Consider the optimal subspace expansion problem for the matrix eigenvalue problem $Ax=λx$: Which vector $w$ in the current subspace $\mathcal{V}$, after multiplied by $A$, provides an optimal subspace expansion for approximating a desired eigenvector $x$ in the sense that $x$ has the smallest angle with the expanded subspace $\mathcal{V}_w=\mathcal{V}+{\rm span}\{Aw\}$, i.e., $w_{opt}=\arg\max_{w\in\mathcal{V}}\cos\angle(\mathcal{V}_w,x)$? This problem is important as many iterative methods construct nested subspaces that successively expand $\mathcal{V}$ to $\mathcal{V}_w$. An expression of $w_{opt}$ by Ye (Linear Algebra Appl., 428 (2008), pp. 911--918) for $A$ general, but it could not be exploited to construct a computable (nearly) optimally expanded subspace. He turns to deriving a maximization characterization of $\cos\angle(\mathcal{V}_w,x)$ for a {\em given} $w\in \mathcal{V}$ when $A$ is Hermitian. We generalize Ye's maximization characterization to the general case and find its maximizer. Our main contributions consist of explicit expressions of $w_{opt}$, $(I-P_V)Aw_{opt}$ and the optimally expanded subspace $\mathcal{V}_{w_{opt}}$ for $A$ general, where $P_V$ is the orthogonal projector onto $\mathcal{V}$. These results are fully exploited to obtain computable optimally expanded subspaces within the framework of the standard, harmonic, refined, and refined harmonic Rayleigh--Ritz methods. We show how to efficiently implement the proposed subspace expansion approaches. Numerical experiments demonstrate the effectiveness of our computable optimal expansions.
For compact self-adjoint operators in Hilbert spaces, two algorithms are proposed to provide fully computable a posteriori error estimate for eigenfunction approximation. Both algorithms apply well to the case of tight clusters and multiple eigenvalues, under the settings of target eigenvalue problems. Algorithm I is based on the Rayleigh quotient and the min-max principle that characterizes the eigenvalue problems. The formula for the error estimate provided by Algorithm I is easy to compute and applies to problems with limited information of Rayleigh quotients. Algorithm II, as an extension …
In this paper, we present a nonnested augmented subspace algorithm and its multilevel correction method for solving eigenvalue problems with curved interfaces. The augmented subspace algorithm and the corresponding multilevel correction method are designed based on a coarse finite element space which is not the subset of the finer finite element space. The nonnested augmented subspace method can transform the eigenvalue problem solving on the finest mesh to the solving linear equation on the same mesh and small scale eigenvalue problem on the low dimensional augmented subspace. The correspondi…
Subspace method based on neural networks for eigenvalue problems
Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）；National Key Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics（计算物理国家重点实验室，应用物理与计算研究所）；HEDPS, Center for Applied Physics and Technology, and College of Engineering, Peking University（北京大学工程与应用技术学院，应用物理与技术中心）
In this paper, by introducing a class of relaxed filtered Krylov subspaces, we propose the relaxed filtered Krylov subspace method for computing the eigenvalues with the largest real parts and the corresponding eigenvectors of non-symmetric matrices. As by-products, the generalizations of the filtered Krylov subspace method and the Chebyshev-Davidson method for solving non-symmetric eigenvalue problems are also presented. We give the convergence analysis of the complex Chebyshev polynomial, which plays a significant role in the polynomial acceleration technique. In addition, numerical experime…
We propose an a posteriori error estimator for high-order $p$- or $hp$-finite element discretizations of selfadjoint linear elliptic eigenvalue problems that is appropriate for estimating the error in the approximation of an eigenvalue cluster and the corresponding invariant subspace. The estimator is based on the computation of approximate error functions in a space that complements the one in which the approximate eigenvectors were computed. These error functions are used to construct estimates of collective measures of error, such as the Hausdorff distance between the true and approximate c…
