Eigenvalue Clusters Research Articles

Spectral discretization errors in filtered subspace iteration

We consider filtered subspace iteration for approximating a cluster of eigenvalues (and its associated eigenspace) of a (possibly unbounded) selfadjoint operator in a Hilbert space. The algorithm is motivated by a quadrature approximation of an operator-valued contour integral of the resolvent. Resolvents on infinite dimensional spaces are discretized in computable finite-dimensional spaces before the algorithm is applied. This study focuses on how such discretizations result in errors in the eigenspace approximations computed by the algorithm. The computed eigenspace is then used to obtain approximations of the eigenvalue cluster. Bounds for the Hausdorff distance between the computed and exact eigenvalue clusters are obtained in terms of the discretization parameters within an abstract framework. A realization of the proposed approach for a model second-order elliptic operator using a standard finite element discretization of the resolvent is described. Some numerical experiments are conducted to gauge the sharpness of the theoretical estimates.

Physics-based preconditioning of Jacobian free Newton Krylov for Burgers’ equation using modified nodal integral method

The application of Nodal Integral Method (NIM) is mostly limited to the low Reynolds number problems; due to lack of suitable nonlinear solvers. This necessitates the development of advanced nonlinear solvers for higher nonlinearity. Newton’s method is a well-established solver for such equations. However, the rate of convergence in Newton methods is dependent on the initial guess. Additionally, the condition number of Jacobian matrix dictates the time required for matrix inversion. In this work, for the first time a preconditioner is developed for fluid flow problem using Modified-NIM (MNIM). This preconditioned Jacobian free Newton Krylov algorithm uses the linearized Modified-MNIM (M2NIM) as the preconditioner, resulting in better eigenvalue clustering, reducing Krylov iterations. The effectiveness of proposed method is demonstrated using 1-D Burgers' equation for larger time steps and higher Reynolds number up to 2500. The proposed preconditioner drastically reduces the spectral radius, CPU run-time and Krylov iterations.

Spectral and convergence analysis of the Discrete ALIF method

The Adaptive Local Iterative Filtering (ALIF) method is a recently proposed iterative procedure to decompose a signal into a finite number of “simple components” called intrinsic mode functions. It is an alternative to the well-known and widely used empirical mode decomposition method, and it has proved to be a powerful and promising algorithm. However, so far, no convergence analysis is available for the ALIF algorithm both in the continuous and discrete settings. In the present paper we focus on the discrete version of the ALIF method and we tackle the problem of studying its convergence properties. Using recent results about sampling matrices and the theory of generalized locally Toeplitz sequences — which we extend in this paper — we perform a spectral analysis of the ALIF iteration matrices, with a special attention to the eigenvalue clustering and the eigenvalue distribution. Based on the eigenvalue distribution, we formulate a necessary condition for the convergence of the Discrete ALIF method. Moreover, we provide a simple criterion to construct appropriate filters satisfying this condition. We also present several numerical examples in support of the theoretical study. Our contribution represents a first significant step toward a complete convergence analysis of the ALIF method, an analysis which appears to be rather difficult from a mathematical viewpoint as the ALIF iteration matrices possess a peculiar structure that, to the best of the authors' knowledge, has never been investigated in the literature.

Weighted Block Golub-Kahan-Lanczos Algorithms for Linear Response Eigenvalue Problem

In order to solve all or some eigenvalues lied in a cluster, we propose a weighted block Golub-Kahan-Lanczos algorithm for the linear response eigenvalue problem. Error bounds of the approximations to an eigenvalue cluster, as well as their corresponding eigenspace, are established and show the advantages. A practical thick-restart strategy is applied to the block algorithm to eliminate the increasing computational and memory costs, and the numerical instability. Numerical examples illustrate the effectiveness of our new algorithms.

TRPL+K: Thick-Restart Preconditioned Lanczos+K Method for Large Symmetric Eigenvalue Problems

The Lanczos method is one of the standard approaches for computing a few eigenpairs of a large, sparse, symmetric matrix. It is typically used with restarting to avoid unbounded growth of memory and computational requirements. Thick-restart Lanczos is a popular restarted variant because of its simplicity and numerical robustness. However, convergence can be slow for highly clustered eigenvalues so more effective restarting techniques and the use of preconditioning is needed. In this paper, we present a thick-restart preconditioned Lanczos method, TRPL+K, that combines the power of locally optimal restarting (+K) and preconditioning techniques with the efficiency of the thick-restart Lanczos method. TRPL+K employs an inner-outer scheme where the inner loop applies Lanczos on a preconditioned operator while the outer loop augments the resulting Lanczos subspace with certain vectors from the previous restart cycle to obtain eigenvector approximations with which it thick restarts the outer subspace. We first identify the differences from various relevant methods in the literature. Then, based on an optimization perspective, we show an asymptotic global quasi optimality of a simplified TRPL+K method compared to an unrestarted global optimal method. Finally, we present extensive experiments showing that TRPL+K either outperforms or matches other state-of-the-art eigenmethods in both matrix-vector multiplications and computational time.

