Designing rational filter functions for solving eigenvalue problems by contour integration

Barel, Marc Van

doi:10.1016/j.laa.2015.05.029

Cited by 28 publications

(40 citation statements)

References 26 publications

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“…Similar conclusion has been obtained for the CIA; see Theorem 3 in [16]. It turns out that the eigenspace of the moment scheme is the same as the one used in the SS-RI algorithm; see (41), (39) and (40). However the derivation here is directly from Theorem 1 and the interpolation theory; one does not need to refer to the derivation of the SS-RI algorithm in Section 3.2.…”

Section: Resolvent Moment Schemesupporting

confidence: 64%

“…Both circles and ellipses are frequently used. Theoretical results based on rational filter theory indicate that circular contours should lead to better accuracy since in this case the filters are more closer to the indicator function of the interval [19,40]. Numerical results, however, demonstrate that flat elliptical contours often achieves better results.…”

Section: Rational Interpolation Approachmentioning

confidence: 96%

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Solving large‐scale nonlinear eigenvalue problems by rational interpolation and resolvent sampling based Rayleigh–Ritz method

Xiao

Zhang

Huang

et al. 2016

Numerical Meth Engineering

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Numerical solution of nonlinear eigenvalue problems (NEPs) is frequently encountered in computational science and engineering. The applicability of most existing methods is limited by the matrix structures, properties of the eigen-solutions, sizes of the problems, etc. This paper aims to remove those limitations and develop robust and universal NEP solvers for large-scale engineering applications. The novelty lies in two aspects. First, a rational interpolation approach (RIA) is proposed based on the Keldysh theorem for holomorphic matrix functions. Comparing with the existing contour integral approach, the RIA provides the possibility to select sampling points in more general regions and has advantages in improving the accuracy and reducing the computational cost. Second, a resolvent sampling scheme using the RIA is proposed to construct reliable search spaces for the Rayleigh-Ritz procedure, based on which a robust eigen-solver, called resolvent sampling based Rayleigh-Ritz method (RSRR), is developed for solving general NEPs. The RSRR can be easily implemented and parallelized. The advantages of the RIA and the performance of the RSRR are demonstrated by a variety of benchmark and application examples.Generally speaking, although there exist a number of numerical methods for NEPs in literature, most of them are restricted by the matrix structures, properties of eigenvalues, computational costs, etc. Accurate and efficient methods that can robustly and reliably calculate all the eigenpairs within a given region, and at the same time, can be applied to large-scale computations are still lacking [3,4,[7][8][9][10][11].In recent years, the contour integral approach (CIA) based on contour integrals of the probed resolvent T .´/ 1 U has attracted great attention [12][13][14][15][16][17][18][19], where the constant matrix U 2 C n L consists of L random column vectors used to probe T .´/ 1 . The CIA is first developed for solving generalized eigenvalue problems [12,13], and later, extended to NEPs in [20] and [15] based on the Smith form and Keldysh's theorem for analytic matrix-valued functions, respectively. Because [20] and [15] result in similar algorithms, we consider the block Sakurai-Sugiura (SS) algorithm in [20]. This algorithm transforms the original NEP into a small-sized generalized eigenvalue problem involving block Hankel matrices by using the monomial moments of the probed resolvent T .´/ 1 U on a given contour. It becomes unstable and inaccurate when higher order contour moments of T .´/ 1 U are used.To circumvent this problem, the SS method with the Rayleigh-Ritz projection (SSRR) has been recently proposed [16]. The eigenspace of the SSRR is constructed from a matrix M 2 C n K L collecting all the monomial moments of T .´/ 1 U up to a given order K. We refer to this as the resolvent moment scheme (or simply moment scheme) for generating eigenspaces. Numerical results in [16] show that the SSRR has a much better stability and accuracy than the SS algorithm when a small value of L is used. Besides, the...

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Section: Resolvent Moment Schemesupporting

confidence: 64%

Section: Rational Interpolation Approachmentioning

confidence: 96%

Solving large‐scale nonlinear eigenvalue problems by rational interpolation and resolvent sampling based Rayleigh–Ritz method

Xiao

Zhang

Huang

et al. 2016

Numerical Meth Engineering

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“…More recently the filter itself has been treated as a parameter that can be designed via optimization methods. Van Barel [2] proposed a non-linear Least-Squares approach for non-Hermitian filters within the SS-H framework, while Xi and Saad [26] described linear Least-Squares optimized filters for the Hermitian FEAST solver. Since Van Barel's approach is geared towards non-Hermitian eigenproblems, it is not possible to make use of the conjugate or reflection symmetry.…”

Section: Related Workmentioning

confidence: 99%

Non-linear Least-Squares Optimization of Rational Filters for the Solution of Interior Hermitian Eigenvalue Problems

Winkelmann

Napoli

2019

Front. Appl. Math. Stat.

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Rational filter functions can be used to improve convergence of contour-based eigensolvers, a popular family of algorithms for the solution of the interior eigenvalue problem. We present a framework for the optimization of rational filters based on a non-convex weighted Least-Squares scheme. When used in combination with the FEAST library, our filters out-perform existing ones on a large and representative set of benchmark problems. This work provides a detailed description of : (1) a set up of the optimization process that exploits symmetries of the filter function for Hermitian eigenproblems, (2) a formulation of the gradient descent and Levenberg-Marquardt algorithms that exploits the symmetries, (3) a method to select the starting position for the optimization algorithms that reliably produces effective filters, (4) a constrained optimization scheme that produces filter functions with specific properties that may be beneficial to the performance of the eigensolver that employs them.

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“…2,3,22 The poles of this scalar function are the eigenvalues of A. See also the work of Austin et al 23 for a pole-finding eigenvalue solver based on rational interpolation that exploits real arithmetic only, and other works 15,24,25 for applications of contour integration to the solution of nonlinear eigenvalue problems.…”

Section: Contour Integration-based Eigenvalue Solversmentioning

confidence: 99%

Domain decomposition approaches for accelerating contour integration eigenvalue solvers for symmetric eigenvalue problems

Kalantzis

Kestyn

Polizzi

et al. 2018

Numerical Linear Algebra App

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Summary This paper discusses techniques for computing a few selected eigenvalue–eigenvector pairs of large and sparse symmetric matrices. A recently developed class of techniques to solve this type of problems is based on integrating the matrix resolvent operator along a complex contour that encloses the interval containing the eigenvalues of interest. This paper considers such contour integration techniques from a domain decomposition viewpoint and proposes two schemes. The first scheme can be seen as an extension of domain decomposition linear system solvers in the framework of contour integration methods for eigenvalue problems, such as FEAST. The second scheme focuses on integrating the resolvent operator primarily along the interface region defined by adjacent subdomains. A parallel implementation of the proposed schemes is described, and results on distributed computing environments are reported. These results show that domain decomposition approaches can lead to reduced run times and improved scalability.

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Designing rational filter functions for solving eigenvalue problems by contour integration

Cited by 28 publications

References 26 publications

Solving large‐scale nonlinear eigenvalue problems by rational interpolation and resolvent sampling based Rayleigh–Ritz method

Solving large‐scale nonlinear eigenvalue problems by rational interpolation and resolvent sampling based Rayleigh–Ritz method

Non-linear Least-Squares Optimization of Rational Filters for the Solution of Interior Hermitian Eigenvalue Problems

Domain decomposition approaches for accelerating contour integration eigenvalue solvers for symmetric eigenvalue problems

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