for large scale polynomial or rational Eigenvalue problems. We present the currently available solution methods such as the Jacobi-Davidson, Arnoldi or the rational Krylov method and analyze their properties. We briefly introduce a new linearization technique and demonstrate how it can be used to improve structure preservation and with this the accuracy and efficiency of linearization based methods. We present several recent applications where structured and unstructured Nonlinear Eigenvalue problems arise and some numerical results. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
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an arnoldi method for Nonlinear Eigenvalue problems
Bit Numerical Mathematics, 2004Co-Authors: Heinrich VossAbstract:For the Nonlinear Eigenvalue problem T(λ)x=0 we propose an iterative projection method for computing a few Eigenvalues close to a given parameter. The current search space is expanded by a generalization of the shift-and-invert Arnoldi method. The resulting projected eigenproblems of small dimension are solved by inverse iteration. The method is applied to a rational Eigenvalue problem governing damped vibrations of a structure.
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a jacobi davidson type projection method for Nonlinear Eigenvalue problems
Future Generation Computer Systems, 2004Co-Authors: Timo Betcke, Heinrich VossAbstract:This paper discusses a projection method for Nonlinear Eigenvalue problems. The subspace of approximants is constructed by a Jacobi-Davidson-type approach, and the arising eigenproblems of small dimension are solved by safeguarded iteration. The method is applied to a rational Eigenvalue problem governing the vibrations of tube bundle immersed in an inviscid compressible fluid.
Karl Meerbergen - One of the best experts on this subject based on the ideXlab platform.
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automatic rational approximation and linearization of Nonlinear Eigenvalue problems
arXiv: Numerical Analysis, 2018Co-Authors: Pieter Lietaert, Javier Perez, Bart Vandereycken, Karl MeerbergenAbstract:We present a method for solving Nonlinear Eigenvalue problems using rational approximation. The method uses the AAA method by Nakatsukasa, S\`{e}te, and Trefethen to approximate the Nonlinear Eigenvalue problem by a rational Eigenvalue problem and is embedded in the state space representation of a rational polynomial by Su and Bai. The advantage of the method, compared to related techniques such as NLEIGS and infinite Arnoldi, is the efficient computation by an automatic procedure. In addition, a set-valued approach is developed that allows building a low degree rational approximation of a Nonlinear Eigenvalue problem. The method perfectly fits the framework of the Compact rational Krylov methods (CORK and TS-CORK), allowing to efficiently solve large scale Nonlinear Eigenvalue problems. Numerical examples show that the presented framework is competitive with NLEIGS and usually produces smaller linearizations with the same accuracy but with less effort for the user.
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Computation of pseudospectral abscissa for large-scale Nonlinear Eigenvalue problems
IMA Journal of Numerical Analysis, 2017Co-Authors: Karl Meerbergen, Wim Michiels, Emre Mengi, Roel Van BeeumenAbstract:Author(s): Meerbergen, K; Michiels, W; Van Beeumen, R; Mengi, E | Abstract: © The authors 2017. We present an algorithm to compute the pseudospectral abscissa for a Nonlinear Eigenvalue problem. The algorithm relies on global under-estimator and over-estimator functions for the Eigenvalue and singular value functions involved. These global models follow from Eigenvalue perturbation theory. The algorithm has three particular features. First, it converges to the globally rightmost point of the pseudospectrum, and it is immune to nonsmoothness. The global convergence assertion is under the assumption that a global lower bound is available for the second derivative of a singular value function depending on one parameter. It may not be easy to deduce such a lower bound analytically, but assigning large negative values works robustly in practice. Second, it is applicable to large-scale problems since the dominant cost per iteration stems from computing the smallest singular value and associated singular vectors, for which efficient iterative solvers can be used. Furthermore, a significant increase in computational efficiency can be obtained by subspace acceleration, that is, by restricting the domains of the linear maps associated with the matrices involved to small but suitable subspaces, and solving the resulting reduced problems. Occasional restarts of these subspaces further enhance the efficiency for large-scale problems. Finally, in contrast to existing iterative approaches based on constructing low-rank perturbations and rightmost Eigenvalue computations, the algorithm relies on computing only singular values of complex matrices. Hence, the algorithm does not require solutions of Nonlinear Eigenvalue problems, thereby further increasing efficiency and reliability. This work is accompanied by a robust implementation of the algorithm that is publicly available.