Calculation of Fullerene Parameters by the Implemented One-Dimensional Method for Determination of Eigenvalues and Eigenfunctions in One-Dimensional Clusters of Planar, Cylindrical, and Spherical Geometry

The eigenvalues and eigenfunctions for clusters with planar, cylindrical, and spherical geometry with arbitrary potential energy profiles were calculated by means of an implemented algorithm. The results of numerical and analytical solutions for clusters of various geometry were compared. The proposed algorithm for determination of cluster eigenvalues and eigenfunctions shows a power law rate of convergence of the solution towards the target eigenfunction coinciding with the rate of convergence in the modified Wielandt method. This algorithm was used to calculate the geometrical potential of giant fullerene as a function of radius for the state with l = 0. The numerical results are in good agreement with the theoretical results.

Asymptotic quadratic convergence of the parallel block-Jacobi EVD algorithm with dynamic ordering for Hermitian matrices

The proof of the asymptotic quadratic convergence is provided for the parallel two-sided block-Jacobi EVD algorithm with dynamic ordering for Hermitian matrices. The discussion covers the case of well-separated eigenvalues as well as clusters of eigenvalues. Having p processors, each parallel iteration step consists of zeroing 2p off-diagonal blocks chosen by dynamic ordering with the aim to maximize the decrease of the off-diagonal Frobenius norm. Numerical experiments illustrate and confirm the developed theory.

Convergence estimates of nonrestarted and restarted block‐Lanczos methods

SummaryThe block‐Lanczos method serves to compute a moderate number of eigenvalues and the corresponding invariant subspace of a symmetric matrix. In this paper, the convergence behavior of nonrestarted and restarted versions of the block‐Lanczos method is analyzed. For the nonrestarted version, we improve an estimate by Saad by means of a change of the auxiliary vector so that the new estimate is much more accurate in the case of clustered or multiple eigenvalues. For the restarted version, an estimate by Knyazev is generalized by extending our previous results on block steepest descent iterations and single‐vector restarted Krylov subspace iterations. The new estimates can also be reformulated and applied to invert‐block‐Lanczos methods for solving generalized matrix eigenvalue problems.

Estimating electromechanical oscillation modes from synchrophasor measurements in bulk power grids using FSSI

Multi-order stochastic subspace identification (MOSSI) has been extensively used to estimate the electromechanical oscillation modes from probing, ambient and ringdown data. It has been validated with good performances, while the computational burden is still a major obstacle. This study develops a fast iterative MOSSI (FSSI) approach for computational enhancement of MOSSI in mode estimation. In the proposed approach, an initial cluster of eigenvalues is formulated through FSSI with repetitive calculations (RCs), and electromechanical oscillation mode separation (EOMS) is utilised to discriminate the electromechanical modes. The RCs within the FSSI is calculated through changing the model order successively over the defined range given by a mean of singular values based order determination strategy. Additionally, the proposed approach is highly reliable against prevalent measurement noises owing to RCs and the EOMS. The performance of the proposed method is evaluated in Kundur's two-area test system by comparing with MOSSI, Prony and autoregressive moving average exogenous. Its applicability for both the ringdown and ambient data is also demonstrated with the phasor measurement units field-measurement data from the China Southern Power Grid. The results confirmed the accuracy, robustness and efficiency of the proposed approach for oscillation mode estimation.

Telescopic Projective Integration for Linear Kinetic Equations with Multiple Relaxation Times

We study a general, high-order, fully explicit numerical method for simulating kinetic equations with a BGK-type collision model with multiple relaxation times. In that case, the problem is stiff and its spectrum consists of multiple separated eigenvalue clusters. Projective integration methods are explicit integration schemes that first take a few small (inner) steps with a simple, explicit method, after which the solution is extrapolated forward in time over a large (outer) time step. These are very efficient schemes, provided there are only two clusters of eigenvalues, one corresponding to a single fast relaxation time scale, and one corresponding to the slow macroscopic dynamics. Here, we show how telescopic projective integration can be used to efficiently integrate kinetic equations with multiple relaxation times. Telescopic projective integration generalizes the idea of projective integration by constructing a hierarchy of projective levels. The main idea is to adjust the size of the inner time step at each level to one of the relaxation time scales. We show that the size of the outer time step, as well as the required number of inner steps at each level, does not depend on the stiffness of the problem. The computational cost of the method depends on the stiffness of the problem only via the number of projective levels. For problems with a fixed number of well-separated spectral clusters, the number of projective levels is independent of the stiffness, and the computational cost of telescopic projective integration is independent of the stiffness. For problems with a time-varying spectrum, the number of projective levels grows logarithmically with the stiffness. We illustrate numerically that, also in that case, the resulting computational cost is acceptable.