It is needed to solve generalized eigenvalue problems (GEP) in many applications, such as the numerical simulation of vibration analysis, quantum mechanics, electronic structure, etc. The subspace iteration is a kind of widely used algorithm to solve eigenvalue problems. To solve the generalized eigenvalue problem, one kind of subspace iteration method, Chebyshev-Davidson algorithm, is proposed recently. In Chebyshev-Davidson algorithm, the Chebyshev polynomial filter technique is incorporated in the subspace iteration. In this paper, based on Chebyshev-Davidson algorithm, a new subspace itera…
A type of parallel augmented subspace scheme for eigenvalue problems is proposed by using coarse space in the multigrid method. With the help of coarse space in multigrid method, solving the eigenvalue problem in the finest space is decomposed into solving the standard linear boundary value problems and very low dimensional eigenvalue problems. The computational efficiency can be improved since there is no direct eigenvalue solving in the finest space and the multigrid method can act as the solver for the deduced linear boundary value problems. Furthermore, for different eigenvalues, the corre…
Energy Error Estimates of Subspace Method and Multigrid Algorithm for Eigenvalue Problems
Memorial University of Newfoundland（纽芬兰纪念大学）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；University of Chinese Academy of Sciences（中国科学院大学）
In this paper we solve $m$-parameter eigenvalue problems ($m$EPs), with $m$ any natural number by representing the problem using Tensor-Trains (TT) and designing a method based on this format. $m$EPs typically arise when separation of variables is applied to separable boundary value problems. Often, methods for solving $m$EP are restricted to $m = 3$, due to the fact that, to the best of our knowledge, no available solvers exist for $m>3$ and reasonable size of the involved matrices. In this paper, we prove that computing the eigenvalues of a $m$EP can be recast into computing the eigenvalues …
The FEAST eigenvalue algorithm is a subspace iteration algorithm that uses contour integration in the complex plane to obtain the eigenvectors of a matrix for the eigenvalues that are located in any user-defined search interval. By computing small numbers of eigenvalues in specific regions of the complex plane, FEAST is able to naturally parallelize the solution of eigenvalue problems by solving for multiple eigenpairs simultaneously. The traditional FEAST algorithm is implemented by directly solving collections of shifted linear systems of equations; in this paper, we describe a variation of …
Three new Arnoldi-type methods are presented to accelerate the modal analysis and critical speed analysis of the damped rotor dynamics finite element (FE) model. They are the linearized quadratic eigenvalue problem (QEP) Arnoldi method, the QEP Arnoldi method, and the truncated generalized standard eigenvalue problem (SEP) Arnoldi method. And, they correspond to three reduction subspaces, including the linearized QEP Krylov subspace, the QEP Krylov subspace, and the truncated generalized SEP Krylov subspace, where the first subspace is also used in the existing Arnoldi-type methods. The numeri…
We first consider the problem of approximating a few eigenvalues of a rational matrix-valued function closest to a prescribed target. It is assumed that the proper rational part of the rational matrix-valued function is expressed in the transfer function form $H(s) = C (sI - A)^{-1} B$, where the middle factor is large, whereas the number of rows of $C$ and the number of columns of $B$ are equal and small. We propose a subspace framework that performs two-sided or one-sided projections on the state-space representation of $H(\cdot)$, commonly employed in model reduction and giving rise to a re…
A posteriori error estimates for discontinuous Galerkin methods using non-polynomial basis functions. Part II: Eigenvalue problems
University of California, Berkeley（加州大学伯克利分校）；Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）；RWTH Aachen University（亚琛工业大学）；Forschungszentrum Jülich（于利希研究中心）
We consider a minimal realization of a rational matrix functions. We perturb the polynomial part and one of the constant matrices from the realization part. We derive explicit computable expressions of backward errors of approximate eigenvalue of rational matrix function. We also determine minimal perturbations for which approximate eigenvalue are exact eigenvalue of the perturbed matrix rational functions.