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compact rational krylov methods for Nonlinear Eigenvalue problems
SIAM Journal on Matrix Analysis and Applications, 2015Co-Authors: Roel Van Beeumen, Karl Meerbergen, Wim MichielsAbstract:We propose a new uniform framework of compact rational Krylov (CORK) methods for solving large-scale Nonlinear Eigenvalue problems $A(\lambda) x = 0$. For many years, linearizations were used for solving polynomial and rational Eigenvalue problems. On the other hand, for the general Nonlinear case, $A(\lambda)$ can first be approximated by a (rational) matrix polynomial and then a convenient linearization is used. However, the major disadvantage of linearization-based methods is the growing memory and orthogonalization costs with the iteration count, i.e., in general they are proportional to the degree of the polynomial. Therefore, the CORK family of rational Krylov methods exploits the structure of the linearization pencils by using a generalization of the compact Arnoldi decomposition. In this way, the extra memory and orthogonalization costs due to the linearization of the original Eigenvalue problem are negligible for large-scale problems. Furthermore, we prove that each CORK step breaks down into an ...
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nleigs a class of fully rational krylov methods for Nonlinear Eigenvalue problems
SIAM Journal on Scientific Computing, 2014Co-Authors: Stefan Güttel, Karl Meerbergen, Roel Van Beeumen, Wim MichielsAbstract:A new rational Krylov method for the efficient solution of Nonlinear Eigenvalue problems is proposed. This iterative method, called fully rational Krylov method for Nonlinear Eigenvalue problems (abbreviated as NLEIGS), is based on linear rational interpolation and generalizes the Newton rational Krylov method proposed in [R. Van Beeumen, K. Meerbergen, and W. Michiels, SIAM J. Sci. Comput., 35 (2013), pp. A327-A350]. NLEIGS utilizes a dynamically constructed rational interpolant of the Nonlinear operator and a new companion-type linearization for obtaining a generalized Eigenvalue problem with special structure. This structure is particularly suited for the rational Krylov method. A new approach for the computation of rational divided differences using matrix functions is presented. It is shown that NLEIGS has a computational cost comparable to the Newton rational Krylov method but converges more reliably, in particular, if the Nonlinear operator has singularities nearby the target set. Moreover, NLEIGS implements an automatic scaling procedure which makes it work robustly independent of the location and shape of the target set, and it also features low-rank approximation techniques for increased computational efficiency. Small- and large-scale numerical examples are included.
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Fast algorithms for computing the distance to instability of Nonlinear Eigenvalue problems, with application to time-delay systems
International Journal of Dynamics and Control, 2014Co-Authors: Dries Verhees, Karl Meerbergen, Roel Beeumen, Nicola Guglielmi, Wim MichielsAbstract:A continuous dynamical system is stable if all Eigenvalues lie strictly in the left half of the complex plane. However, this is not a robust measure because stability is no longer guaranteed when the system parameters are slightly perturbed. Therefore, the pseudospectrum of a matrix and its pseudospectral abscissa are studied. Mostly, one is often interested in computing the distance to instability, because it is a robust measure for stability against perturbations. As a first contribution, this paper presents two iterative methods for computing the distance to instability, considering complex perturbations. The first one is based on locating a zero of the pseudospectral abscissa function. This method is particularly suitable for large sparse matrices as it is based on repeated Eigenvalue computations, where the original matrix is perturbed with a rank one matrix. The second method is based on a recently proposed global optimization technique. The advantages of both methods can be combined in a hybrid algorithm. As a second contribution we show that the methods apply to a broad class of Nonlinear Eigenvalue problems, in particular Eigenvalue problems inferred from linear delay-differential equations, and, therefore, they are useful for a wide range of problems. In the numerical examples the standard Eigenvalue problem, the quadratic Eigenvalue problem and the delay Eigenvalue problem are addressed.
L V Stepanova - One of the best experts on this subject based on the ideXlab platform.
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stress strain state near the crack tip under mixed mode loading asymptotic approach and numerical solutions of Nonlinear Eigenvalue problems
MECHANICS RESOURCE AND DIAGNOSTICS OF MATERIALS AND STRUCTURES (MRDMS-2016): Proceedings of the 10th International Conference on Mechanics Resource an, 2016Co-Authors: L V Stepanova, Ekaterina YakovlevaAbstract:Creep crack problems in damaged materials under mixed-mode loading in the creep-damage coupled formulation are considered. The class of self-similar solutions to the plane creep crack problems in a damaged medium under mixed-mode loading is given. With the similarity variable and the self-similar representation of the solution for a power-law creeping material and the Kachanov-Rabotnov power-law damage evolution equation, the near-crack-tip stresses, creep strain rates and continuity (integrity) distributions for plane stress conditions are obtained. The self-similar solutions are based on the hypothesis of the existence of a completely damaged zone near the crack tip. It is shown that the asymptotical analysis of the near crack-tip fields gives rise to Nonlinear Eigenvalue problems. A technique enabling all the Eigenvalues to be found numerically is proposed, and numerical solutions to Nonlinear Eigenvalue problems arising from the mixed-mode crack problems in a power-law medium under plane stress condit...
