Quadratic Eigenvalue Problem Research Articles

Conjugate gradient-like algorithms for constrained operator equation related to quadratic inverse eigenvalue problems

As is well known, linear operator equations have wide applications in many areas of engineering and applied mathematics. In the present paper, we are interested in solving the linear operator equation $$\begin{aligned} {\mathscr {F}}(J)+{\mathscr {G}}(K)+{\mathscr {H}}(L)=N, \end{aligned}$$ where J, K and L should be partially bisymmetric under a prescribed submatrix constraint. Three conjugate gradient-like algorithms are derived for solving this constrained operator equation including the Lyapunov, Stein and Sylvester matrix equations and the quadratic inverse eigenvalue problem as special cases. The algorithms converge to the solutions of the linear operator equation within a finite number of iterations in the absence of round-off errors. At the end, the accuracy and efficiency of the introduced algorithms are demonstrated numerically with three examples.

Backward error analysis of linearizing–balancing strategies for heavily damped quadratic eigenvalue problem

A classical approach for solving quadratic eigenvalue problem (QEP) is via linearization. However, it can suffer from numerical instability when the norms of coefficient matrices vary widely. Two strategies of Betcke’s balancing for heavily damped QEP are considered. One strategy is to balance matrix polynomial before linearizing (called balancing–linearizing for short). The other strategy is first to linearize the matrix polynomial, then balancing (called linearizing–balancing for short). We analyze the backward error of approximate eigenpairs computed by these two strategies, and find that the backward error of approximate eigenpairs by linearizing–balancing is smaller than that by balancing–linearizing under relatively mild conditions. Numerical experiments are presented to demonstrate the advantages of linearizing-balancing.

Computation of normal modes based on a coupled finite/infinite element formulation

The computation of eigenfrequencies and corresponding normal modes in exterior acoustic domains is presented based on a coupled finite/infinite element approach. First, the eigenvalue problem is solved in 2D elliptical domains. The applicability of normal modes is discussed in the example of sonic crystals. It can be shown that the sound-insulating effect of sonic crystals and individual meta-atoms in the free field can be related to certain normal modes in exterior acoustics. The numerical results provide a new point of view and physical insights into the effects and underlying physics of sound insulation by finite periodic sonic crystals and acoustic meta-atoms in the free field. Second, the quadratic eigenvalue problem of a hollow sphere submerged in water is addressed by a contour integral method based on resolvent moments. The accuracy of the proposed numerical framework is shown by comparing the eigenfrequencies to those obtained by boundary element discretization as well as finite element discretization in combination with perfectly matched layers. Extensive parameter studies demonstrate the performance of the method with regard to both projection and discretization parameters. The contour integral method is compared to the Rayleigh-Ritz procedure with second-order Krylov subspaces via the residuals of the computed eigenpairs.

Spectral theory for self-adjoint quadratic eigenvalue problems - a review

Many physical problems require the spectral analysis of quadratic matrix polynomials $M\lambda^2+D\lambda +K$, $\lambda \in \mathbb{C}$, with $n \times n$ Hermitian matrix coefficients, $M,\;D,\;K$. In this largely expository paper, we present and discuss canonical forms for these polynomials under the action of both congruence and similarity transformations of a linearization and also $\lambda$-dependent unitary similarity transformations of the polynomial itself. Canonical structures for these processes are clarified, with no restrictions on eigenvalue multiplicities. Thus, we bring together two lines of attack: (a) analytic via direct reduction of the $n \times n$ system itself by $\lambda$-dependent unitary similarity and (b) algebraic via reduction of $2n \times 2n$ symmetric linearizations of the system by either congruence (Section 4) or similarity (Sections 5 and 6) transformations which are independent of the parameter $\lambda$. Some new results are brought to light in the process. Complete descriptions of associated canonical structures (over $\mathbb{R}$ and over $\mathbb{C}$) are provided -- including the two cases of real symmetric coefficients and complex Hermitian coefficients. These canonical structures include the so-called sign characteristic. This notion appears in the literature with different meanings depending on the choice of canonical form. These sign characteristics are studied here and connections between them are clarified. In particular, we consider which of the linearizations reproduce the (intrinsic) signs associated with the analytic (Rellich) theory (Sections 7 and 9).