In this paper we extend the Residual Arnoldi method for calculating an extreme eigenvalue (e.g. largest real part, dominant,...) to the case where the matrices depend on parameters. The difference between this Arnoldi method and the classical Arnoldi algorithm is that in the former the residual is added to the subspace. We develop a Tensor-Krylov method that applies the Residual Arnoldi method (RA) for a grid of parameter points at the same time. The subspace contains an approximate Krylov space for all these points. Instead of adding the residuals for all parameter values to the subspace we c…
This paper develops a new class of algorithms for general linear systems and eigenvalue problems. These algorithms apply fast randomized sketching to accelerate subspace projection methods, such as GMRES and Rayleigh--Ritz. This approach offers great flexibility in designing the basis for the approximation subspace, which can improve scalability in many computational environments. The resulting algorithms outperform the classic methods with minimal loss of accuracy. For model problems, numerical experiments show large advantages over MATLAB's optimized routines, including a $100 \times$ speedu…
The purpose of this article is to approximately compute the eigenvalues of the symmetric Dirichlet Laplacian within an interval $(0,Λ)$. A novel domain decomposition Ritz method, partition of unity condensed pole interpolation method, is proposed. This method can be used in distributed computing environments where communication is expensive, e.g., in clusters running on cloud computing services or networked workstations. The Ritz space is obtained from local subspaces consistent with a decomposition of the domain into subdomains. These local subspaces are constructed independently of each othe…
We propose an eigensolver and the corresponding package, GCGE, for solving large scale eigenvalue problems. This method is the combination of damping idea, subspace projection method and inverse power method with dynamic shifts. To reduce the dimensions of projection subspaces, a moving mechanism is developed when the number of desired eigenpairs is large. The numerical methods, implementing techniques and the structure of the package are presented. Plenty of numerical results are provided to demonstrate the efficiency, stability and scalability of the concerned eigensolver and the package GCG…
Accelerating Eigenvalue Dataset Generation via Chebyshev Subspace Filter
Department of Computer Science（计算机科学系）；Cranberry-Lemon University（Cranberry-Lemon 大学）；Department of Computational Neuroscience（计算神经科学系）；University of the Witwatersrand（沃图瓦特沃兹兰大学）；University of Science and Technology of China（中国科学技术大学）；CAS Key Laboratory of Technology in GIPAS, University of Science and Technology of China（中国科学技术大学GIPAS技术关键实验室）
Randomized block Krylov subspace methods form a powerful class of algorithms for computing the extreme eigenvalues of a symmetric matrix or the extreme singular values of a general matrix. The purpose of this paper is to develop new theoretical bounds on the performance of randomized block Krylov subspace methods for these problems. For matrices with polynomial spectral decay, the randomized block Krylov method can obtain an accurate spectral norm estimate using only a constant number of steps (that depends on the decay rate and the accuracy). Furthermore, the analysis reveals that the behavio…
A thick-restart Lanczos type algorithm is proposed for Hermitian $J$-symmetric matrices. Since Hermitian $J$-symmetric matrices possess doubly degenerate spectra or doubly multiple eigenvalues with a simple relation between the degenerate eigenvectors, we can improve the convergence of the Lanczos algorithm by restricting the search space of the Krylov subspace to that spanned by one of each pair of the degenerate eigenvector pairs. We show that the Lanczos iteration is compatible with the $J$-symmetry, so that the subspace can be split into two subspaces that are orthogonal to each other. The…
We continue the work of [Camano, Lackner, Monk, SIAM J. Math. Anal., Vol. 49, No. 6, pp. 4376-4401 (2017)] on electromagnetic Stekloff eigenvalues. The authors recognized that in general the eigenvalues due not correspond to the spectrum of a compact operator and hence proposed a modified eigenvalue problem with the desired properties. The present article considers the original and the modified electromagnetic Stekloff eigenvalue problem. We cast the problems as eigenvalue problem for a holomorphic operator function $A(\cdot)$. We construct a "test function operator function" $T(\cdot)$ so tha…
A thorough backward stability analysis of Hotelling's deflation, an explicit external deflation procedure through low-rank updates for computing many eigenpairs of a symmetric matrix, is presented. Computable upper bounds of the loss of the orthogonality of the computed eigenvectors and the symmetric backward error norm of the computed eigenpairs are derived. Sufficient conditions for the backward stability of the explicit external deflation procedure are revealed. Based on these theoretical results, the strategy for achieving numerical backward stability by dynamically selecting the shifts is…
This paper is devoted to the study of preservation of eigenvalues, Jordan structure and complementary invariant subspaces of structured matrices under structured perturbations. Perturbations and structure-preserving perturbations are determined such that a perturbed matrix reproduces a given subspace as an invariant subspace and preserves a pair of complementary invariant subspaces of the unperturbed matrix. These results are further utilized to obtain structure-preserving perturbations which modify certain eigenvalues of a given structured matrix and reproduce a set of desired eigenvalues whi…
Convergence rates of block iterations for solving eigenvalue problems typically measure errors of Ritz values approximating eigenvalues. The errors of the Ritz values are commonly bounded in terms of principal angles between the initial or iterative subspace and the invariant subspace associated with the target eigenvalues. Such bounds thus cannot be applied repeatedly as needed for restarted block eigensolvers, since the left- and right-hand sides of the bounds use different terms. They must be combined with additional bounds which could cause an overestimation. Alternative repeatable bounds …
