Polynomial Eigenvalue Problem Research Articles

The tropical scaling for the polynomial eigenvalue problem solved by a contour integral method

AbstractThe contour integral method is a class of efficient methods for computing partial eigenvalues of polynomial eigenvalue problem (PEP). Among them, the recently developed Sakurai–Sugiura method with Rayleigh–Ritz projection (SS‐RR) method has received much attention for its effectiveness. However, the SS‐RR method may suffer from numerical instability when the coefficient matrices of the projected PEP vary widely in norm. To improve the numerical stability, we incorporate the tropical scaling technique into the SS‐RR method and establish upper bounds for the backward error of an approximate eigenpair of the original eigenvalue problem. These bounds shed light on mechanism that the tropical scaling improves the numerical stability of the original SS‐RR method. Numerical experiments show that the actual backward errors can be successfully reduced by scaling and the bounds can predict well the errors occurring before and after scaling.

Computing leaky modes of optical fibers using a FEAST algorithm for polynomial eigenproblems

An efficient contour integral technique to approximate a cluster of nonlinear eigenvalues of a polynomial eigenproblem, circumventing certain large inversions from a linearization, is presented. It is applied to the nonlinear eigenproblem that arises from a frequency-dependent perfectly matched layer. This approach is shown to result in an accurate method for computing leaky modes of optical fibers. Extensive computations on an antiresonant fiber with a complex transverse microstructure are reported. This structure is found to present substantial computational difficulties: Even when employing over one million degrees of freedom, the fiber model appears to remain in a preasymptotic regime where computed confinement loss values are likely to be off by orders of magnitude. Other difficulties in computing mode losses, together with practical techniques to overcome them, are detailed.

NEP

SLEPc is a parallel library for the solution of various types of large-scale eigenvalue problems. Over the past few years, we have been developing a module within SLEPc, called NEP, that is intended for solving nonlinear eigenvalue problems. These problems can be defined by means of a matrix-valued function that depends nonlinearly on a single scalar parameter. We do not consider the particular case of polynomial eigenvalue problems (which are implemented in a different module in SLEPc) and focus here on rational eigenvalue problems and other general nonlinear eigenproblems involving square roots or any other nonlinear function. The article discusses how the NEP module has been designed to fit the needs of applications and provides a description of the available solvers, including some implementation details such as parallelization. Several test problems coming from real applications are used to evaluate the performance and reliability of the solvers.

The subspace iteration method for nonlinear eigenvalue problems occurring in the dynamics of structures with viscoelastic elements

The paper presents an extension of the subspace iteration method for application in systems with viscoelastic damping elements, which are described by both classical and fractional models. The presented method enables determination of only a certain number of eigenvalues and associated eigenvectors. This is very useful, especially when the considered problem has many degrees of freedom, in which case it would be very time-consuming or even impossible to determine all the eigenvalues. At the same time, from the engineering point of view, it is not necessary to know all the eigenvalues. In this paper, the subspace iteration method is used for the first time to solve the nonlinear eigenproblems which appear in the dynamic analysis of systems with viscoelastic damping elements. The proposed solution consists of assuming the number of eigenvalues to be determined, taking the initial point of iteration and then solving the nonlinear reduced eigenproblem with Hermitian matrices in each iterative loop. The solution to the reduced eigenproblem is obtained using the continuation method, and it is an additional novelty in the paper. The correctness and effectiveness of the proposed approach are illustrated with numerical examples.

Calculation of conditions for maintaining an ICRF-plasma using a self-consistent model

The article proposes a new approach to calculating the strength of the magnetic field on the inner wall of the discharge chamber, which is necessary to maintain a steady state of a low-pressure ICRF discharge. The model is treate as a nonlinear eigenproblem. The influence of the third type boundary conditions for electron density as well as and the nonlinear boundary conditions for electrical strength is considered. This approach makes it possible solving two problems of designing ICRF plasma torches: for a given electron density in the discharge find the magnetic field strength that ensures the maintenance of the discharge, or, conversely, at a given magnetic field strength, determine the value of the electron density that can be created in the discharge. In addition, the radial distributions of the electric and magnetic fields and the electron concentration can be determined.

