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A harmonic FEAST algorithm for non-Hermitian generalized eigenvalue problems

LINEAR ALGEBRA AND ITS APPLICATIONS 2019年 578卷 75-94页

作者： Yin, Guojian Sun Yat Sen Univ Sch Math Guangzhou Guangdong Peoples R China

The FEAST algorithm, a contour-integral based eigensolver, was developed for computing the eigenvalues inside a given interval, along with their eigenvectors, of a Hermitian generalized eigenproblem. The FEAST algorithm is a fast and stable technique, and is easily parallelizable. However, it was shown that this algorithm may fail to find the desired eigenpairs when applied to non-Hermitian problems. Efforts have been made to adapt FEAST to non-Hermitian problems. In this work, we aim at formulating a new non-Hermitian scheme for the FEAST algorithm. Our new scheme is based on the partial generalized Schur decomposition. Unlike the existing non Hermitian FEAST methods, our method seelcs approximation to the generalized Schur vectors instead of the potentially ill-conditioned eigenvectors. Our new method can compute not only the eigenvectors but also the right and left generalized Schur vectors corresponding to target eigenvalues. We present standard benchmark numerical experiments in which our method achieves better stability and efficiency comparing with the existing non-Hermitian FEAST algorithms. (C) 2019 Elsevier Inc. All rights reserved.

关键词： generalized eigenvalue problems generalized Schur form Contour integral Spectral projection

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A Contour-Integral Based Method with Schur-Rayleigh-Ritz Procedure for generalized eigenvalue problems

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JOURNAL OF SCIENTIFIC COMPUTING 2019年第1期81卷 252-270页

作者： Yin, Guojian Sun Yat Sen Univ Sch Math Guangzhou Guangdong Peoples R China

Recently, a class of eigensolvers based on contour integrals has been developed for computing the eigenvalues inside a given region in the complex plane. The CIRR method (a Rayleigh-Ritz type method with contour integrals) is a classic example among this kind of methods. It first constructs a subspace to contain the eigenspace of interest via a set of contour integrals, and then uses the standard Rayleigh-Ritz procedure to extract desired eigenpairs. However, it was shown that the CIRR method may fail to find the desired eigenpairs when the considered eigenproblem is non-Hermitian. This fact motivates us to develop a non-Hermitian scheme for the CIRR method. To this end, we formulate a Schur-Rayleigh-Ritz procedure to extract the desired eigenpairs. The theoretical analysis shows that our new extraction scheme can make the CIRR method also applicable for the non-Hermitian problems. Some implementation issues arising in practical applications are also studied. Numerical experiments are reported to illustrate the numerical performance of our new method.

关键词： generalized eigenvalue problems Contour integral QZ method generalized Schur decomposition

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FAST COMPUTATION OF SPECTRAL DENSITIES FOR generalized eigenvalue problems

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SIAM JOURNAL ON SCIENTIFIC COMPUTING 2018年第4期40卷 A2749-A2773页

作者： Xi, Yuanzhe Li, Ruipeng Saad, Yousef Emory Univ Dept Math & Comp Sci Atlanta GA 30322 USA Lawrence Livermore Natl Lab Ctr Appl Sci Comp PoB 808L-561 Livermore CA 94551 USA Univ Minnesota Dept Comp Sci & Engn Minneapolis MN 55455 USA

The distribution of the eigenvalues of a Hermitian matrix (or of a Hermitian matrix pencil) reveals important features of the underlying problem, whether a Hamiltonian system in physics or a social network in behavioral sciences. However, computing all the eigenvalues explicitly is prohibitively expensive for real-world applications. This paper presents two types of methods to efficiently estimate the spectral density of a matrix pencil (A, B) where both A and B are sparse Hermitian and B is positive definite. The methods are targeted at the situation when the matrix B scaled by its diagonal is very well conditioned, as is the case when the problem arises from some finite element discretizations of certain partial differential equations. The first method is an adaptation of the kernel polynomial method (KPM) and the second is based on Gaussian quadrature by the Lanczos procedure. By employing Chebyshev polynomial approximation techniques, we can avoid direct factorizations in both methods, making the resulting algorithms suitable for large matrices. Under some assumptions, we prove bounds that suggest that the Lanczos method converges twice as fast as the KPM method. Numerical examples further indicate that the Lanczos method can provide more accurate spectral densities when the eigenvalue distribution is highly nonuniform. As an application, we show how to use the computed spectral density to partition the spectrum into intervals that contain roughly the same number of eigenvalues. This procedure, which makes it possible to compute the spectrum by parts, is a key ingredient in the new breed of eigensolvers that exploit "spectrum slicing."

