Interior Eigenvalue Problems Research Articles

Complex moment-based eigensolver coupled with two Krylov subspaces

Complex moment-based eigensolvers have been well studied for solving interior eigenvalue problems because of their high parallel efficiency. Recently, as a time-efficient complex moment-based eigensolver, the block SS–CAA method was proposed that is based on the communication-avoiding Arnoldi procedure regarding the high-order complex moments. These complex moment-based eigensolvers, including the block SS–CAA method, usually use with a subspace iteration technique for improving the accuracy. In this paper, we aim to improve the convergence behavior of the block SS–CAA method using a block Arnoldi iteration instead of the subspace iteration. The proposed method is based on a special subspace coupled with two Krylov subspaces: one is for the high-order complex moments, and the other is for the iteration technique. Numerical experiments indicate that the proposed method has a higher convergence rate than the block SS–CAA method and the FEAST eigensolver.

Flexible subspace iteration with moments for an effective contour integration‐based eigensolver

AbstractContour integration schemes are a valuable tool for the solution of difficult interior eigenvalue problems. However, the solution of many large linear systems with multiple right hand sides may prove a prohibitive computational expense. The number of right hand sides, and thus, computational cost may be reduced if the projected subspace is created using multiple moments. In this work, we explore heuristics for the choice and application of moments with respect to various other important parameters in a contour integration scheme. We provide evidence for the expected performance, accuracy, and robustness of various schemes, showing that good heuristic choices can provide a scheme featuring good properties in all three of these measures.

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

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 a contour based eigensolvers 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.

A Golub--Kahan Davidson Method for Accurately Computing a Few Singular Triplets of Large Sparse Matrices

Obtaining high accuracy singular triplets for large sparse matrices is a significant challenge, especially when searching for the smallest triplets. Due to the difficulty and size of these problems, efficient methods must function iteratively, with preconditioners, and under strict memory constraints. In this research, we present a Golub--Kahan Davidson method (GKD), which satisfies these requirements and includes features such as soft-locking with orthogonality guarantees, an inner correction equation similar to Jacobi--Davidson, locally optimal +k restarting, and the ability to find real zero singular values in both square and rectangular matrices. Additionally, our method achieves full accuracy while avoiding the augmented matrix, which often converges slowly for the smallest triplets due to the difficulty of interior eigenvalue problems. We describe our method in detail, including implementation issues that arise. Our experimental results confirm the efficiency and stability of our method over the current implementation of PHSVDS in the PRIMME software package.

Localized spectrum slicing

Given a sparse Hermitian matrix $A$ and a real number $\mu$, we construct a set of sparse vectors, each approximately spanned only by eigenvectors of $A$ corresponding to eigenvalues near $\mu$. This set of vectors spans the column space of a localized spectrum slicing (LSS) operator, and is called an LSS basis set. The sparsity of the LSS basis set is related to the decay properties of matrix Gaussian functions. We present a divide-and-conquer strategy with controllable error to construct the LSS basis set. This is a purely algebraic process using only submatrices of $A$, and can therefore be applied to general sparse Hermitian matrices. The LSS basis set leads to sparse projected matrices with reduced sizes, which allows the projected problems to be solved efficiently with techniques using sparse linear algebra. As an example, we demonstrate that the LSS basis set can be used to solve interior eigenvalue problems for a discretized second order partial differential operator in one-dimensional and two-dimensional domains, as well as for a matrix of general sparsity pattern.

Block Krylov-type complex moment-based eigensolvers for solving generalized eigenvalue problems

Complex moment-based eigensolvers for solving interior eigenvalue problems have been studied because of their high parallel efficiency. Recently, we proposed the block Arnoldi-type complex moment-based eigensolver without a low-rank approximation. A low-rank approximation plays a very important role in reducing computational cost and stabilizing accuracy in complex moment-based eigensolvers. In this paper, we develop the method and propose block Krylov-type complex moment-based eigensolvers with a low-rank approximation. Numerical experiments indicate that the proposed methods have higher performance than the block SS---RR method, which is one of the most typical complex moment-based eigensolvers.

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

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.

Harmonic and refined harmonic shift-invert residual Arnoldi and Jacobi–Davidson methods for interior eigenvalue problems

This paper concerns the harmonic shift-invert residual Arnoldi (HSIRA) and Jacobi–Davidson (HJD) methods as well as their refined variants RHSIRA and RHJD for the interior eigenvalue problem. Each method needs to solve an inner linear system to expand the subspace successively. When the linear systems are solved only approximately, we are led to the inexact methods. We prove that the inexact HSIRA, RHSIRA, HJD and RHJD methods mimic their exact counterparts well when the inner linear systems are solved with only low or modest accuracy. We show that (i) the exact HSIRA and HJD expand subspaces better than the exact SIRA and JD and (ii) the exact RHSIRA and RHJD expand subspaces better than the exact HSIRA and HJD. Based on the theory, we design stopping criteria for inner solves. To be practical, we present restarted HSIRA, HJD, RHSIRA and RHJD algorithms. Numerical results demonstrate that these algorithms are much more efficient than the restarted standard SIRA and JD algorithms and furthermore the refined harmonic algorithms outperform the harmonic ones very substantially.