A Novel Analysis Method Based on Quadratic Eigenvalue Problem for Multirelay Magnetic Coupling Wireless Power Transfer

To simplify the analysis process of the multirelay magnetic coupling wireless power transfer system, a novel analysis method based on the quadratic eigenvalue problem (QEP) has been proposed. The second-order underlying equation of the system is given. The eigensolution represents the inherent properties of the entire system and provides important and useful information. The analytical solutions of currents induced in each coil are expressed in terms of the eigensolution of the corresponding QEP. Real parts and imaginary parts of the eigenvalues represent attenuation coefficients and system resonance frequencies, respectively. The effect of the load resistance on the eigenvalues is studied. It is found that the attenuation coefficient of an eigenvalue increases as the load resistance increases so that it can be ignored. The relationships between eigenvalues and some key frequencies are discussed, such as system resonance frequencies, zero-phase angular frequencies, and constant voltage/current frequencies. The results show that fixed zero-phase angular frequencies and constant current frequencies are equal to the imaginary parts of the eigenvalues with load resistance is equal to zero, and the constant voltage frequencies are equal to the imaginary parts of the complex eigenvalues in a strong damping stage. Finally, experimental results verify the theoretical analysis.

Flexural edge waves in a Kirchhoff plate carrying periodic edge resonators and resting on a Winkler foundation

Flexural edge waves in a Kirchhoff plate, carrying periodically spaced spring–mass​ resonators at its edge and laid on a Winkler foundation, are considered. A spectral element method is applied on a representative unit cell to develop the dynamic stiffness matrix of the structure. In light of structural periodicity, Bloch wave theorem is used to derive a quadratic eigenvalue problem of the wave propagation constants, which are numerically solved for. Frequency bands are analyzed using the attenuation and phase constants. Two types of bandgaps are identified: one is attributed to the existence of attached resonators and the other is due to the fact that these resonators are arranged periodically. It is found that these bandgaps are influenced by the value of the resonator’s natural frequency and the stiffness of the elastic foundation. The analytically realized bandgap structures agree very well with the finite-element obtained dispersion curves by COMSOL.

Relative perturbation theory for quadratic Hermitian eigenvalue problems

In this paper, we derive new relative perturbation bounds for eigenvectors and eigenvalues for regular quadratic eigenvalue problems of the form (λ2M+λC+K)x=0, where M and K are nonsingular Hermitian matrices and C is a general Hermitian matrix. These results are based on new relative perturbation bounds for an equivalent regular Hermitian matrix pair A−λB. The new bounds can be applied to quadratic eigenvalue problems appearing in many relevant applications, such as mechanical models with indefinite damping. The quality of our bounds is demonstrated by several numerical experiments.

Solution of the symmetric band partial inverse eigenvalue problem for the damped mass spring system

ABSTRACT The structured partial quadratic inverse eigenvalue problem (SPQIEP) is to construct the structured quadratic matrix polynomial using the partial eigendata. The structures arising in physical applications include symmetry, band (tridiagonal, diagonal, pentagonal) etc. The construction of the structured matrix polynomial is the most difficult aspect of this problem and the research on structured inverse eigenvalue problem is rare. In this paper, the symmetric band partial quadratic inverse eigenvalue problem (SBPQIEP) for the damped mass spring system is considered. This problem concerns in finding the symmetric band matrices , and C with bandwidth p from m ( ) prescribed eigenpairs so that the corresponding quadratic matrix polynomial has the given eigenpairs as its eigenvalues and eigenvectors. In general, SBPQIEP is very hard to solve due to the additional band structure constraint. We propose a novel method, based on the matrix-vectorization and Kronecker product of matrices for solving this problem. Furthermore, explicit expressions for general solutions are presented. Numerical experiments on a spring mass problem are presented to illustrate the applicability and the practical usefulness of the proposed method.