关键词： spectral density density of states generalized eigenvalue problems spectrum slicing Chebyshev approximation perturbation theory

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A FEAST algorithm with oblique projection for generalized eigenvalue problems

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NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS 2017年第4期24卷 n/a-N.PAG页

作者： Yin, Guojian Chan, Raymond H. Yeung, Man-Chung Sun Yat Sen Univ Sch Math Guangzhou Guangdong Peoples R China Chinese Univ Hong Kong Dept Math Shatin Hong Kong Peoples R China Univ Wyoming Dept Math Dept 3036 1000 East Univ Ave Laramie WY 82071 USA

The contour integral-based eigensolvers are the recent efforts for computing the eigenvalues inside a given region in the complex plane. The best-known members are the Sakurai-Sugiura method, its stable version CIRR, and the FEAST algorithm. An attractive computational advantage of these methods is that they are easily parallelizable. The FEAST algorithm was developed for the generalized Hermitian eigenvalue problems. It is stable and accurate. However, it may fail when applied to non-Hermitian problems. Recently, a dual subspace FEAST algorithm was proposed to extend the FEAST algorithm to non-Hermitian problems. In this paper, we instead use the oblique projection technique to extend FEAST to the non-Hermitian problems. Our approach can be summarized as follows: (a) construct a particular contour integral to form a search subspace containing the desired eigenspace and (b) use the oblique projection technique to extract desired eigenpairs with appropriately chosen test subspace. The related mathematical framework is established. Comparing to the dual subspace FEAST algorithm, we can save the computational cost roughly by a half if only the eigenvalues or the eigenvalues together with their right eigenvectors are needed. We also address some implementation issues such as how to choose a suitable starting matrix and design-efficient stopping criteria. Numerical experiments are provided to illustrate that our method is stable and efficient.

关键词： contour integral generalized eigenvalue problems spectral projection

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Relationships among contour integral-based methods for solving generalized eigenvalue problems

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JAPAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS 2016年第3期33卷 721-750页

作者： Imakura, Akira Du, Lei Sakurai, Tetsuya Univ Tsukuba Grad Sch Syst & Informat Engn 1-1-1 Tennodai Tsukuba Ibaraki 3058573 Japan Dalian Univ Technol Sch Math Sci Dalian 11024 Peoples R China

Recently, contour integral-based methods have been actively studied for solving interior eigenvalue problems that find all eigenvalues located in a certain region and their corresponding eigenvectors. In this paper, we reconsider the algorithms of the five typical contour integral-based eigensolvers from the viewpoint of projection methods, and then map the relationships among these methods. From the analysis, we conclude that all contour integral-based eigensolvers can be regarded as projection methods and can be categorized based on their subspace used, the type of projection and the problem to which they are applied implicitly.

关键词： generalized eigenvalue problems Contour integral-based eigensolvers Projection methods

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A block Arnoldi-type contour integral spectral projection method for solving generalized eigenvalue problems

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APPLIED MATHEMATICS LETTERS 2014年第1期32卷 22-27页

作者： Imakura, Akira Du, Lei Sakurai, Tetsuya Univ Tsukuba Tsukuba Ibaraki 3058573 Japan JST CREST Tokyo Japan

For generalized eigenvalue problems, we consider computing all eigenvalues located in a certain region and their corresponding eigenvectors. Recently, contour integral spectral projection methods have been proposed for solving such problems. In this study, from the analysis of the relationship between the contour integral spectral projection and the Krylov subspace, we conclude that the Rayleigh-Ritz-type of the contour integral spectral projection method is mathematically equivalent to the Arnoldi method with the projected vectors obtained from the contour integration. By this Arnoldi-based interpretation, we then propose a block Arnoldi-type contour integral spectral projection method for solving the eigenvalue problem. (C) 2014 Elsevier Ltd. All rights reserved.

关键词： generalized eigenvalue problems Contour integral spectral projection Krylov subspace Block Arnoldi method

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RI biorthogonal polynomials of Hahn type

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JOURNAL OF DIFFERENCE EQUATIONS AND APPLICATIONS 2024年第11期30卷 1765-1781页

作者： Vinet, Luc Zaimi, Meri Zhedanov, Alexei Univ Montreal Ctr Rech Math POB 6128Ctr Ville Stn Montreal PQ H3C 3J7 Canada Insitut Valorisat Donnees IVADO Montreal PQ Canada Renmin Univ China Sch Math Beijing Peoples R China

A finite family of R-I polynomials is introduced and studied. It consists in a set of polynomials of F-3(2) form whose biorthogonality to an ensemble of rational functions is spelled out. These polynomials are shown to satisfy two generalized eigenvalue problems: in addition to their recurrence relation of R-I type, they are also found to obey a difference equation. Underscoring this bispectrality is a triplet of operators with tridiagonal actions. The algebra associated to these operators is provided.

