Showing papers on "Divide-and-conquer eigenvalue algorithm published in 2012"
Journal Article•
TL;DR: A novel metric learning approach called DML-eig is introduced which is shown to be equivalent to a well-known eigen value optimization problem called minimizing the maximal eigenvalue of a symmetric matrix.
Abstract: The main theme of this paper is to develop a novel eigenvalue optimization framework for learning a Mahalanobis metric. Within this context, we introduce a novel metric learning approach called DML-eig which is shown to be equivalent to a well-known eigenvalue optimization problem called minimizing the maximal eigenvalue of a symmetric matrix (Overton, 1988; Lewis and Overton, 1996). Moreover, we formulate LMNN (Weinberger et al., 2005), one of the state-of-the-art metric learning methods, as a similar eigenvalue optimization problem. This novel framework not only provides new insights into metric learning but also opens new avenues to the design of efficient metric learning algorithms. Indeed, first-order algorithms are developed for DML-eig and LMNN which only need the computation of the largest eigenvector of a matrix per iteration. Their convergence characteristics are rigorously established. Various experiments on benchmark data sets show the competitive performance of our new approaches. In addition, we report an encouraging result on a difficult and challenging face verification data set called Labeled Faces in the Wild (LFW).
348 citations
TL;DR: In this paper, a numerical method for computing all eigenvalues (and the corresponding eigenvectors) of a nonlinear holomorphic eigenvalue problem that lie within a given contour in the complex plane is proposed.
284 citations
TL;DR: It is proved that the eigenvalue spacing of the Erdős-Rényi graph in the bulk of the spectrum has the same distribution as that of the Gaussian orthogonal ensemble.
Abstract: We consider the ensemble of adjacency matrices of Erdős-Renyi random graphs, i.e. graphs on N vertices where every edge is chosen independently and with probability p ≡ p(N). We rescale the matrix so that its bulk eigenvalues are of order one. Under the assumption $${p N \gg N^{2/3}}$$
, we prove the universality of eigenvalue distributions both in the bulk and at the edge of the spectrum. More precisely, we prove (1) that the eigenvalue spacing of the Erdős-Renyi graph in the bulk of the spectrum has the same distribution as that of the Gaussian orthogonal ensemble; and (2) that the second largest eigenvalue of the Erdős-Renyi graph has the same distribution as the largest eigenvalue of the Gaussian orthogonal ensemble. As an application of our method, we prove the bulk universality of generalized Wigner matrices under the assumption that the matrix entries have at least 4 + e moments.
268 citations
TL;DR: This work proposes the use of polynomial approximation combined with non-monomial linearizations for matrix eigenvalue problems that are nonlinear in the eigenvalues of interest or on a pre-specified curve in the complex plane.
Abstract: This work is concerned with numerical methods for matrix eigenvalue problems that are nonlinear in the eigenvalue parameter. In particular, we focus on eigenvalue problems for which the evaluation of the matrix-valued function is computationally expensive. Such problems arise, e.g., from boundary integral formulations of elliptic PDE eigenvalue problems and typically exclude the use of established nonlinear eigenvalue solvers. Instead, we propose the use of polynomial approximation combined with non-monomial linearizations. Our approach is intended for situations where the eigenvalues of interest are located on the real line or, more generally, on a pre-specified curve in the complex plane. A first-order perturbation analysis for nonlinear eigenvalue problems is performed. Combined with an approximation result for Chebyshev interpolation, this shows exponential convergence of the obtained eigenvalue approximations with respect to the degree of the approximating polynomial. Preliminary numerical experiments demonstrate the viability of the approach in the context of boundary element methods.
85 citations
TL;DR: This work provides a characterization of problems that can be efficiently solved as polynomial eigenvalue problems (PEPs) and presents a resultant-based method for transforming a system of poynomial equations to a polynometric eigen value problem.
