Showing papers on "Divide-and-conquer eigenvalue algorithm published in 2014"

The ELPA library: scalable parallel eigenvalue solutions for electronic structure theory and computational science

Abstract: Obtaining the eigenvalues and eigenvectors of large matrices is a key problem in electronic structure theory and many other areas of computational science. The computational effort formally scales as O(N(3)) with the size of the investigated problem, N (e.g. the electron count in electronic structure theory), and thus often defines the system size limit that practical calculations cannot overcome. In many cases, more than just a small fraction of the possible eigenvalue/eigenvector pairs is needed, so that iterative solution strategies that focus only on a few eigenvalues become ineffective. Likewise, it is not always desirable or practical to circumvent the eigenvalue solution entirely. We here review some current developments regarding dense eigenvalue solvers and then focus on the Eigenvalue soLvers for Petascale Applications (ELPA) library, which facilitates the efficient algebraic solution of symmetric and Hermitian eigenvalue problems for dense matrices that have real-valued and complex-valued matrix entries, respectively, on parallel computer platforms. ELPA addresses standard as well as generalized eigenvalue problems, relying on the well documented matrix layout of the Scalable Linear Algebra PACKage (ScaLAPACK) library but replacing all actual parallel solution steps with subroutines of its own. For these steps, ELPA significantly outperforms the corresponding ScaLAPACK routines and proprietary libraries that implement the ScaLAPACK interface (e.g. Intel's MKL). The most time-critical step is the reduction of the matrix to tridiagonal form and the corresponding backtransformation of the eigenvectors. ELPA offers both a one-step tridiagonalization (successive Householder transformations) and a two-step transformation that is more efficient especially towards larger matrices and larger numbers of CPU cores. ELPA is based on the MPI standard, with an early hybrid MPI-OpenMPI implementation available as well. Scalability beyond 10,000 CPU cores for problem sizes arising in the field of electronic structure theory is demonstrated for current high-performance computer architectures such as Cray or Intel/Infiniband. For a matrix of dimension 260,000, scalability up to 295,000 CPU cores has been shown on BlueGene/P.

A multi-level correction scheme for eigenvalue problems

Abstract: In this paper, a new type of multi-level correction scheme is proposed forsolving eigenvalue problems by finite element method. With this new scheme,the accuracy of eigenpair approximations can be improved after each correc-tion step which only needs to solve a source problem on finer finite elementspace and an eigenvalue problem on the coarsest finite element space. Thiscorrection scheme can improve the efficiency of solving eigenvalue problemsby finite element method.Keywords. Eigenvalue problem, multi-level correction scheme, finite ele-ment method, multi-space, multi-grid.AMS subject classifications. 65N30, 65B99, 65N25, 65L15 1 Introduction The purpose of this paper is to propose a new type of multi-level correction schemebased on finite element discretization to solve eigenvalue problems. The two-gridmethod for solving eigenvalue problems has been proposed and analyzed by Xu andZhou in [21]. The idea of the two-grid comes from [19, 20] for nonsymmetric orindefinite problems and nonlinear elliptic equations. Since then, there have existedmany numerical methods for solving eigenvalue problems based on the idea of two-grid method ([1, 6, 17]).

Guaranteed lower eigenvalue bounds for the biharmonic equation

Abstract: The computation of lower eigenvalue bounds for the biharmonic operator in the buckling of plates is vital for the safety assessment in structural mechanics and highly on demand for the separation of eigenvalues for the plate's vibrations. This paper shows that the eigenvalue provided by the nonconforming Morley finite element analysis, which is perhaps a lower eigenvalue bound for the biharmonic eigenvalue in the asymptotic sense, is not always a lower bound. A fully-explicit error analysis of the Morley interpolation operator with all the multiplicative constants enables a computable guaranteed lower eigenvalue bound. This paper provides numerical computations of those lower eigenvalue bounds and studies applications for the vibration and the stability of a biharmonic plate with different lower-order terms.