DDoS Attack Detection Based on Self-organizing Mapping Network in Software Defined Networking

The software defined networking is a new kind of network architecture, the programmability of SDN enables hackers to easily launch DDoS attack on the network through software programming. To solve the problem, a DDoS attack detection scheme based on self-organizing mapping network in the software defined networking was proposed. The first is to give an early warning according to the probability of occurrence of Packet_In event. If the threshold value is exceeded, the characteristics of the flow are calculated and the self-organizing mapping network is used for clustering of eigenvalues to finally detect the DDoS attack flow. The experimental results showed that the DDoS attack detection scheme based on self-organizing mapping network was superior to the comparison scheme in detection rate and false alarm rate.

A Korovkin-type theory for non-self-adjoint Toeplitz operators

We consider the Frobenius optimal preconditioners for Toeplitz operators with complex-valued symbols. We prove Korovkin-type theorems for C⁎-algebra generated by complex-valued continuous periodic symbols corresponding to Toeplitz operators under various modes of convergence in eigenvalue cluster sense. This generalizes the existing results for C⁎-algebra generated by real-valued symbols. In some special cases, the optimal preconditioners can be chosen from matrix algebras with faster convergence rate for the corresponding preconditioned linear systems. We will also consider the linear positive operators associated with the eigenvalues of these preconditioners and obtain a Korovkin-type theorem for such operators as an application of our results.

Semiclassical Szegö Limit of Eigenvalue Clusters for the Hydrogen Atom Zeeman Hamiltonian

We prove a limiting eigenvalue distribution theorem (LEDT) for suitably scaled eigenvalue clusters around the discrete negative eigenvalues of the hydrogen atom Hamiltonian formed by the perturbation by a weak constant magnetic field. We study the hydrogen atom Zeeman Hamiltonian $H_V(h,B) = (1/2)( - i h {\mathbf \nabla} - {\mathbf A}(h))^2 - |x|^{-1}$, defined on $L^2 (R^3)$, in a constant magnetic field ${\mathbf B}(h) = {\mathbf \nabla} \times {\mathbf A}(h)=(0,0,\epsilon(h)B)$ in the weak field limit $\epsilon(h) \rightarrow 0$ as $h\rightarrow{0}$. We consider the Planck's parameter $h$ taking values along the sequence $h=1/(N+1)$, with $N=0,1,2,\ldots$, and $N\rightarrow\infty$. We prove a semiclassical $N \rightarrow \infty$ LEDT of the Szeg\"o-type for the scaled eigenvalue shifts and obtain both ({\bf i}) an expression involving the regularized classical Kepler orbits with energy $E=-1/2$ and ({\bf ii}) a weak limit measure that involves the component $\ell_3$ of the angular momentum vector in the direction of the magnetic field. This LEDT extends results of Szeg\"o-type for eigenvalue clusters for bounded perturbations of the hydrogen atom to the Zeeman effect. The new aspect of this work is that the perturbation involves the unbounded, first-order, partial differential operator $w(h, B) = \frac{(\epsilon(h)B)^2}{8} (x_1^2 + x_2^2) - \frac{ \epsilon(h)B}{2} hL_3 ,$ where the operator $hL_3$ is the third component of the usual angular momentum operator and is the quantization of $\ell_3$. The unbounded Zeeman perturbation is controlled using localization properties of both the hydrogen atom coherent states $\Psi_{\alpha,N}$, and their derivatives $L_3(h)\Psi_{\alpha,N}$, in the large quantum number regime $N\rightarrow\infty$.

An a posteriori estimator of eigenvalue/eigenvector error for penalty-type discontinuous Galerkin methods

We provide an abstract framework for analyzing discretization error for eigenvalue problems discretized by discontinuous Galerkin methods such as the local discontinuous Galerkin method and symmetric interior penalty discontinuous Galerkin method. The analysis applies to clusters of eigenvalues that may include degenerate eigenvalues. We use asymptotic perturbation theory for linear operators to analyze the dependence of eigenvalues and eigenspaces on the penalty parameter. We first formulate the DG method in the framework of quadratic forms and construct a companion infinite dimensional eigenvalue problem. With the use of the companion problem, the eigenvalue/vector error is estimated as a sum of two components. The first component can be viewed as a “non-conformity” error that we argue can be neglected in practical estimates by properly choosing the penalty parameter. The second component is estimated a posteriori using auxiliary subspace techniques, and this constitutes the practical estimate.