Abstract: We present a method for solving systems of polynomial equations appearing in computer vision. This method is based on polynomial eigenvalue solvers and is more straightforward and easier to implement than the state-of-the-art Grobner basis method since eigenvalue problems are well studied, easy to understand, and efficient and robust algorithms for solving these problems are available. We provide a characterization of problems that can be efficiently solved as polynomial eigenvalue problems (PEPs) and present a resultant-based method for transforming a system of polynomial equations to a polynomial eigenvalue problem. We propose techniques that can be used to reduce the size of the computed polynomial eigenvalue problems. To show the applicability of the proposed polynomial eigenvalue method, we present the polynomial eigenvalue solutions to several important minimal relative pose problems.
82 citations
TL;DR: An iterative algorithm using restarted Arnoldi method to solve the resulting non-Hermitian eigenvalue problem and can be easily used in the qualitative methods in inverse scattering and modified to compute transmission eigenvalues for other models such as elasticity problem.
Abstract: Transmission eigenvalue problem has important applications in inverse scattering. Since the problem is non-self-adjoint, the computation of transmission eigenvalues needs special treatment. Based on a fourth-order reformulation of the transmission eigenvalue problem, a mixed finite element method is applied. The method has two major advantages: 1) the formulation leads to a generalized eigenvalue problem naturally without the need to invert a related linear system, and 2) the nonphysical zero transmission eigenvalue, which has an infinitely dimensional eigenspace, is eliminated. To solve the resulting non-Hermitian eigenvalue problem, an iterative algorithm using restarted Arnoldi method is proposed. To make the computation efficient, the search interval is decided using a Faber-Krahn type inequality for transmission eignevalues and the interval is updated at each iteration. The algorithm is implemented using Matlab. The code can be easily used in the qualitative methods in inverse scattering and modified to compute transmission eigenvalues for other models such as elasticity problem.
59 citations
TL;DR: This work introduces some ways to compute the lower and upper bounds of the Laplace eigenvalue problem by using the special nonconforming finite elements, i.e., enriched Crouzeix-Raviart element and extended Q1rot, to get the lower bound of the eigen value.
Abstract: We introduce some ways to compute the lower and upper bounds of the Laplace eigenvalue problem. By using the special nonconforming finite elements, i.e., enriched Crouzeix-Raviart element and extended Q1rot we get the lower bound of the eigenvalue. Additionally, we use conforming finite elements to do the postprocessing to get the upper bound of the eigenvalue, which only needs to solve the corresponding source problems and a small eigenvalue problem if higher order postprocessing method is implemented. Thus, we can obtain the lower and upper bounds of the eigenvalues simultaneously by solving eigenvalue problem only once. Some numerical results are also presented to demonstrate our theoretical analysis.
57 citations
TL;DR: In this article, the authors derived a contour integral formula for the Harish-Chandra Itzykson-Zuber integral when X and Y are real symmetric and Y is a rank 1 matrix.
Abstract: In this paper, we consider N-dimensional real Wishart matrices Y in the class \input amssym $W_{\Bbb R} (\Sigma ,M)$ in which all but one eigenvalue of Σ is 1. Let the nontrivial eigenvalue of Σ be 1+τ; then as N, M ∞, with M/N γ2 finite and nonzero, the eigenvalue distribution of Y will converge into the Marchenko-Pastur distribution inside a bulk region. When τ increases from 0, one starts to see a stray eigenvalue of Y outside of the support of the Marchenko-Pastur density. As this stray eigenvalue leaves the bulk region, a phase transition will occur in the largest eigenvalue distribution of the Wishart matrix.
In this paper we will compute the asymptotics of the largest eigenvalue distribution when the phase transition occurs. We will first establish the results that are valid for all N and M and will use them to carry out the asymptotic analysis. In particular, we have derived a contour integral formula for the Harish-Chandra Itzykson-Zuber integral when X and Y are real symmetric and Y is a rank 1 matrix. This allows us to write down a Fredholm determinant formula for the largest eigenvalue distribution and analyze it using orthogonal polynomial techniques. As a result, we obtain an integral formula for the largest eigenvalue distribution in the large- N limit characterized by Painleve transcendents. The approach used in this paper is very different from a recent paper by Bloemenal and Virag, in which the largest eigenvalue distribution was obtained using a stochastic operator method. In particular, the Painleve formula for the largest eigenvalue distribution obtained in this paper is new. © 2012 Wiley Periodicals, Inc.