Matrix LSQR algorithms for solving constrained quadratic inverse eigenvalue problem

The inverse eigenvalue problem appears in many applications such as control design, seismic tomography, exploration and remote sensing, molecular spectroscopy, particle physics, structural analysis, and mechanical system simulation. This paper investigates the matrix form of LSQR methods for solving the quadratic inverse eigenvalue problem with partially bisymmetric matrices under a prescribed submatrix constraint. In order to illustrate the effectiveness and feasibility of our results, one numerical example is presented.

On the Robustness of the Superdirective Beamformer

In microphone array beamforming, a high directional gain is always desired for acoustic noise and reverberation suppression; as a result, the superdirective beamformer has been of great interest in many applications. However, this beamformer is well known to be very sensitive to array imperfections. While much effort has been made to improve its robustness, it is still a major problem. This paper is essentially devoted to the study of the robustness of the superdirective beamformer and derivation of better ways to deal with this important issue. We first prove that any distortionless fixed beamformer can be written as the sum of two orthogonal beamformers, i.e., the sum of the classical delay-and-sum (DS) beamformer and a reduced-rank beamformer. Based on this property, different kinds of robust superdirective beamformers are then developed. We also show that the robust design problem can be transformed into a quadratic eigenvalue problem (QEP), which leads to a solution that achieves the maximum possible directivity factor (DF) while meets the white noise gain (WNG) constraint over a frequency band of interest.

Linear Constrained Rayleigh Quotient Optimization: Theory and Algorithms

We consider the following constrained Rayleigh quotient optimization problem (CRQopt) $$ \min_{x\in \mathbb{R}^n} x^{T}Ax\,\,\mbox{subject to}\,\, x^{T}x=1\,\mbox{and}\,C^{T}x=b, $$ where $A$ is an $n\times n$ real symmetric matrix and $C$ is an $n\times m$ real matrix. Usually, $m\ll n$. The problem is also known as the constrained eigenvalue problem in the literature because it becomes an eigenvalue problem if the linear constraint $C^{T}x=b$ is removed. We start by equivalently transforming CRQopt into an optimization problem, called LGopt, of minimizing the Lagrangian multiplier of CRQopt, and then an problem, called QEPmin, of finding the smallest eigenvalue of a quadratic eigenvalue problem. Although such equivalences has been discussed in the literature, it appears to be the first time that these equivalences are rigorously justified. Then we propose to numerically solve LGopt and QEPmin by the Krylov subspace projection method via the Lanczos process. The basic idea, as the Lanczos method for the symmetric eigenvalue problem, is to first reduce LGopt and QEPmin by projecting them onto Krylov subspaces to yield problems of the same types but of much smaller sizes, and then solve the reduced problems by some direct methods, which is either a secular equation solver (in the case of LGopt) or an eigensolver (in the case of QEPmin). The resulting algorithm is called the Lanczos algorithm. We perform convergence analysis for the proposed method and obtain error bounds. The sharpness of the error bound is demonstrated by artificial examples, although in applications the method often converges much faster than the bounds suggest. Finally, we apply the Lanczos algorithm to semi-supervised learning in the context of constrained clustering.

Homotopy perturbation technique for improving solutions of large quadratic eigenvalue problems: Application to friction-induced vibration

This paper puts forward a projection technique for accurately calculating solutions of large Quadratic Eigenvalue Problem. The aim here is to stabilize the complex eigensolutions whilst reducing residual errors, especially when considering significant damping contribution or asymmetric stiffness matrices. Hence, more confident results can be obtained in the frequency band of interest. To achieve this, high order modes, calculated using the homotopy perturbation technique, are introduced in the projection step of the classical method. This numerical proposal is a generalization of the classical projection, based only on normal modes of the associated undamped problem. To evaluate the efficiency of the suggested method, a finite element application dedicated to a friction-induced vibration problem is investigated.

