Quadratic Eigenvalue Problem Research Articles

A Robust Numerical Algorithm for Computing Maxwell's Transmission Eigenvalue Problems

We study a robust and efficient eigensolver for computing a few smallest positive eigenvalues of the three-dimensional Maxwell's transmission eigenvalue problem. The discretized governing equations by the Nedelec edge element result in a large-scale quadratic eigenvalue problem (QEP) for which the spectrum contains many zero eigenvalues and the coefficient matrices consist of patterns in the matrix form $XY^{-1}Z$, both of which prevent existing eigenvalue solvers from being efficient. To remedy these difficulties, we rewrite the QEP as a particular nonlinear eigenvalue problem and develop a secant-type iteration, together with an indefinite locally optimal block preconditioned conjugate gradient (LOBPCG) method, to sequentially compute the desired positive eigenvalues. Furthermore, we propose a novel method to solve the linear systems in each iteration of LOBPCG. Intensive numerical experiments show that our proposed method is robust, although the desired real eigenvalues are surrounded by complex eigenv...

Definite Quadratic Eigenvalue Problems

Free vibrations of fluid-solids structures are governed by a nonsymmetric eigenvalue problem. This problem can be transformed into a definite quadratic eigenvalue problem. The present paper considers properties of definite problems and variational characterization for definite quadratic eigenvalue problem. We propose improvement of existing methods which determine whether a quadratic pencil Q(λ) is definite. The improved method is reflected in a better localization of start parameters μ and ξ where Q(μ) > 0 >Q(ξ) and μ >ξ.

A Linear-Complexity Finite-Element-Based Eigenvalue Solver for Efficient Analysis of 3-D On-Chip Integrated Circuits

This letter presents a linear-complexity finite-element-based eigenvalue solver for efficient analysis of 3-D on-chip integrated circuits. In this solver, from the original 3-D quadratic eigenvalue problem governing the on-chip circuits, we formulate a new generalized eigenvalue problem to efficiently compute the eigenvalues and eigenvectors of physical interest. We also develop an efficient linear-complexity solution for the matrix equation involved in the solution of the generalized eigenvalue problem. Numerical results demonstrate the accuracy and efficiency of the proposed eigenvalue solver.

A new look at the doubling algorithm for a structured palindromic quadratic eigenvalue problem

SummaryRecently, Guo and Lin [SIAM J. Matrix Anal. Appl., 31 (2010), 2784–2801] proposed an efficient numerical method to solve the palindromic quadratic eigenvalue problem (PQEP) (λ2AT+λQ + A)z = 0 arising from the vibration analysis of high speed trains, where have special structures: both Q and A are, among others, m × m block matrices with each block being k × k (thus, n = mk), and moreover, Q is block tridiagonal, and A has only one nonzero block in the (1,m)th block position. The key intermediate step of the method is the computation of the so‐called stabilizing solution to the n × n nonlinear matrix equation X + ATX−1A = Q via the doubling algorithm. The aim of this article is to propose an improvement to this key step through solving a new nonlinear matrix equation having the same form but of only k × k in size. This new and much smaller matrix equation can also be solved by the doubling algorithm. For the same accuracy, it takes the same number of doubling iterations to solve both the larger and the new smaller matrix equations, but each doubling iterative step on the larger equation takes about 4.8 as many flops than the step on the smaller equation. Replacing Guo's and Lin's key intermediate step by our modified one leads to an alternative method for the PQEP. This alternative method is faster, but the improvement in speed is not as dramatic as just for solving the respective nonlinear matrix equations and levels off as m increases. Numerical examples are presented to show the effectiveness of the new method. Copyright © 2014 John Wiley & Sons, Ltd.

On Spectral Analysis and a Novel Algorithm for Transmission Eigenvalue Problems

The transmission eigenvalue problem, besides its critical role in inverse scattering problems, deserves special interest of its own due to the fact that the corresponding differential operator is neither elliptic nor self-adjoint. In this paper, we provide a spectral analysis and propose a novel iterative algorithm for the computation of a few positive real eigenvalues and the corresponding eigenfunctions of the transmission eigenvalue problem. Based on approximation using continuous finite elements, we first derive an associated symmetric quadratic eigenvalue problem (QEP) for the transmission eigenvalue problem to eliminate the nonphysical zero eigenvalues while preserve all nonzero ones. In addition, the derived QEP enables us to consider more refined discretization to overcome the limitation on the number of degrees of freedom. We then transform the QEP to a parameterized symmetric definite generalized eigenvalue problem (GEP) and develop a secant-type iteration for solving the resulting GEPs. Moreover, we carry out spectral analysis for various existence intervals of desired positive real eigenvalues, since a few lowest positive real transmission eigenvalues are of practical interest in the estimation and the reconstruction of the index of refraction. Numerical experiments show that the proposed method can find those desired smallest positive real transmission eigenvalues accurately, efficiently, and robustly.