Low-Rank Tensor Methods with Subspace Correction for Symmetric Eigenvalue Problems

Abstract: We consider the solution of large-scale symmetric eigenvalue problems for which it is known that the eigenvectors admit a low-rank tensor approximation. Such problems arise, for example, from the discretization of high-dimensional elliptic PDE eigenvalue problems or in strongly correlated spin systems. Our methods are built on imposing low-rank (block) tensor train (TT) structure on the trace minimization characterization of the eigenvalues. The common approach of alternating optimization is combined with an enrichment of the TT cores by (preconditioned) gradients, as recently proposed by Dolgov and Savostyanov for linear systems. This can equivalently be viewed as a subspace correction technique. Several numerical experiments demonstrate the performance gains from using this technique.

Numerical Optimization of Eigenvalues of Hermitian Matrix Functions

Abstract: This work concerns the global minimization of a prescribed eigenvalue or a weighted sum of prescribed eigenvalues of a Hermitian matrix-valued function depending on its parameters analytically in a box. We describe how the analytical properties of eigenvalue functions can be put into use to derive piecewise quadratic functions that underestimate the eigenvalue functions. These piecewise quadratic underestimators lead us to a global minimization algorithm, originally due to Breiman and Cutler. We prove the global convergence of the algorithm and show that it can be effectively used for the minimization of extreme eigenvalues, e.g., the largest eigenvalue or the sum of the largest specified number of eigenvalues. This is particularly facilitated by the analytical formulas for the first derivatives of eigenvalues, as well as analytical lower bounds on the second derivatives that can be deduced for extreme eigenvalue functions. The applications that we have in mind also include the ${\rm H}_\infty$-norm of a ...

The first Robin eigenvalue with negative boundary parameter

Abstract: We give a counterexample to the long standing conjecture that the ball maximises the first eigenvalue of the Robin eigenvalue problem with negative parameter among domains of the same volume. Furthermore, we show that the conjecture holds in two dimensions provided that the boundary parameter is small. This is the first known example within the class of isoperimetric spectral problems for the first eigenvalue of the Laplacian where the ball is not an optimiser.

Two-Grid Methods for Maxwell Eigenvalue Problems

Abstract: Two new two-grid algorithms are proposed for solving the Maxwell eigenvalue problem. The new methods are based on the two-grid methodology recently proposed by Xu and Zhou [Math. Comp., 70 (2001), pp. 17--25] and further developed by Hu and Cheng [Math. Comp., 80 (2011), pp. 1287--1301] for elliptic eigenvalue problems. The new two-grid schemes reduce the solution of the Maxwell eigenvalue problem on a fine grid to one linear indefinite Maxwell equation on the same fine grid and an original eigenvalue problem on a much coarser grid. The new schemes, therefore, save total computational cost. The error estimates reveals that the two-grid methods maintain asymptotically optimal accuracy, and the numerical experiments presented confirm the theoretical results.

Bounds on Eigenvalues of a Spatial Correlation Matrix

Abstract: It is critical to understand the properties of spatial correlation matrices in massive multiple-input-multiple-output (MIMO) systems. We derive new bounds on the extreme eigenvalues of a spatial correlation matrix that is characterized by the exponential model in this paper. The new upper bound on the maximum eigenvalue is tighter than the previously known bound. Moreover, numerical studies show that our new lower bound on the maximum eigenvalue is close to the true maximum eigenvalue in most cases. We also derive an upper bound on the minimum eigenvalue that is also tight. These bounds can be exploited to analyze many wireless communication scenarios including uniform planar arrays, which are expected to be widely used for massive MIMO systems.

A Hamiltonian–Krein (Instability) Index Theory for Solitary Waves to KdV-Like Eigenvalue Problems

Abstract: The Hamiltonian–Krein (instability) index is concerned with determining the number of eigenvalues with positive real part for the Hamiltonian eigenvalue problem , where is skew-symmetric and is self-adjoint. If has a bounded inverse the index is well established, and it is given by the number of negative eigenvalues of the operator constrained to act on some finite-codimensional subspace. There is an important class of problems—namely, those of KdV-type—for which does not have a bounded inverse. In this paper, we overcome this difficulty and derive the index for eigenvalue problems of KdV-type. We use the index to discuss the spectral stability of homoclinic traveling waves for KdV-like problems and Benjamin—Bona—Mahony-type problems.