Linear Prediction-Based DOA Estimator in Non-Uniform Noise

We study the DOA estimation problem in the presence of non-uniform white noise and propose a new method based on a linear prediction approach. The problem is first converted to the uniform noise case by averaging autocorrelations of the array output. The resulting equations can be seen as a quadratic eigenvalue problem, whose solution is the vector of DOA estimations. This new approach is iteration-free and has a low complexity. Simulation results show the effectiveness of the proposed algorithm.

Homogenization approach and Floquet-Bloch theory for wave analysis in fluid-saturated porous media with mesoscopic heterogeneities

We consider fluid-saturated poroelastic media whose the mechanical response is governed by the Biot model relevant to a mesoscopic scale. Assuming the material properties being described by periodic functions, to analyze wave propagation in such heterogeneous and anisotropic media, we derive a formulation based on the Floquet-Bloch (FB) wave decomposition which enables to analyze waves within the whole first Brillouin zone associated with the periodic structure. The wave dispersion results obtained by the FB approach are compared with those computed using a model derived by the homogenization based on the asymptotic analysis with respect to the scale parameter. As another new ingredient, the homogenized model is extended to describe media saturated simultaneously by multiple different fluids, so that the model involves new permeability tensors and differs in structure from the model derived earlier. The dispersion analysis by the FB approach leads to a cumbersome quadratic eigenvalue problem to be solved for complex wave numbers. We suggest an efficient filtration strategy to identify the principle propagating modes (the fast and slow compressional waves and the shear waves). For comparison with results of the FB transformation applied at the mesoscopic heterogeneity scale, the homogenized model responses are reconstructed using the corrector results of the homogenization with fixing a finite scale. Numerical examples illustrate very good correspondence of the dispersion results, as computed by both the approaches.

Eigenvalues of the non-backtracking operator detached from the bulk

We describe the non-backtracking spectrum of a stochastic block model with connection probabilities [Formula: see text]. In this regime we answer a question posed in [L. Dall’Amico, R. Couillet and N. Tremblay, Revisiting the Bethe–Hessian: Improved community detection in sparse heterogeneous graphs, in Advances in Neural Information Processing Systems (2019), pp. 4039–4049] regarding the existence of a real eigenvalue “inside” the bulk, close to the location [Formula: see text]. We also introduce a variant of the Bauer–Fike theorem well suited for perturbations of quadratic eigenvalue problems, which could be of independent interest.

Topological split-ring resonator based metamaterials with PT symmetry relying on gain and loss

A one-dimensional metamaterial with parity-time (${\cal PT}$) symmetry that relies on balanced gain and loss is introduced, comprising of magnetically coupled split-ring resonators (SRRs). A particular topology that combines a non-trivial (topological) dimer configuration with a trivial (non-topological) dimer configuration which are separated by a central SRR with neither gain or loss, is investigated. By focusing on the dynamical aspects of such a topological ${\cal PT}$ metamaterial (PTMM), the existence of {\em topologically protected interface states} which are localized at the central SRR is demonstrated numerically. The solution of the corresponding {\em quadratic eigenvalue problem} reveals that the protected state is actually a robust eigenmode of the topological PTMM, whose eigenvalue is isolated in the middle of the gap (mid-gap state) of the two-band frequency spectrum. Direct numerical simulations have been further used to determine the robustness and dynamic stability of these states in the parameter space of the {\em dimerization strength} and the {\em gain-loss coefficient}.