Multi-level homotopy perturbation and projection techniques for the reanalysis of quadratic eigenvalue problems: The application of stability analysis

Complex eigenvalue analysis is widely used to investigate the stability of a dynamical system with frictional contact. For finite element models, iterative solvers are needed to precisely calculate complex modes and eigenvalues. However, in cases such as reanalysis studies, optimization or uncertainty propagation processes, computational cost can quickly become too time consuming. For multiple samplings, two methods combining homotopy perturbation and projection techniques are proposed for the reanalysis of quadratic eigenvalue problems. To highlight the efficiency of the proposed methods, a complete numerical application including nominal and perturbed solution calculations, coalescence graph and parametric analysis, is performed. The precision of results and computational time are compared with those obtained using commercial software.

Calculation of eigenpair derivatives for symmetric quadratic eigenvalue problem with repeated eigenvalues

In this paper, we consider computing the derivatives of the semisimple eigenvalues and corresponding eigenvectors of symmetric quadratic eigenvalue problem. In the proposed method, the eigenvector derivatives of the symmetric quadratic eigenvalue problem are divided into a particular solution and a homogeneous solution; a simplified method is given to calculate the particular solution by solving a linear system with nonsingular coefficient matrix, the method is numerically stable and efficient. Two numerical examples are included to illustrate the validity of the proposed method.

Stress concentration phenomena in the vicinity of composite plate-doubler junctions by a layerwise analysis approach

This paper presents an analysis method for the determination of displacements, strains and stresses in a laminated composite plate subjected to tensile load that is reinforced by a doubler. The analysis approach consists of two parts. Firstly, a ‘global’ solution that is based on Classical Laminated Plate Theory (CLPT) is introduced. Secondly, a ‘local’ model is derived that enables the assessment of the three-dimensional stress state near the plate-doubler junction. The local solution employs a discretization of the physical layers into a number of mathematical layers which necessitates the numerical treatment of a quadratic eigenvalue problem. Consequently, the current approach can be classified as being a semi-analytical layerwise analysis method. Results are generated for several different plate-doubler configurations, and it is found that the current analysis model delivers excellent results when compared to finite element simulations, however with only a fraction of the computational time and effort that are needed for the FEM analyses.

Parametric studies on bending vibration of axially-loaded twisted Timoshenko beams with locally distributed Kelvin–Voigt damping

Using the Timoshenko beam theory and applying Hamilton׳s principle, the present study establishes bending–bending vibrations of an axially loaded twisted beam with locally distributed Kelvin–Voigt damping. The equations of motion of the twisted beam are derived in the twist coordinate frame. Then, a finite element method is used to reduce the equations of motion into linear second-order ordinary differential equations with constant coefficients. A quadratic eigenvalue problem of a damped system is formulated to study the effects of the twist angle, damping amount, size and location of damped segment, axial load and restraint types on the eigenfrequency of the damped twisted beams. It is believed that the present model is valuable for the parametric studies to understand better the various variables of the damped twisted beam affecting its vibration characteristics.

Application of the alternating direction method for an inverse monic quadratic eigenvalue problem

This paper shows that the alternating direction method can be used to solve the inverse monic quadratic eigenvalue problem (IMQEP) with the semidefiniteness and sparsity requirements. The results of numerical examples show that the proposed method works well.

Two dimensional autoregressive estimation from noisy observations as a quadratic eigenvalue problem

We consider the problem of two dimensional (2-D) autoregressive (AR) parameter estimation in the presence of observation noise. The proposed method is based on Yule-Walker Equations. We express the Yule Walker equations as a quadratic eigenvalue problem then by solving these equations, the parameters of the signal and noise are estimated. We also apply the proposed method to (2-D) spectrum estimation of the sinusoidal signals in noise. The performance of the proposed method is evaluated by computer simulation examples.

Accurate vectorial finite element mode solver for magneto-optic and anisotropic waveguides.

In this work, a dielectric waveguide mode solver is presented considering a general nonreciprocal permittivity tensor. The proposed method allows us to investigate important cases of practical interest in the field of integrated optics, such as magneto-optical isolators and anisotropic waveguides. Unlike the earlier developed mode solver, our approach allows for the precise computation of both forward and backward propagating modes in the nonreciprocal case, ensuring high accuracy and computational efficiency. As a result, the nonreciprocal loss/phase shift can be directly computed, avoiding the use of the perturbation method. To compute the electromagnetic modes, the Rayleigh-Ritz functional is derived for the non-self adjoint case, it is discretized using the node-based finite element method and the penalty function is added to remove the spurious solutions. The resulting quadratic eigenvalue problem is linearized and solved in terms of the propagation constant for a given frequency (i.e., γ-formulation). The main benefits of this formulation are that it avoids the time-consuming iterations and preserves the matrix sparsity. Finally, the method is used to study two examples of integrated optical isolators based on nonreciprocal phase shift and nonreciprocal loss effect, respectively. The developed method is then compared with the perturbation approach and its simplified formulation based on semivectorial approximation.