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asymptotics of Eigenvalues of the Nonlinear Eigenvalue problem arising from the near mixed mode crack tip stress strain field problems
Numerical Analysis and Applications, 2016Co-Authors: L V Stepanova, Ekaterina YakovlevaAbstract:In the present paper, approximate analytical and numerical solutions to Nonlinear Eigenvalue problems arising in Nonlinear fracture mechanics in studying stress-strain fields near a crack tip under mixed-mode loading are presented. Asymptotic solutions are obtained by the perturbation method (the artificial small parameter method). The artificial small parameter is the difference between the Eigenvalue corresponding to the Nonlinear Eigenvalue problem and the Eigenvalue related to the linear “undisturbed” problem. It is shown that the perturbation technique is an effective method of solving Nonlinear Eigenvalue problems in Nonlinear fracture mechanics. A comparison of numerical and asymptotic results for different values of the mixity parameter and hardening exponent shows good agreement. Thus, the perturbation theory technique for studying Nonlinear Eigenvalue problems is offered and applied to Eigenvalue problems arising in fracture mechanics analysis in the case of mixed-mode loading.
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asymptotics of the near crack tip stress field of a growing fatigue crack in damaged materials numerical experiment and analytical solution
Numerical Analysis and Applications, 2015Co-Authors: L V Stepanova, S A IgoninAbstract:In this paper, an asymptotic analysis of growing fatigue near-crack-tip fields in a damaged material is performed. The integrity parameter describing the damage accumulation process in the vicinity of the crack tip is incorporated into the constitutive law of an isotropic linear elastic material. An asymptotic solution based on the eigenfunction expansion method is obtained. It is shown that the problem is reduced to a Nonlinear Eigenvalue problem. An analytical solution of the Nonlinear Eigenvalue problem is found by the artificial small parameter method. The perturbation theory approach allows us to derive the analytical presentation of the stress and integrity distributions near the crack tip. The technique proposed permits us to find higher-order terms of the asymptotic expansions of the stress components and the integrity parameter.
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perturbation method for solving the Nonlinear Eigenvalue problem arising from fatigue crack growth problem in a damaged medium
Applied Mathematical Modelling, 2014Co-Authors: L V Stepanova, S A IgoninAbstract:Abstract An analytical solution of the Nonlinear Eigenvalue problem arising from the fatigue crack growth problem in a damaged medium in coupled formulation is obtained. The perturbation technique for solving the Nonlinear Eigenvalue problem is used. The method allows to find the analytical formula expressing the Eigenvalue as the function of parameters of the damage evolution law. It is shown that the Eigenvalues of the Nonlinear Eigenvalue problem are fully determined by the exponents of the damage evolution law. In the paper the third-order (four-term) asymptotic expansions of the angular functions determining the stress and continuity fields in the neighborhood of the crack tip are given. The asymptotic expansions of the angular functions permit to find the closed-form solution for the problem considered.
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Eigenvalue analysis for a crack in a power law material
Computational Mathematics and Mathematical Physics, 2009Co-Authors: L V StepanovaAbstract:A Nonlinear Eigenvalue problem related to determining the stress and strain fields near the tip of a transverse crack in a power-law material is studied. The Eigenvalues are found by a perturbation method based on representations of an Eigenvalue, the corresponding eigenfunction, and the material Nonlinearity parameter in the form of series expansions in powers of a small parameter equal to the difference between the Eigenvalues in the linear and Nonlinear problems. The resulting Eigenvalues are compared with the accurate numerical solution of the Nonlinear Eigenvalue problem.
Luigi De Pascale - One of the best experts on this subject based on the ideXlab platform.
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asymptotic behavior of non linear Eigenvalue problems involving p laplacian type operators
Proceedings of The Royal Society A: Mathematical Physical and Engineering Sciences, 2007Co-Authors: Thierry Champion, Luigi De PascaleAbstract:We study the asymptotic behaviour of two Nonlinear Eigenvalue problems which involve $p$ -Laplacian-type operators. In the first problem we consider the limit as $p\to\infty$ of the sequences of the $k$ th Eigenvalues of the $p$ -Laplacian operators. The second problem we study is the homogenization of Nonlinear Eigenvalue problems for some $p$ -Laplacian-type operators with $p$ fixed. Our asymptotic analysis relies on a convergence result for particular critical values of a class of Rayleigh quotients, stated in a unified framework, and on the notion of $\varGamma$ -convergence.
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asymptotic behavior of non linear Eigenvalue problems involving p laplacian type operators
Proceedings of The Royal Society A: Mathematical Physical and Engineering Sciences, 2007Co-Authors: Thierry Champion, Luigi De PascaleAbstract:We study the asymptotic behaviour of two Nonlinear Eigenvalue problems which involve -convergence.
