Showing papers on "Divide-and-conquer eigenvalue algorithm published in 2013"

NLEVP: A Collection of Nonlinear Eigenvalue Problems

Abstract: We present a collection of 52 nonlinear eigenvalue problems in the form of a MATLAB toolbox. The collection contains problems from models of real-life applications as well as ones constructed specifically to have particular properties. A classification is given of polynomial eigenvalue problems according to their structural properties. Identifiers based on these and other properties can be used to extract particular types of problems from the collection. A brief description of each problem is given. NLEVP serves both to illustrate the tremendous variety of applications of nonlinear eigenvalue problems and to provide representative problems for testing, tuning, and benchmarking of algorithms and codes.

Stable and Efficient Spectral Divide and Conquer Algorithms for the Symmetric Eigenvalue Decomposition and the SVD

Abstract: Spectral divide and conquer algorithms solve the eigenvalue problem for all the eigenvalues and eigenvectors by recursively computing an invariant subspace for a subset of the spectrum and using it to decouple the problem into two smaller subproblems. A number of such algorithms have been developed over the last 40 years, often motivated by parallel computing and, most recently, with the aim of achieving minimal communication costs. However, none of the existing algorithms has been proved to be backward stable, and they all have a significantly higher arithmetic cost than the standard algorithms currently used. We present new spectral divide and conquer algorithms for the symmetric eigenvalue problem and the singular value decomposition that are backward stable, achieve lower bounds on communication costs recently derived by Ballard, Demmel, Holtz, and Schwartz, and have operation counts within a small constant factor of those for the standard algorithms. The new algorithms are built on the polar decompos...

Fluctuations of Linear Eigenvalue Statistics of β Matrix Models in the Multi-cut Regime

Abstract: We study the asymptotic expansion in n for the partition function of β matrix models with real analytic potentials in the multi-cut regime up to the O(n −1) terms. As a result, we find the limit of the generating functional of linear eigenvalue statistics and the expressions for the expectation and the variance of linear eigenvalue statistics, which in the general case contain the quasi periodic in n terms.

Verified Eigenvalue Evaluation for the Laplacian over Polygonal Domains of Arbitrary Shape

Abstract: The finite element method (FEM) is applied to bound leading eigenvalues of the Laplace operator over polygonal domains. Compared with classical numerical methods, most of which can only give concrete eigenvalue bounds over special domains of symmetry, our proposed algorithm can provide concrete eigenvalue bounds for domains of arbitrary shape, even when the eigenfunction has a singularity. The problem of eigenvalue estimation is solved in two steps. First, we construct a computable a priori error estimation for the FEM solution of Poisson's problem, which holds even for nonconvex domains with reentrant corners. Second, new computable lower bounds are developed for the eigenvalues. Because the interval arithmetic is implemented throughout the computation, the desired eigenvalue bounds are expected to be mathematically correct. We illustrate several computation examples, such as the cases of an L-shaped domain and a crack domain, to demonstrate the efficiency and flexibility of the proposed method.

A POD reduced‐order model for eigenvalue problems with application to reactor physics

Abstract: SUMMARY A reduced-order model based on proper orthogonal decomposition (POD) has been presented and applied to solving eigenvalue problems. The model is constructed via the method of snapshots, which is based upon the singular value decomposition of a matrix containing the characteristics of a solution as it evolves through time. Part of the novelty of this work is in how this snapshot data are generated, and this is through the recasting of eigenvalue problem, which is time independent, into a time-dependent form. Instances of time-dependent eigenfunction solutions are therefore used to construct the snapshot matrix. The reduced order model's capabilities in efficiently resolving eigenvalue problems that typically become computationally expensive (using standard full model discretisations) has been demonstrated. Although the approach can be adapted to most general eigenvalue problems, the examples presented here are based on calculating dominant eigenvalues in reactor physics applications. The approach is shown to reconstruct both the eigenvalues and eigenfunctions accurately using a significantly reduced number of unknowns in comparison with ‘full’ models based on finite element discretisations. The novelty of this paper therefore includes a new approach to generating snapshots, POD's application to large-scale eigenvalue calculations, and reduced-order model's application in reactor physics.Copyright © 2013 John Wiley & Sons, Ltd.