56 citations
TL;DR: Two theoretical results are presented that mirror the well-known trace minimization principle and Cauchy's interlacing inequalities for the symmetric eigenvalue problem.
Abstract: We present two theoretical results for the linear response eigenvalue problem. The first result is a minimization principle for the sum of the smallest eigenvalues with the positive sign. The second result is Cauchy-like interlacing inequalities. Although the linear response eigenvalue problem is a nonsymmetric eigenvalue problem, these results mirror the well-known trace minimization principle and Cauchy's interlacing inequalities for the symmetric eigenvalue problem.
54 citations
TL;DR: Computational results show the robustness, efficiency, and high speed of the proposed algorithms for solving large scale Eigenvalue Complementarity Problems with real symmetric matrices.
Abstract: In this paper, we investigate a DC (Difference of Convex functions) programming technique for solving large scale Eigenvalue Complementarity Problems (EiCP) with real symmetric matrices. Three equivalent formulations of EiCP are considered. We first reformulate them as DC programs and then use DCA (DC Algorithm) for their solution. Computational results show the robustness, efficiency, and high speed of the proposed algorithms.
48 citations
TL;DR: A rigorous convergence and error analysis of a Galerkin boundary element method for the Dirichlet Laplacian eigenvalue problem is presented and quasi-optimal error estimates are presented.
Abstract: In this paper, a rigorous convergence and error analysis of a Galerkin boundary element method for the Dirichlet Laplacian eigenvalue problem is presented. The formulation of the eigenvalue problem in terms of a boundary integral equation yields a nonlinear boundary integral operator eigenvalue problem. This nonlinear eigenvalue problem and its Galerkin approximation are analyzed in the framework of eigenvalue problems for holomorphic Fredholm operator-valued functions. The convergence of the approximation is shown and quasi-optimal error estimates are presented. Numerical experiments are given confirming the theoretical results.
TL;DR: This paper proposes an error estimator of residual type and shows it is reliable and efficient for each eigenvalue problem in the family of periodic Hermitian eigen value problems on a unit cell, parametrised by a two-dimensional parameter - the quasimomentum.
Abstract: In this paper we propose and analyse adaptive finite element methods for computing the band structure of 2D periodic photonic crystals. The problem can be reduced to the computation of the discrete spectra of each member of a family of periodic Hermitian eigenvalue problems on a unit cell, parametrised by a two-dimensional parameter - the quasimomentum. These eigenvalue problems involve non-coercive elliptic operators with generally discontinuous coefficients and are solved by adaptive finite elements. We propose an error estimator of residual type and show it is reliable and efficient for each eigenvalue problem in the family. In particular we prove that if the error estimator converges to zero then the distance of the computed eigenfunction from the true eigenspace also converges to zero and the computed eigenvalue converges to a true eigenvalue with double the rate. We also prove that if the distance of a computed sequence of approximate eigenfunctions from the true eigenspace approaches zero, then so must the error estimator. The results hold for eigenvalues of any multiplicity. We illustrate the benefits of the resulting adaptive method in practice, both for fully periodic structures and also for the computation of eigenvalues in the band gap of structures with defect, using the supercell method.
TL;DR: The asymptotic quasi-optimal adaptive finite element eigenvalue solver (AFEMES) involves a proper termination criterion for the algebraic eigen value solver and does not need any coarsening.
Abstract: This paper presents a combined adaptive finite element method with an iterative algebraic eigenvalue solver for a symmetric eigenvalue problem of asymptotic quasi-optimal computational complexity. The analysis is based on a direct approach for eigenvalue problems and allows the use of higher-order conforming finite element spaces with fixed polynomial degree. The asymptotic quasi-optimal adaptive finite element eigenvalue solver (AFEMES) involves a proper termination criterion for the algebraic eigenvalue solver and does not need any coarsening. Numerical evidence illustrates the asymptotic quasi-optimal computational complexity in 2 and 3 dimensions.