Preconditioned Locally Harmonic Residual Method for Computing Interior Eigenpairs of Certain Classes of Hermitian Matrices

We propose a Preconditioned Locally Harmonic Residual (PLHR) method for computing several interior eigenpairs of a generalized Hermitian eigenvalue problem, without traditional spectral transformations, matrix factorizations, or inversions. PLHR is based on a short-term recurrence, easily extended to a block form, computing eigenpairs simultaneously. PLHR can take advantage of Hermitian positive definite preconditioning, e.g., based on an approximate inverse of an absolute value of a shifted matrix, introduced in [SISC, 35 (2013), pp. A696-A718]. Our numerical experiments demonstrate that PLHR is efficient and robust for certain classes of large-scale interior eigenvalue problems, involving Laplacian and Hamiltonian operators, especially if memory requirements are tight.

A modification of the factorization method for the classical acoustic inverse scattering problems

It is well-known that sampling type methods for solving inverse scattering problems fail if the wave number is an eigenvalue of a corresponding interior eigenvalue problem. By adding the far field patterns corresponding to an artificial ball lying within the obstacle and imposing an impedance boundary condition on the boundary of this ball we propose a modification of the factorization method which provides the characterization of the unknown obstacle for all wave numbers. Some numerical experiments are presented to demonstrate the feasibility and effectiveness of our method.

An interior eigenvalue problem from electronic structure calculations

We consider the generalized eigenvalue problem Ax = λBx, where A and B are real symmetric matrices and B is also positive definite. All the eigenvalues of this problem are real, and it is often necessary to compute only a few eigenvalues which are important for applications. In electronic structure calculations of materials, specific interior eigenvalues are of fundamental interest, since they play crucial roles in various industrial applications. In this paper, we propose an approach based on the inertia of the linear matrix pencil of A and B. The eigenvalue problem is restated, and a class of algorithms is presented for separating the target eigenvalues from the others.

Fast iterative interior eigensolver for millions of atoms

We show that a combination of the Generalized Davidson method and harmonic Ritz values (called harmonic Davidson) is well-suited for solving large interior eigenvalue problems using a plane wave basis. The algorithm enables us to calculate impurity and band edge states for systems of 100,000 atoms in about one day on 32 cores. We demonstrate the capabilities of the method by calculating the electronic states of a large GaAs quantum dot embedded in an AlAs matrix.

A global harmonic Arnoldi method for large non-Hermitian eigenproblems with an application to multiple eigenvalue problems

The global Arnoldi method can be used to compute exterior eigenpairs of a large non-Hermitian matrix A , but it does not work well for interior eigenvalue problems. Based on the global Arnoldi process that generates an F -orthonormal basis of a matrix Krylov subspace, we propose a global harmonic Arnoldi method for computing certain harmonic F -Ritz pairs that are used to approximate some interior eigenpairs. We propose computing the F -Rayleigh quotients of the large non-Hermitian matrix with respect to harmonic F -Ritz vectors and taking them as new approximate eigenvalues. They are better and more reliable than the harmonic F -Ritz values. The global harmonic Arnoldi method inherits convergence properties of the harmonic Arnoldi method applied to a larger matrix whose distinct eigenvalues are the same as those of the original given matrix. Some properties of the harmonic F -Ritz vectors are presented. As an application, assuming that A is diagonalizable, we show that the global harmonic Arnoldi method is able to solve multiple eigenvalue problems both in theory and in practice. To be practical, we develop an implicitly restarted global harmonic Arnoldi algorithm with certain harmonic F -shifts suggested. In particular, this algorithm can be adaptively used to solve multiple eigenvalue problems. Numerical experiments show that the algorithm is efficient for the eigenproblem and is reliable for quite ill-conditioned multiple eigenproblems.

A method based on Rayleigh quotient gradient flow for extreme and interior eigenvalue problems

Recently, a continuous method has been proposed by Golub and Liao as an alternative way to solve the minimum and interior eigenvalue problems. According to their numerical results, their method seems promising. This article is an extension along this line. In this article, firstly, we convert an eigenvalue problem to an equivalent constrained optimization problem. Secondly, using the Karush–Kuhn–Tucker conditions of this equivalent optimization problem, we obtain a variant of the Rayleigh quotient gradient flow, which is formulated by a system of differential-algebraic equations. Thirdly, based on the Rayleigh quotient gradient flow, we give a practical numerical method for the minimum and interior eigenvalue problems. Finally, we also give some numerical experiments of our method, the Golub and Liao method, and EIGS (a Matlab implementation for computing eigenvalues using restarted Arnoldi’s method) for some typical eigenvalue problems. Our numerical experiments indicate that our method seems promising for most test problems.