A projection method for nonlinear eigenvalue problems using contour integrals

Abstract: In this paper, we indicate that the Sakurai-Sugiura method with Rayleigh-Ritz projection technique, a numerical method for generalized eigenvalue problems, can be extended to nonlinear eigenvalue problems. The target equation is T (λ)v = 0, where T is a matrix-valued function. The method can extract only the eigenvalues within a Jordan curve Γ by converting the original problem to a problem with a smaller dimension. Theoretical validation of the method is discussed, and we describe its application using numerical examples.

Minimization Principles for the Linear Response Eigenvalue Problem II: Computation

Abstract: In Part I of this paper we presented minimization principles and related theoretical results for the linear response eigenvalue problem. Here we develop best approximations for the few smallest eigenvalues with the positive sign via a structure-preserving subspace projection. Then we present four-dimensional subspace search conjugate gradient-like algorithms for simultaneously computing these eigenvalues and their associated eigenvectors. Finally, we present numerical examples to illustrate convergence behaviors of the proposed methods with and without preconditioning.

Robust Successive Computation of Eigenpairs for Nonlinear Eigenvalue Problems

Abstract: Newton-based methods are well-established techniques for solving nonlinear eigenvalue problems. If a larger portion of the spectrum is sought, however, their tendency to reconverge to previously determined eigenpairs is a hindrance. To overcome this limitation, we propose and analyze a deflation strategy for nonlinear eigenvalue problems, based on the concept of minimal invariant pairs. We develop this strategy into a Jacobi--Davidson-type method and discuss its various algorithmic details. Finally, the efficiency of our approach is demonstrated by a sequence of numerical examples.

On the asymptotics of the principal eigenvalue for a Robin problem with a large parameter in planar domains

Abstract: :It is easy to check that E( ) 0The study of the principal eigenvalue arises in several applications: work [1] discussesthe stochastic meaning of the Robin eigenvalues, paper [2] shows that the eigenvalue problem

Computation of Maxwell’s transmission eigenvalues and its applications in inverse medium problems

Abstract: We present an iterative method to compute the Maxwell’s transmission eigenvalue problem which has importance in non-destructive testing of anisotropic materials. The transmission eigenvalue problem is first written as a quad-curl eigenvalue problem. Then we show that the real transmission eigenvalues are the roots of a nonlinear function whose value is the generalized eigenvalue of a related self-adjoint quad-curl eigenvalue problem which is computed using a mixed finite element method. A secant method is used to compute the roots of the nonlinear function. Numerical examples are presented to validate the method. Moreover, the method is employed to study the dependence of the transmission eigenvalue on the anisotropy and to reconstruct the index of refraction of an inhomogeneous medium.

Adaptive Normalized Quasi-Newton Algorithms for Extraction of Generalized Eigen-Pairs and Their Convergence Analysis

Abstract: The main contributions of this paper are to propose and analyze fast and numerically stable adaptive algorithms for the generalized Hermitian eigenvalue problem (GHEP), which arises in many signal processing applications. First, for given explicit knowledge of a matrix pencil, we formulate two novel deterministic discrete-time (DDT) systems for estimating the generalized eigen-pair (eigenvector and eigenvalue) corresponding to the largest/smallest generalized eigenvalue. By characterizing a generalized eigen-pair as a stationary point of a certain function, the proposed DDT systems can be interpreted as natural combinations of the normalization and quasi-Newton steps for finding the solution. Second, we present adaptive algorithms corresponding to the proposed DDT systems. Moreover, we establish rigorous analysis showing that, for a step size within a certain range, the sequence generated by the DDT systems converges to the orthogonal projection of the initial estimate onto the generalized eigensubspace corresponding to the largest/smallest generalized eigenvalue. Numerical examples demonstrate the practical applicability and efficacy of the proposed adaptive algorithms.