Geometric Measure of Entanglement and U-Eigenvalues of Tensors

Abstract: We study tensor analysis problems motivated by the geometric measure of quantum entanglement. We define the concept of the unitary eigenvalue (U-eigenvalue) of a complex tensor, the unitary symmetric eigenvalue (US-eigenvalue) of a symmetric complex tensor, and the best complex rank-one approximation. We obtain an upper bound on the number of distinct US-eigenvalues of symmetric tensors and count all US-eigenpairs with nonzero eigenvalues of symmetric tensors. We convert the geometric measure of the entanglement problem to an algebraic equation system problem. A numerical example shows that a symmetric real tensor may have a best complex rank-one approximation that is better than its best real rank-one approximation, which implies that the absolute-value largest Z-eigenvalue is not always the geometric measure of entanglement.

Rayleigh Quotient Based Optimization Methods For Eigenvalue Problems

Abstract: Four classes of eigenvalue problems that admit similar min-max principles and the Cauchy interlacing inequalities as the symmetric eigenvalue problem famously does are investigated. These min-max principles pave ways for efficient numerical solutions for extreme eigenpairs by optimizing the so-called Rayleigh quotient functions. In fact, scientists and engineers have already been doing that for computing the eigenvalues and eigenvectors of Hermitian matrix pencils A − λB with B positive definite, the first class of our eigenvalue problems. But little attention has gone to the other three classes: positive semidefinite pencils, linear response eigenvalue problems, and hyperbolic eigenvalue problems, in part because most min-max principles for the latter were discovered only very recently and some more are being discovered. It is expected that they will drive the effort to design better optimization based numerical methods for years to come.

Eigenvalue statistics for product complex Wishart matrices

Abstract: The eigenvalue statistics for complex $N \times N$ Wishart matrices $X_{r,s}^\dagger X_{r,s}$, where $ X_{r,s}$ is equal to the product of $r$ complex Gaussian matrices, and the inverse of $s$ complex Gaussian matrices, are considered. In the case $r=s$ the exact form of the global density is computed. The averaged characteristic polynomial for the corresponding generalized eigenvalue problem is calculated in terms of a particular generalized hypergeometric function ${}_{s+1} F_r$. For finite $N$ the eigenvalue probability density function is computed, and is shown to be an example of a biorthogonal ensemble. A double contour integral form of the corresponding correlation kernel is derived, which allows the hard edge scaled limit to be computed. The limiting kernel is given in terms of certain Meijer G-functions, and is identical to that found in the recent work of Kuijlaars and Zhang in the case $s=0$. Properties of the kernel and corresponding correlation functions are discussed.

Memory‐efficient Arnoldi algorithms for linearizations of matrix polynomials in Chebyshev basis

Abstract: Novel memory-efficient Arnoldi algorithms for solving matrix polynomial eigenvalue problems are presented. More specifically, we consider the case of matrix polynomials expressed in the Chebyshev basis, which is often numerically more appropriate than the standard monomial basis for a larger degree d. The standard way of solving polynomial eigenvalue problems proceeds by linearization, which increases the problem size by a factor d. Consequently, the memory requirements of Krylov subspace methods applied to the linearization grow by this factor. In this paper, we develop two variants of the Arnoldi method that build the Krylov subspace basis implicitly, in a way that only vectors of length equal to the size of the original problem need to be stored. The proposed variants are generalizations of the so-called quadratic Arnoldi method and two-level orthogonal Arnoldi procedure methods, which have been developed for the monomial case. We also show how the typical ingredients of a full implementation of the Arnoldi method, including shift-and-invert and restarting, can be incorporated. Numerical experiments are presented for matrix polynomials up to degree 30 arising from the interpolation of nonlinear eigenvalue problems, which stem from boundary element discretizations of PDE eigenvalue problems. Copyright (C) 2013 John Wiley & Sons, Ltd.