关键词： Biorthogonal polynomials bispectrality generalized eigenvalue problems tridiagonal operators Hahn polynomials

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A symmetry-protected exceptional ring in a photonic crystal with negative index media

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NANOPHOTONICS 2023年第13期12卷 2335-2346页

作者： Isobe, Takuma Yoshida, Tsuneya Hatsugai, Yasuhiro Univ Tsukuba Grad Sch Pure & Appl Sci Tsukuba Ibaraki Japan Kyoto Univ Dept Phys Kyoto Japan Univ Tsukuba Dept Phys Tsukuba Japan

Non-Hermitian topological band structures such as symmetry-protected exceptional rings (SPERs) can emerge for systems described by the generalized eigenvalue problem (GEVP) with Hermitian matrices. In this paper, we numerically analyze a photonic crystal with negative index media, which is described by the GEVP with Hermitian matrices. Our analysis using COMSOL Multiphysics((R)) demonstrates that a SPER emerges for photonic crystals composed of split-ring resonators and metal-wire structures. We expect that the above SPER can be observed in experiments as it emerges at a finite frequency.

关键词： generalized eigenvalue problems non-Hermitian topology topological photonics

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FAME: Fast Algorithms for Maxwell's Equations for Three-dimensional Photonic Crystals

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ACM TRANSACTIONS ON MATHEMATICAL SOFTWARE 2021年第3期47卷 26-26页

作者： Lyu, Xing-long Li, Tiexiang Huang, Tsung-ming Lin, Jia-wei Lin, Wen-wei Wang, Sheng Southeast Univ Sch Math Nanjing 211189 Peoples R China Nanjing Ctr Appl Math Nanjing 211135 Peoples R China Natl Taiwan Normal Univ Dept Math Taipei 116 Taiwan Natl Chiao Tung Univ Dept Appl Math Hsinchu 300 Taiwan

In this article, we propose the Fast Algorithms for Maxwell's Equations (FAME) package for solving Maxwell's equations for modeling three-dimensional photonic crystals. FAME combines the null-space free method with fast Fourier transform (FFT)-based matrix-vector multiplications to solve the generalized eigenvalue problems (GEPs) arising from Yee's discretization. The GEPs are transformed into a null-space free standard eigenvalue problem with a Hermitian positive-definite coefficient matrix. The computation times for FFT-based matrix-vector multiplications with matrices of dimension 7 million are only 0.33 and 3.6 x 10(-3) seconds using MATLAB with an Intel Xeon CPU and CUDA C++ programming with a single NVIDIA Tesla P100 GPU, respectively. Such multiplications significantly reduce the computational costs of the conjugate gradient method for solving linear systems. We successfully use FAME on a single P100 GPU to solve a set of GEPs with matrices of dimension more than 19 million, in 127 to 191 seconds per problem. These results demonstrate the potential of our proposed package to enable large-scale numerical simulations for novel physical discoveries and engineering applications of photonic crystals.

关键词： Maxwell's equations three-dimensional photonic crystals generalized eigenvalue problems null-space free method fast Fourier transform

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An Eigenmode Correlation-Based Algorithm for Approaching Antenna Optimal Currents With Multiple Feeds

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RADIO SCIENCE 2020年第5期55卷 e2019RS006957-e2019RS006957页

作者： Zhang, Wei-Quan Zhang, Guo-Yang Wang, Bing-Zhong Xiong, Jiang Univ Elect Sci & Technol China Inst Appl Phys Computat Electromagnet Lab Chengdu Peoples R China

A feeding algorithm has been proposed for the excitation of the optimal electric current of Rayleigh quotient form optimization problems. Modal weighting coefficients of the corresponding generalized eigenvalue problem are calculated to form the correlation information between the desired and all other unwanted modes. Based on this information, multiple feeds are arranged to approximate the desired current distributions. The eigencurrent distribution is also utilized for some shape modification to the radiator for a possible feed number reduction. Three canonical optimization problems for electrically small antennas, minimum Q, maximum G/Q, and maximum G are studied and discussed. Several numerical examples with a certain arbitrary shape and electrical size are given. With the proposed feeding algorithm, calculated and simulated current distributions and performance indices are close to the theoretical optimum solution. Finally, a practical feeding network design for a classic characteristic mode problem is shown as a validation.

关键词： antenna optimization electrically small antennas generalized eigenvalue problems method of moments Q factor

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