Mode decomposition of nonlinear eigenvalue problems and application in flow stability

Direct numerical simulations are carried out with different disturbance forms introduced into the inlet of a flat plate boundary layer with the Mach number 4.5. According to the biorthogonal eigenfunction system of the linearized Navier-Stokes equations and the adjoint equations, the decomposition of the direct numerical simulation results into the discrete normal mode is easily realized. The decomposition coefficients can be solved by doing the inner product between the numerical results and the eigenfunctions of the adjoint equations. For the quadratic polynomial eigenvalue problem, the inner product operator is given in a simple form, and it is extended to an Nth-degree polynomial eigenvalue problem. The examples illustrate that the simplified mode decomposition is available to analyze direct numerical simulation results.

Convergence behavior and steady state response of a rib-stiffened, layered plate structure subjected to high frequency acoustic loading

An existing analytical, frequency domain solution for wave propagation in coated, ribbed, three-dimensional elastic layered plates excited by acoustic plane waves provides fast solutions for high frequency excitations. The solution methodology, which is found to be numerically unstable under certain conditions, contains an Ansatz for a particular wave number expansion in the direction of periodicity. Evidence is presented to show that the numerical instability is due to the specific choice of the wave number basis. In order to provide a remedy while retaining the positive aspects of the solution methodology, we determine the set of admissible propagating (and attenuating) waves via an eigenvalue analysis. Several approaches exist to determine the admissible waves of structures with periodicity. The Wave Guide Finite Element (WFE) method leads to a two parameter, nonlinear eigenvalue problem, which is difficult to solve. The Scale Independent Element (SIE) formulation results in a two-parameter quadratic eigenvalue problem and overcomes the numerical issues of the WFE method. This study examines available methods in determining the admissible waves and compares them with those established using the dispersion relation. The computed admissible waves are then compared with the aforementioned Ansatz.

An augmented Lagrangian dual optimization approach to the [formula omitted]-weighted model updating problem

Model updating for the quadratic eigenvalue problem aims to update the model Q(λ):=λ2M+λC+K by given eigendata. In this paper, we consider the H-weighted model updating problem which can not only preserve the symmetry and definiteness of the original model but also express our confidence in the original model through assigning different confidence weights. We propose an augmented Lagrangian dual method for the H-weighted model updating problem. Under some mild assumptions, our method is shown to converge at least linearly. Numerical results illustrate the effectiveness of our method. In addition, we compare our method with the semi-definite programming (SDP) method. Numerical results illustrate that when the scale of the model becomes large our method still works but the SDP method failed to converge.

Fast algorithms for computing the distance to instability of nonlinear eigenvalue problems, with application to time-delay systems

A continuous dynamical system is stable if all eigenvalues lie strictly in the left half of the complex plane. However, this is not a robust measure because stability is no longer guaranteed when the system parameters are slightly perturbed. Therefore, the pseudospectrum of a matrix and its pseudospectral abscissa are studied. Mostly, one is often interested in computing the distance to instability, because it is a robust measure for stability against perturbations. As a first contribution, this paper presents two iterative methods for computing the distance to instability, considering complex perturbations. The first one is based on locating a zero of the pseudospectral abscissa function. This method is particularly suitable for large sparse matrices as it is based on repeated eigenvalue computations, where the original matrix is perturbed with a rank one matrix. The second method is based on a recently proposed global optimization technique. The advantages of both methods can be combined in a hybrid algorithm. As a second contribution we show that the methods apply to a broad class of nonlinear eigenvalue problems, in particular eigenvalue problems inferred from linear delay-differential equations, and, therefore, they are useful for a wide range of problems. In the numerical examples the standard eigenvalue problem, the quadratic eigenvalue problem and the delay eigenvalue problem are addressed.

General Solutions for a Class of Inverse Quadratic Eigenvalue Problems

AbstractBased on various matrix decompositions, we compare different techniques for solving the inverse quadratic eigenvalue problem, where n × n real symmetric matrices M, C and K are constructed so that the quadratic pencil Q(λ) = λ2M + λC + K yields good approximations for the given k eigenpairs. We discuss the case where M is positive definite for 1 ≤ k ≤ n, and a general solution to this problem for n + 1 ≤ k ≤ 2n. The efficiency of our methods is illustrated by some numerical experiments.

Quadratic inverse eigenvalue problem for damped gyroscopic systems

In this paper, we discuss the quadratic inverse eigenvalue problem for damped gyroscopic systems, that is, we are to construct a damped gyroscopic second order system having given 2 n eigenvalues a...

On the Inverse Symmetric Quadratic Eigenvalue Problem

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.

Solutions of a Quadratic Inverse Eigenvalue Problem for Damped Gyroscopic Second-Order Systems

Givenkpairs of complex numbers and vectors (closed under conjugation), we consider the inverse quadratic eigenvalue problem of constructingn×nreal matricesM,D,G, andK, whereM>0,KandDare symmetric, andGis skew-symmetric, so that the quadratic pencilQ(λ)=λ2M+λ(D+G)+Khas the givenkpairs as eigenpairs. First, we construct a general solution to this problem withk≤n. Then, with the special propertiesD=0andK<0, we construct a particular solution. Numerical results illustrate these solutions.