Free products of large random matrices - a short review of recent developments

Abstract: We review methods to calculate eigenvalue distributions of products of large random matrices. We discuss a generalization of the law of free multiplication to non-Hermitian matrices and give a couple of examples illustrating how to use these methods in practice. In particular we calculate eigenvalue densities of products of Gaussian Hermitian and non-Hermitian matrices including combinations of GUE and Ginibre matrices.

Eigenvalue analysis for acoustic problem in 3D by boundary element method with the block Sakurai–Sugiura method

Abstract: This paper presents accurate numerical solutions for nonlinear eigenvalue analysis of three-dimensional acoustic cavities by boundary element method (BEM). To solve the nonlinear eigenvalue problem (NEP) formulated by BEM, we employ a contour integral method, called block Sakurai–Sugiura (SS) method, by which the NEP is converted to a standard linear eigenvalue problem and the dimension of eigenspace is reduced. The block version adopted in present work can also extract eigenvalues whose multiplicity is larger than one, but for the complex connected region which includes a internal closed boundary, the methodology yields fictitious eigenvalues. The application of the technique is demonstrated through the eigenvalue calculation of sphere with unique homogenous boundary conditions, cube with mixed boundary conditions and a complex connected region formed by cubic boundary and spherical boundary, however, the fictitious eigenvalues can be identified by Burton–Miller's method. These numerical results are supported by appropriate convergence study and comparisons with close form.

Approximation of differential eigenvalue problems

Abstract: To solve a one-dimensional second-order differential eigenvalue problem, we use the finite-element method with numerical integration. We analyze the influence of the quadrature formulas used on the error in the approximate eigenvalues and eigenelements. Theoretical results are illustrated by experiments for a model problem.

Efficient Rearrangement Algorithms for Shape Optimization on Elliptic Eigenvalue Problems

Abstract: In this paper, several efficient rearrangement algorithms are proposed to find the optimal shape and topology for elliptic eigenvalue problems with inhomogeneous structures The goal is to solve minimization and maximization of the k-th eigenvalue and maximization of spectrum ratios of the second order elliptic differential operator Physically, these problems are motivated by the frequency control based on density distribution of vibrating membranes The methods proposed are based on Rayleigh quotient formulation of eigenvalues and rearrangement algorithms which can handle topology changes automatically Due to the efficient rearrangement strategy, the new proposed methods are more efficient than classical level set approaches based on shape and/or topological derivatives Numerous numerical examples are provided to demonstrate the robustness and efficiency of new approach

Efficient algorithms for computing the largest eigenvalue of a nonnegative tensor

Abstract: Consider the problem of computing the largest eigenvalue for nonnegative tensors. In this paper, we establish the Q-linear convergence of a power type algorithm for this problem under a weak irreducibility condition. Moreover, we present a convergent algorithm for calculating the largest eigenvalue for any nonnegative tensors.

A novel method for computing small-signal stability boundaries of large-scale power systems

Abstract: This paper proposes a new method to compute the varying value of a controller parameter that causes the eigenvalue to cross the small-signal stability boundaries. In the method, the crossing point of the eigenvalue locus and the boundary can be determined, without having to calculate the eigenvalues or eigenvectors of the state matrix of power system. The models are established and the unreduced Jacobian matrix is used to improve the computing efficiency.

The Dominant Eigenvalue of an Essentially Nonnegative Tensor

Abstract: It is well known that the dominant eigenvalue of a real essentially nonnegative matrix is a convex function of its diagonal entries. This convexity is of practical importance in population biology, graph theory, demography, analytic hierarchy process and so on. In this paper, the concept of essentially nonnegativity is extended frommatrices to higher order tensors, and the convexity and log convexity of dominant eigenvalues for such a class of tensors are established. Particularly, for any nonnegative tensor, the spectral radius turns out to be the dominant eigenvalue and hence possesses these convexities. Finally, an algorithm is given to calculate the dominant eigenvalue, and numerical results are reported to show the effectiveness of the proposed algorithm. Copyright c ⃝ 2013 John Wiley & Sons, Ltd.

