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

Imakura, Akira; Du, Lei; Sakurai, Tetsuya

doi:10.1007/s13160-016-0224-x

Cited by 29 publications

(29 citation statements)

References 25 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Clearly, C N can be bounded independently of h, since }pz´S h N q´1} V Ñ }pz´S N q´1} V . Thus, by virtue of (16), δ h Ñ 0 as h Ñ 0. In particular, for sufficiently small h, we have δ h ă 1.…”

Section: Discretization Errors In Eigenspacementioning

confidence: 97%

“…Following [2], we identify two different classes of methods in the existing literature that use contour integrals for computation of a targeted cluster of matrix eigenvalues. One class of methods, that often goes by the name SSM [24] (see also [4,16]), approximates Λ by the eigenvalues of a system of moment matrices based on contour integrals. The moment matrices are obtained by approximating the integrals by a quadrature, and the spectral approximation error depends on the accuracy of the quadrature.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Spectral discretization errors in filtered subspace iteration

2019

View full text Add to dashboard Cite

We consider filtered subspace iteration for approximating a cluster of eigenvalues (and its associated eigenspace) of a (possibly unbounded) selfadjoint operator in a Hilbert space. The algorithm is motivated by a quadrature approximation of an operator-valued contour integral of the resolvent. Resolvents on infinite dimensional spaces are discretized in computable finite-dimensional spaces before the algorithm is applied. This study focuses on how such discretizations result in errors in the eigenspace approximations computed by the algorithm. The computed eigenspace is then used to obtain approximations of the eigenvalue cluster. Bounds for the Hausdorff distance between the computed and exact eigenvalue clusters are obtained in terms of the discretization parameters within an abstract framework. A realization of the proposed approach for a model second-order elliptic operator using a standard finite element discretization of the resolvent is described. Some numerical experiments are conducted to gauge the sharpness of the theoretical estimates. arXiv:1709.06694v4 [math.NA]

show abstract

Section: Discretization Errors In Eigenspacementioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

Spectral discretization errors in filtered subspace iteration

2019

View full text Add to dashboard Cite

show abstract

“…Today, there are several complex moment-based eigensolvers including direct extensions of the Sakurai-Sugiura's approach [16][17][18][19]21] and FEAST eigensolver [38] and its variants [44,53,54]. For details, we refer [20] and reference therein. e Sakurai-Sugiura methods have been developed to nonlinear eigenvalue problems, unlike the FEAST eigensolver.…”

Section: Sakurai-sugiura Methodsmentioning

confidence: 99%

Efficient and scalable calculation of complex band structure using Sakurai-Sugiura method

Iwase

Futamura

Imakura

et al. 2017

Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis

Self Cite

View full text Add to dashboard Cite

Complex band structures (CBSs) are useful to characterize the static and dynamical electronic properties of materials. Despite the intensive developments, the first-principles calculation of CBS for over several hundred atoms is still computationally demanding. We here propose an efficient and scalable computational method to calculate CBSs. e basic idea is to express the Kohn-Sham equation of the real-space grid scheme as a quadratic eigenvalue problem and compute only the solutions which are necessary to construct the CBS by Sakurai-Sugiura method. e serial performance of the proposed method shows a significant advantage in both runtime and memory usage compared to the conventional method. Furthermore, owing to the hierarchical parallelism in Sakurai-Sugiura method and the domain-decomposition technique for real-space grids, we can achieve an excellent scalability in the CBS calculation of a boron and nitrogen doped carbon nanotube consisting of more than 10,000 atoms using 2,048 nodes (139,264 cores) of Oakforest-PACS.

show abstract

“…In this study, we develop a verification method for partial eigenvalues using the block Hankeltype Sakurai-Sugiura method [7], which receives attentions in recent years by virtue of the scalability in parallel and versatility [8]. We shed light on a new perspective of this method.…”

Section: Introductionmentioning

confidence: 99%

Verified partial eigenvalue computations using contour integrals for Hermitian generalized eigenproblems

Imakura

Morikuni

Takayasu

2020

Journal of Computational and Applied Mathematics

Self Cite

View full text Add to dashboard Cite

We propose a verified computation method for partial eigenvalues of a Hermitian generalized eigenproblem. The block Sakurai-Sugiura Hankel method, a contour integral-type eigensolver, can reduce a given eigenproblem into a generalized eigenproblem of block Hankel matrices whose entries consist of complex moments. In this study, we evaluate all errors in computing the complex moments. We derive a truncation error bound of the quadrature. Then, we take numerical errors of the quadrature into account and rigorously enclose the entries of the block Hankel matrices. Each quadrature point gives rise to a linear system, and its structure enables us to develop an efficient technique to verify the approximate solution. Numerical experiments show that the proposed method outperforms a standard method and infer that the proposed method is potentially efficient in parallel.

show abstract

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

Cited by 29 publications

References 25 publications

Spectral discretization errors in filtered subspace iteration

Spectral discretization errors in filtered subspace iteration

Efficient and scalable calculation of complex band structure using Sakurai-Sugiura method

Verified partial eigenvalue computations using contour integrals for Hermitian generalized eigenproblems

Contact Info

Product

Resources

About