Strengthened Linear Sampling Method with a Reference Ball

By adding a reference ball as an extra artificial obstacle component to the underlying scattering system, we propose a simple but robust and effective technique to choose the crucial cut-off value required in the linear sampling method (LSM) for inverse scattering problems. The reference ball technique causes few extra computational costs to the LSM, but brings in a practically important byproduct, i.e., it eliminates the interior eigenvalue problem automatically, a well-known barrier when applying the LSM. Some mathematical justifications of the new technique are provided, and numerical experiments are also presented to demonstrate its feasibility and effectiveness.

Modified Combined Field Integral Equations for Electromagnetic Scattering

The boundary integral formulation of exterior boundary value problems for the Maxwell system may not be equivalent to the original uniquely solvable problem if the wave number corresponds to an eigenvalue of an associated interior eigenvalue problem. To avoid these spurious modes one may use a combined boundary integral approach. To analyze the resulting boundary integral equations in the energy function spaces suitable regularizations have to be introduced. Here we formulate and analyze a modified boundary integral equation which is based on the use of standard boundary integral operators only. A first numerical example shows the applicability of the proposed approach.

A generalized projective dynamic for solving extreme and interior eigenvalue problems

In [18] (Golub and Liao), a continuous-time system which is based on the projective dynamic is proposed to solve some concave optimization problems (with the unit ball constraint) resulted from extreme and interior eigenvalue problems. The convergence inside the unit ball is established; however, neither further convergence result outside the unit ball nor the stability analysis is available. Moreover, preliminary numerical experience indicates that this method is sensitive to a parameter whose optimal value is still difficult to determine. After analyzing the stability of this dynamic, in this paper, we develop a generalized model and analyze the convergence of the new model both inside and outside the unit ball. The flow of the generalized model is proved to converge almost globally to some eigenvector corresponding to the smallest eigenvalue, and share many surprisingly analogous properties with the Rayleigh quotient gradient flow. Links of our generalized projective dynamical system with other related works are also discussed. The efficiency of our new model is both addressed in theory and verified in numerical testing.

Continuous methods for extreme and interior eigenvalue problems

In this paper, continuous methods are introduced to compute both the extreme and interior eigenvalues and their corresponding eigenvectors for real symmetric matrices. The main idea is to convert the extreme and interior eigenvalue problems into some optimization problems. Then a continuous method which includes both a merit function and an ordinary differential equation (ODE) is introduced for each resulting optimization problem. The convergence of each ODE solution is proved for any starting point. The limit of each ODE solution for any starting point is fully studied. Both the extreme and the interior eigenvalues and their corresponding eigenvectors can be easily obtained under a very mild condition. Promising numerical results are also presented.

The refined harmonic Arnoldi method and an implicitly restarted refined algorithm for computing interior eigenpairs of large matrices

The harmonic Arnoldi method can be used to compute some eigenpairs of a large matrix, and it is more suitable for finding interior eigenpairs. However, the harmonic Ritz vectors obtained by the method may converge erratically and may even fail to converge, so that resulting algorithms may not perform well. To improve convergence, a refined harmonic Arnoldi method is proposed that replaces the harmonic Ritz vectors by certain refined eigenvector approximations. The implicit restart technique advocated by Sorensen is applied to the refined harmonic Arnoldi method and an implicitly restarted refined harmonic Arnoldi algorithm (IRRHA) is presented. For the refined algorithm a new shift scheme, called refined shifts, is proposed that is shown to be better than the exact shifts scheme used in literature. The refined shifts can be computed reliably and cheaply. Extensive numerical experiments are made on real world problems, and comparisons are drawn on IRRHA and the implicitly restarted harmonic Arnoldi algorithm (IRHA) of Morgan as well as the implicitly restarted Arnoldi algorithm (IRA) of Sorensen and the implicitly restarted refined Arnoldi algorithm (IRRA) of Jia. They indicate that the refined algorithms outperform their standard counterparts considerably. Furthermore, for exterior eigenvalue problems IRRA is preferable and more powerful, and for interior eigenvalue problems IRRHA is more attractive and can be much more efficient.

Boundary integral methods in low frequency acoustics

This expository paper is concerned with the direct integral formulations for boundary value problems of the Helmholtz equation. We discuss unique solvability for the corresponding boundary integral equations and its relations to the interior eigenvalue problems of the Laplacian. Based on the integral representations, we study the asymptotic behaviors of the solutions to the boundary value problems when the wave number tends to zero. We arrive at the asymptotic expansions for the solutions, and show that in all the cases, the leading terms in the expansions are always the corresponding potentials for the Laplacian. Our integral equation procedures developed here are general enough and can be adapted for treating similar low frequency scattering problems.