Convergence Orders of Iterative Methods for Nonlinear Eigenvalue Problems

Abstract: The convergence analysis of iterative methods for nonlinear eigenvalue problems is in the most cases restricted either to algebraically simple eigenvalues or to polynomial eigenvalue problems. In this paper we consider two classical methods for general holomorphic eigenvalue problems, namely the nonlinear generalized Rayleigh quotient iteration (NGRQI) and the augmented Newton method. The analysis of the convergence order of both methods is based on the representation of the eigenvalues as poles of the resolvent. This approach was already chosen for the analysis of the NGRQI by Langer in [19] for a more general setting where such a representation of the eigenvalues had to be assumed. The convergence orders of both methods depend on the order which an eigenvalue has as pole of the resolvent. Both methods exhibit a local quadratic convergence order for semi-simple eigenvalues. For defective eigenvalues in general only a local linear convergence is possible. In numerical experiments the theoretical results are confirmed.

Computing Derivatives of Repeated Eigenvalues and Corresponding Eigenvectors of Quadratic Eigenvalue Problems

Abstract: We consider quadratic eigenvalue problems in which the coefficient matrices, and hence the eigenvalues and eigenvectors, are functions of a real parameter. Our interest is in cases in which these functions remain differentiable when eigenvalues coincide. Many papers have been devoted to numerical methods for computing derivatives of eigenvalues and eigenvectors, but most require the eigenvalues to be well separated. The few that consider close or repeated eigenvalues place severe restrictions on the eigenvalue derivatives. We propose, analyze, and test new algorithms for computing first and higher order derivatives of eigenvalues and eigenvectors that are valid much more generally. Numerical results confirm the effectiveness of our methods for tightly clustered eigenvalues.

Converting Path Structures Into Block Structures Using Eigenvalue Decompositions of Self-Similarity Matrices.

Abstract: In music structure analysis the two principles of repetition and homogeneity are fundamental for partitioning a given audio recording into musically meaningful structural elements. When converting the audio recording into a suitable self-similarity matrix (SSM), repetitions typically lead to path structures, whereas homogeneous regions yield block structures. In previous research, handling both structural elements at the same time has turned out to be a challenging task. In this paper, we introduce a novel procedure for converting path structures into block structures by applying an eigenvalue decomposition of the SSM in combination with suitable clustering techniques. We demonstrate the effectiveness of our conversion approach by showing that algorithms previously designed for homogeneitybased structure analysis can now be applied for repetitionbased structure analysis. Thus, our conversion may open up novel ways for handling both principles within a unified structure analysis framework.

Optimization of the first eigenvalue of equations with indefinite weights

Abstract: Abstract We investigate minimization and maximization of the principal eigenvalue of the Laplacian under Dirichlet boundary conditions in case the weight has indefinite sign and varies in a class of rearrangements. Biologically, such optimization problems are motivated by the question of determining the most convenient spatial arrangement of favorable and unfavorable resources for a species to survive or to decline. The question may have practical importance in the context of reserve design or pest control.

Local and Parallel Finite Element Discretizations for Eigenvalue Problems

Abstract: Based on the work of Xu and Zhou [Math. Comp., 69 (2000), pp. 881--909], this paper combines the local defect-correction technique and the shifted-inverse power method to establish new local and parallel finite element three-scale schemes for a class of eigenvalue problems. It is proved that with these schemes, the solution of an eigenvalue problem on a fine grid $\pi_{h}$ is reduced to the solution of an eigenvalue problem on a much coarser grid $\pi_{H}$, the solution of a linear algebraic system on a globally mesoscopic grid $\pi_{w}$, and the solutions of linear systems on several locally fine grids in parallel. The principle to determine the diameters of three different scale grids is given. Especially, this paper devises a new local and parallel finite element multiscale discretization scheme. Theoretical analysis and numerical experiments show that the computational approach proposed in this paper is simple and easy to carry out and can be used to solve singular eigenvalue problems efficiently.