01 Jan 2012
TL;DR: For the efficient approximate evaluation of parameter sensitivities of isolated eigenpairs on the entire parameter space, a sparse tensor spectral collocation method on an anisotropic sparse grid in the parameter domain is proposed and analyzed.
Abstract: We design and analyze algorithms for the efficient sensitivity computation of eigenpairs of parametric elliptic self-adjoint eigenvalue problems on high-dimensional parameter spaces. We quantify the analytic dependence of eigenpairs on the parameters. For the efficient approximate evaluation of parameter sensitivities of isolated eigenpairs on the entire parameter space we propose and analyze a sparse tensor spectral collocation method on an anisotropic sparse grid in the parameter domain. The stable numerical implementation of these methods is discussed and their error analysis is given. Applications to parametric elliptic eigenvalue problems with infinitely many parameters arising from elliptic differential operators with random coefficients are presented.
TL;DR: In this paper, the authors considered a two-particle discrete Schrodinger operator corresponding to a system of two identical particles on a lattice interacting via an attractive pairwise zero-range potential.
Abstract: We consider a two-particle discrete Schrodinger operator corresponding to a system of two identical particles on a lattice interacting via an attractive pairwise zero-range potential. We show that there is a unique eigenvalue below the bottom of the essential spectrum for all values of the coupling constant and two-particle quasimomentum. We obtain a convergent expansion for the eigenvalue.
TL;DR: The new method improves and generalizes the SHIRA method of Mehrmann and Watkins (2001) to the case where the skew-symmetric matrix is singular and computes a few eigenvalues and eigenvectors of the matrix pencil close to a given target point.
TL;DR: In this article, a Dirichlet-to-Neumann eigenvalue problem for differential forms on a compact Riemannian manifold with smooth boundary is studied, and upper and lower bounds for the first eigen value in several contexts are derived.
TL;DR: Using the recovered function and its gradient, this work is able to enhance the eigenvalue approximation and increase its convergence rate to $h^{2\alpha}$, where $\alpha > 1$ is the superconvergence rate of the recovered gradient.
Abstract: Function value recovery techniques for linear finite elements are discussed. Using the recovered function and its gradient, we are able to enhance the eigenvalue approximation and increase its convergence rate to $h^{2\alpha}$, where $\alpha > 1$ is the superconvergence rate of the recovered gradient. This is true in both symmetric and nonsymmetric eigenvalue problems.
TL;DR: A new finite element two-scale discretization scheme is proposed by combining the mixed finite element method with the shifted-inverse power method for solving matrix eigenvalue problems.
Abstract: This paper discusses highly efficient discretization schemes for mixed variational formulation of eigenvalue problems. A new finite element two-scale discretization scheme is proposed by combining the mixed finite element method with the shifted-inverse power method for solving matrix eigenvalue problems. With this scheme, the solution of an eigenvalue problem on a fine grid is reduced to the solution of an eigenvalue problem on a much coarser grid and the solution of a linear algebraic system on the fine grid . Theoretical analysis shows that the scheme has high efficiency. For instance, when using the Mini element to solve Stokes eigenvalue problem, the resulting solution can maintain an asymptotically optimal accuracy by taking , and when using the - element to solve eigenvalue problems of electric field, the calculation results can maintain an asymptotically optimal accuracy by taking . Finally, numerical experiments are presented to support the theoretical analysis.
TL;DR: In this article, the properties of the first eigenvalue and its eigenvector were investigated for large sparse random symmetric matrices that are characterized by bimodal degree distributions.
Abstract: The properties of the first (largest) eigenvalue and its eigenvector (first eigenvector) are investigated for large sparse random symmetric matrices that are characterized by bimodal degree distributions. In principle, one should be able to accurately calculate them by solving a functional equation concerning auxiliary fields which come out in an analysis based on replica/cavity methods. However, the difficulty in analytically solving this equation makes an accurate calculation infeasible in practice. To overcome this problem, we develop approximation schemes on the basis of two exceptionally solvable examples. The schemes are reasonably consistent with numerical experiments when the statistical bias of positive matrix entries is sufficiently large, and they qualitatively explain why considerably large finite size effects of the first eigenvalue can be observed when the bias is relatively small.