Fast Enclosure for All Eigenvalues and Invariant Subspaces in Generalized Eigenvalue Problems

Abstract: Two fast algorithms for enclosing all eigenvalues and invariant subspaces in generalized eigenvalue problems are proposed. In these algorithms, individual eigenvectors and invariant subspaces are enclosed when eigenvalues are well separated and closely clustered, respectively. The first algorithm involves only cubic complexity and automatically determines eigenvalue clusters. The second algorithm is applicable even for defective eigenvalues. Numerical results show the properties of the proposed algorithms.

Coupled finite and boundary element methods for fluid-solid interaction eigenvalue problems

Abstract: We analyze the approximation of a vibro-acoustic eigenvalue problem for an elastic body which is submerged in a compressible inviscid fluid in $\mathbb{R}^3$. As a model, the time-harmonic elastodynamic and the Helmholtz equation are used and are coupled in a strong sense via the standard transmission conditions on the interface between the solid and the fluid. Our approach is based on a coupling of the field equations for the solid with boundary integral equations for the fluid. The coupled formulation of the eigenvalue problem leads to a nonlinear eigenvalue problem with respect to the eigenvalue parameter since the frequency occurs nonlinearly in the used boundary integral operators for the Helmholtz equation. The nonlinear eigenvalue problem and its Galerkin discretization are analyzed within the framework of eigenvalue problems for Fredholm operator-valued functions where convergence is shown and error estimates are given. For the numerical solution of the discretized nonlinear matrix eigenvalue prob...

Approximation of differential eigenvalue problems with a nonlinear dependence on the parameter

Abstract: A differential eigenvalue problem with a nonlinear dependence on the parameter is approximated by the finite-element method with numerical integration. We study the error in the approximate eigenvalues and eigenfunctions.

A backward stable algorithm for quadratic eigenvalue problems

Abstract: We present a backward stable algorithm for dense quadratic eigenvalue problems. Our algorithm incorporates a tropical-like scaling, a strategy for choosing linearizations, and an associated strategy for recovering eigentriples. We prove that the growth factor in the translation from conditioning for the quadratic to conditioning for the linearization and the growth factor in the translation from backward error for the linearization to backward error for the quadratic are both of order one in the algorithm.

Several properties of invariant pairs of nonlinear algebraic eigenvalue problems

Abstract: We analyze several important properties of invariant pairs of nonlinear algebraic eigenvalue problems of the form T (λ)v = 0. Invariant pairs are generalizations of invariant subspaces in association with block Rayleigh quotients of square matrices to a nonlinear matrix-valued function T (·). They play an important role in the analysis of nonlinear eigenvalue problems and algorithms. In this paper, we first show that the algebraic, partial, and geometric multiplicities together with the Jordan chains corresponding to an eigenvalue of T (λ)v = 0 are completely represented by the Jordan canonical form of a simple invariant pair that captures this eigenvalue. We then investigate approximation errors and perturbations of a simple invariant pair. We also show that second order accuracy in eigenvalue approximation can be achieved by the two-sided block Rayleigh functional for non-defective eigenvalues. Finally, we study the matrix representation of the Frechet derivative of the eigenproblem, and we discuss the norm estimate of the inverse derivative, which measures the conditioning and sensitivity of simple invariant pairs.

Computation of the minimum eigenvalue for a nonlinear Sturm-Liouville problem

Abstract: A condition for the existence of a minimum eigenvalue corresponding to a positive eigenfunction of the nonlinear eigenvalue problem for an ordinary differential equation is determined. The problem is approximated by a mesh scheme of the finite element method. The convergence of approximate solutions to exact ones is studied. Theoretical results are illustrated by numerical experiments for a model problem.

Minimization principles and computation for the generalized linear response eigenvalue problem

Abstract: The minimization principle and Cauchy-like interlacing inequalities for the generalized linear response eigenvalue problem are presented. Based on these theoretical results, the best approximations through structure-preserving subspace projection and a locally optimal block conjugate gradient-like algorithm for simultaneously computing the first few smallest eigenvalues with the positive sign are proposed. Numerical results are presented to illustrate essential convergence behaviors of the proposed algorithm.