A Dai-Liao conjugate gradient algorithm with clustering of eigenvalues

A new value for the parameter in Dai and Liao conjugate gradient algorithm is presented. This is based on the clustering of eigenvalues of the matrix which determine the search direction of this algorithm. This value of the parameter lead us to a variant of the Dai and Liao algorithm which is more efficient and more robust than the variants of the same algorithm based on minimizing the condition number of the matrix associated to the search direction. Global convergence of this variant of the algorithm is briefly discussed.

Optimal convergence of adaptive FEM for eigenvalue clusters in mixed form

It is shown that the h-adaptive mixed finite element method for the discretization of eigenvalue clusters of the Laplace operator produces optimal convergence rates in terms of nonlinear approximation classes. The results are valid for the typical mixed spaces of Raviart-Thomas or Brezzi-Douglas-Marini type with arbitrary fixed polynomial degree in two and three space dimensions.

Effective and Robust Preconditioning of General SPD Matrices via Structured Incomplete Factorization

For general symmetric positive definite (SPD) matrices, we present a framework for designing effective and robust black-box preconditioners via structured incomplete factorization. In a scaling-and-compression strategy, off-diagonal blocks are first scaled on both sides (by the inverses of the factors of the corresponding diagonal blocks) and then compressed into low-rank approximations. ULV-type factorizations are then computed. A resulting prototype preconditioner is always positive definite. Generalizations to practical hierarchical multilevel preconditioners are given. Systematic analysis of the approximation error, robustness, and effectiveness is shown for both the prototype preconditioner and the multilevel generalization. In particular, we show how local scaling and compression control the approximation accuracy and robustness and how aggressive compression leads to efficient preconditioners that can significantly reduce the condition number and improve the eigenvalue clustering. A result is also ...

Asymptotic quadratic convergence of the serial block-Jacobi EVD algorithm for Hermitian matrices

We provide the proof of the asymptotic quadratic convergence of the classical serial block-Jacobi EVD algorithm for Hermitian matrices with well-separated eigenvalues (including the multiple ones) as well as clusters of eigenvalues. At each iteration step, two off-diagonal blocks with the largest Frobenius norm are eliminated which is an extension of the original Jacobi approach to the block case. Numerical experiments illustrate and confirm the developed theory.

A hybrid Jacobi–Davidson method for interior cluster eigenvalues with large null-space in three dimensional lossless Drude dispersive metallic photonic crystals

We study how to efficiently solve the eigenvalue problems in computing band structure of three-dimensional dispersive metallic photonic crystals with face-centered cubic lattices based on the lossless Drude model. The discretized Maxwell equations result in large-scale standard eigenvalue problems whose spectrum contains many zero and cluster eigenvalues, both prevent existed eigenvalue solver from being efficient. To tackle this computational difficulties, we propose a hybrid Jacobi–Davidson method (hHybrid) that integrates harmonic Rayleigh–Ritz extraction, a new and hybrid way to compute the correction vectors, and a FFT-based preconditioner. Intensive numerical experiments show that the hHybrid outperforms existed eigenvalue solvers in terms of timing and convergence behaviors.

Accelerating the shifted Laplace preconditioner for the Helmholtz equation by multilevel deflation

Many important physical phenomena can be described by the Helmholtz equation. We investigate to what extent the convergence of the shifted Laplacian preconditioner for the Helmholtz equation can be accelerated using deflation with multigrid vectors. We therefore present a unified framework for two published algorithms. The first deflates the preconditioned operator and requires no further preconditioning. The second deflates the original operator and combines deflation and preconditioning in a multiplicative fashion. We pursue two scientific contributions. First we show, using a model problem analysis, that both algorithms cluster the eigenvalues. The new and key insight here is that the near-kernel of the coarse grid operator causes a limited set of eigenvalues to shift away from the center of the cluster with a distance proportional to the wave number. This effect is less pronounced in the first algorithmic variant at the expense of a higher computational cost. In the second contribution we quantify for the first time the large amount of reduction in CPU-time that results from the clustering of eigenvalues and the reduction in iteration count. We report to this end on the findings of an implementation in PETSc on two and three-dimensional problems with constant and variable wave number.