TL;DR: In this paper, the convergence and accuracy of approximate eigenvalues and eigenelements were analyzed by a sample scheme of the finite element method with numerical integration for a one-dimensional sign-indefinite second-order differential eigenvalue problem.
Abstract: A variational sign-indefinite eigenvalue problem in an infinite-dimensional Hilbert space is approximated by a problem in a finite-dimensional subspace. We analyze the convergence and accuracy of approximate eigenvalues and eigenelements. The general results are illustrated by a sample scheme of the finite-element method with numerical integration for a one-dimensional sign-indefinite second-order differential eigenvalue problem.
TL;DR: A discontinuous Galerkin method, with hp-adaptivity based on the approximate solution of appropriate dual problems, is employed for highly-accurate eigenvalue computations on a collection of benchmark examples.
TL;DR: In this paper, the principal eigenvalue of generalised Robin boundary value problems on non-smooth domains was considered and conditions were provided so that the related eigen value problem has a principal Eigenvalue.
Abstract: We consider the principal eigenvalue of generalised Robin boundary value problems on non-smooth domains, where the zero order coefficient of the boundary operator is negative or changes sign. We provide conditions so that the related eigenvalue problem has a principal eigenvalue. We work with the framework involving measure data on the boundary due to Arendt and Warma (Potential Anal 19:341–363, 2003). Examples of simple domains with cusps are used to illustrate all possible phenomena.
TL;DR: In this paper, the quadratic two-parameter eigenvalue problem (QMEP) is transformed into a nonsingular five-parameters eigen value problem.
TL;DR: In this paper, four variants of a method hybridizing perturbation and Polynomial Chaos (PC) expansion approaches are proposed and compared and applied to the problem of an Euler Bernoulli beam and a thin plate with stochastic properties.
TL;DR: In this paper, the spectral properties of doubly stochastic matrices have been studied and sufficient conditions for the inverse eigenvalue problem for doubly polynomial matrices are derived.
TL;DR: It is identified that any Sturm-Liouville problem consisting of such an equation with transmission condition and an arbitrary separated or real coupled self-adjoint boundary condition has a representation as an equivalent finite dimensional matrix eigenvalue problem.
Abstract: We identify a class of Sturm-Liouville equations with transmission conditions such that any Sturm-Liouville problem consisting of such an equation with transmission condition and an arbitrary separated or real coupled self-adjoint boundary condition has a representation as an equivalent finite dimensional matrix eigenvalue problem. Conversely, given any matrix eigenvalue problem of certain type and an arbitrary separated or real coupled self-adjoint boundary condition and transmission condition, we construct a class of Sturm-Liouville problems with this specified boundary condition and transmission condition, each of which is equivalent to the given matrix eigenvalue problem.
TL;DR: This work reduces the matrix to a banded form, which then requires the eigenvalues of the banded matrix to be computed, and combines this insight with tile algorithms that can be scheduled via a dynamic runtime system to multicore architectures.
Abstract: Classical solvers for the dense symmetric eigenvalue problem suffer from the first step, which involves a reduction to tridiagonal form that is dominated by the cost of accessing memory during the panel factorization The solution is to reduce the matrix to a banded form, which then requires the eigenvalues of the banded matrix to be computed The standard divide and conquer algorithm can be modified for this purpose The paper combines this insight with tile algorithms that can be scheduled via a dynamic runtime system to multicore architectures A detailed analysis of performance and accuracy is included Performance improvements of 14-fold and 4-fold speedups are reported relative to LAPACK and Intel's Math Kernel Library
TL;DR: A spectral transformation allowing the computation of the least stable eigenmodes in a prescribed frequency range, based on the filtering of the linearized equations of motion, which has the advantage of low memory requirements and is therefore suitable for large two- or three-dimensional problems.