A parallel implementation of Davidson methods for large-scale eigenvalue problems in SLEPc

Abstract: In the context of large-scale eigenvalue problems, methods of Davidson type such as Jacobi-Davidson can be competitive with respect to other types of algorithms, especially in some particularly difficult situations such as computing interior eigenvalues or when matrix factorization is prohibitive or highly inefficient. However, these types of methods are not generally available in the form of high-quality parallel implementations, especially for the case of non-Hermitian eigenproblems. We present our implementation of various Davidson-type methods in SLEPc, the Scalable Library for Eigenvalue Problem Computations. The solvers incorporate many algorithmic variants for subspace expansion and extraction, and cover a wide range of eigenproblems including standard and generalized, Hermitian and non-Hermitian, with either real or complex arithmetic. We provide performance results on a large battery of test problems.

Restructuring the Tridiagonal and Bidiagonal QR Algorithms for Performance

Abstract: We show how both the tridiagonal and bidiagonal QR algorithms can be restructured so that they become rich in operations that can achieve near-peak performance on a modern processor. The key is a novel, cache-friendly algorithm for applying multiple sets of Givens rotations to the eigenvector/singular vector matrix. This algorithm is then implemented with optimizations that: (1) leverage vector instruction units to increase floating-point throughput, and (2) fuse multiple rotations to decrease the total number of memory operations. We demonstrate the merits of these new QR algorithms for computing the Hermitian eigenvalue decomposition (EVD) and singular value decomposition (SVD) of dense matrices when all eigenvectors/singular vectors are computed. The approach yields vastly improved performance relative to traditional QR algorithms for these problems and is competitive with two commonly used alternatives---Cuppen’s Divide-and-Conquer algorithm and the method of Multiple Relatively Robust Representations---while inheriting the more modest O(n) workspace requirements of the original QR algorithms. Since the computations performed by the restructured algorithms remain essentially identical to those performed by the original methods, robust numerical properties are preserved.

On the Inverse Symmetric Quadratic Eigenvalue Problem

Abstract: The detailed spectral structure of symmetric, algebraic, quadratic eigenvalue problems has been developed recently. In this paper we take advantage of these canonical forms to provide a detailed analysis of inverse problems of the following form: construct the coefficient matrices from the spectral data including the classical eigenvalue/eigenvector data and sign characteristics for the real eigenvalues. An orthogonality condition dependent on these signs plays a vital role in this construction. Special attention is paid to the cases when the leading and trailing coefficients of the quadratic matrix polynomial are prescribed to be positive definite.

Greedy Algorithms for High-Dimensional Eigenvalue Problems

Abstract: In this article, we present two new greedy algorithms for the computation of the lowest eigenvalue (and an associated eigenvector) of a high-dimensional eigenvalue problem, and prove some convergence results for these algorithms and their orthogonalized versions. The performance of our algorithms is illustrated on numerical test cases (including the computation of the buckling modes of a microstructured plate), and compared with that of another greedy algorithm for eigenvalue problems introduced by Ammar and Chinesta.

On the symmetric quadratic eigenvalue complementarity problem

Abstract: In this paper, the solution of the symmetric quadratic eigenvalue complementarity problem QEiCP is addressed. The QEiCP has a solution provided the so-called co-regular and co-hyperbolic properties hold and is said to be symmetric if all the matrices involved in its definition are symmetric. We show that under the two conditions stated above the symmetric QEiCP can be reduced to the problem of computing a stationary point of an appropriate nonlinear program NLP. We also investigate the reduction of the QEiCP to a simpler eigenvalue complementarity problem EiCP. This transformation enables us to show that the co-regular and co-hyperbolic properties are not necessary for the existence of a solution to the QEiCP. Furthermore the QEiCP is shown to be equivalent to the problem of finding a stationary point of a quadratic fractional program QFP under special conditions on the matrices of the QEiCP. The use of the so-called spectral projected-gradient SPG algorithm for dealing with the programs NLP and QFP is also investigated. Some considerations about the implementation of this algorithm are discussed. Computational experience is included to highlight the efficiency of the algorithm for finding a solution of the QEiCP by exploring the NLPs mentioned above.