Effect of wear on frictionally excited thermoelastic instability: A finite element approach

A finite element model for the effect of wear on thermoelastic instability (TEI) is developed by combining the equations of thermoelasticity, the classical wear law, along with the conforming contact conditions. The method is based on a two-dimensional, frictional sliding model with a bimaterial interface and a simplified geometry of finite thickness. An assumption of the solution in the perturbation form leads to a quadratic eigenvalue problem. The existing analytical solutions for two half planes are employed to validate the numerical solutions for several representative scenarios, including a limiting case in the absence of wear. The analytical solutions are also sought for the special cases when one of the materials is a nonconductor and when the two materials are identical, for the purpose of comparison. In general, good agreements between the numerical and analytical approaches have been obtained. However, the discrepancies exist when the wear rates of the two materials are close to each other and when the wear rates are significantly greater than the critical rate. It is confirmed through this study that wear may suppress or amplify the effect of TEI depending on the thermomechanical properties of the materials, which is consistent with the recent research findings on the same topic via an analytical approach.

An analytic methodology to determine generators redispatch for proactive damping of critical electromechanical oscillations

This paper presents a model-based method for applying online proactive generators redispatch to improve damping of the critical electromechanical oscillations of power system. The proposed method comprises two stages: 1) monitoring modal characteristics of oscillatory modes in ambient condition, and 2) applying generators redispatch based on sensitivities of the critical mode to the generators active power changes using a new analytic method. An online identification method such as error feedback lattice recursive least square adaptive filter is applied for online estimation of the oscillatory modes. Then, whenever the damping ratio of an identified mode is less than a preset threshold, its sensitivities to the generators active power changes are calculated in terms of power system model parameters and power flow variables, which can be obtained via state estimation or measured directly by phasor measurement units. The formulae for calculating these sensitivity factors are developed considering 3rd order dynamic model of synchronous generators along with excitation system and power system stabilizer, and ZIP model of loads. The resultant equations are organized as a quadratic eigenvalue problem and the sensitivity factors are calculated in a systematic manner. Nextly, the candidate generators are ranked online by a new index based on sensitivity factors. Finally, using the proposed index, the most effective generators are proactively redispatched to improve the critical mode damping before occurrence of any fault that jeopardizes the system stability. Applicability and accuracy of the proposed formulae are investigated in the IEEE 39-bus New England power system. Effectiveness of the proposed method is also confirmed with time-domain simulation and state space analysis.

Formulation and validation of the shift cell technique for acoustic applications of poro-elastic materials described by the Biot theory

The inclusion of vibroacoustic treatments at early stage of product development through the use of poro-elastic media with periodic inclusions, which exhibit proper dynamic filtering effects, is a powerful strategy for the achievement of lightweight sound packages and represents a convenient solution for manufacturing aspects. This can have different applications in transportation (aerospace, automotive, railway), energy and civil engineering fields, where weight, space and vibroacoustic comfort are still critical challenges. This paper develops the shift cell operator approach as a numerical tool to investigate the dispersion characteristics of periodic poro-elastic media. It belongs to the class of the k(ω) (wave number as a function of the angular frequency) methods and leads to a quadratic eigenvalue problem, even when considering frequency-dependent materials, contrarily to the ω(k) approach that would lead to a non-linear eigenvalue problem for frequency-dependent materials.The full formulation is detailed and the approach is successfully validate for a homogeneous poro-elastic material and a more complex periodic system containing periodic perfectly rigid circular inclusions.

A Benchmark Study on Eigenfrequencies of Fluid-Loaded Structures

In this paper, a coupled finite/infinite element method is applied for computing eigenfrequencies of structures in exterior acoustic domains. The underlying quadratic eigenvalue problem is addressed by a contour integral method based on resolvent moments. The numerical framework is applied to an academic example of a hollow sphere submerged in water. Comparisons of the computed eigenfrequencies to those obtained by boundary element discretizations as well as finite element discretizations in conjunction with perfectly matched layers verify the proposed numerical framework. Furthermore, extensive parameter studies are carried out illustrating the performance of the method with regard to both projection and discretization parameters. Finally, we point out that the proposed method achieves significantly smaller residuals of the computed eigenpairs than the Rayleigh Ritz procedure with second-order Krylov subspaces.