The Homogeneous B 1 Model as Polynomial Eigenvalue Problem

The homogeneous version of the B 1 leakage model is a non-linear eigenvalue problem which is generally solved iteratively by a root-finding algorithm, combined to the supplementary eigenvalue problem of the multiplication factor. This problem is widely used for ordinary cross section preparation in reactor analysis. Our work approximates this problem with a polynomial eigenvalue problem, which can be easily transformed into an ordinary linear generalized eigenproblem of size equal to the initial one multiplied by the polynomial degree used for the approximation of a transcendental function. This procedure avoids recurring to numerical root-finding methods supported by extra eigenvalue problems. The solution of the fundamental buckling with increasing approximation order is compared to the reference value obtained by inverse iterations.

Modal analysis of coupled soil-structure systems

This paper presents an efficient approach for the modal analysis of coupled soil-structure systems, for which the dynamic response is strongly influenced by the embedment in the soil. The methodology is based on a finite element-perfectly matched layer model that allows for the derivation of frequency-independent system matrices and the computation of the modal properties of the coupled system. This is achieved by solving a nonlinear eigenproblem using a Compact Rational Krylov (CoRK) eigensolver. A procedure is developed to sort the computed eigenpairs, filter out the spurious modes of the system which are related to the near-field and truncated far-field soil subdomains and select the physical structural modes of system. The proposed method can be used in the dynamic assessment and structural identification of strongly coupled soil-structure systems such as fully or partially buried structures and allows for the interpretation of experimentally identified modal properties of these systems, especially in the presence of highly damped or closely spaced coupled modes. The applicability and the scalability of the proposed approach for 2D and 3D problems is demonstrated in two case studies.

Structured strong $\boldsymbol{\ell}$-ifications for structured matrix polynomials in the monomial basis

In the framework of Polynomial Eigenvalue Problems (PEPs), most of the matrix polynomials arising in applications are structured polynomials (namely, (skew-)symmetric, (skew-)Hermitian, (anti-)palindromic, or alternating). The standard way to solve PEPs is by means of linearizations. The most frequently used linearizations belong to general constructions, valid for all matrix polynomials of a fixed degree, known as companion linearizations. It is well known, however, that it is not possible to construct companion linearizations that preserve any of the previous structures for matrix polynomials of even degree. This motivates the search for more general companion forms, in particular companion $\ell$-ifications. In this paper, we present, for the first time, a family of (generalized) companion $\ell$-ifications that preserve any of these structures, for matrix polynomials of degree $k=(2d+1)\ell$. We also show how to construct sparse $\ell$-ifications within this family. Finally, we prove that there are no structured companion quadratifications for quartic matrix polynomials.

Revealing stable and unstable modes of denoisers through nonlinear eigenvalue analysis

In this paper, we propose to analyze stable and unstable modes of black-box image denoisers through nonlinear eigenvalue analysis. We aim to find input images for which the denoiser output is proportional to the input. We treat this as a generalized nonlinear eigenproblem. Potential implications are wide, as most image processing algorithms can be viewed as black-box operators. We introduce a generalized nonlinear power-method to solve eigenproblems for such operators. This allows us to reveal stable modes of the denoiser: optimal inputs, achieving superior PSNR in noise removal. Analogously to the linear case, such stable modes show coarse structures and correspond to large eigenvalues. We also provide a method to generate unstable modes, which the denoiser suppresses strongly, which are textural with small eigenvalues. We validate the method using total-variation (TV) and demonstrate it on the EPLL (Zoran–Weiss) and the Non-local means denoisers. Finally, we suggest an encryption–decryption application.

Analytical solutions to some generalized and polynomial eigenvalue problems

AbstractIt is well-known that the finite difference discretization of the Laplacian eigenvalue problem −Δu=λuleads to a matrix eigenvalue problem (EVP)Ax=λxwhere the matrixAis Toeplitz-plus-Hankel. Analytical solutions to tridiagonal matrices with various boundary conditions are given in a recent work of Strang and MacNamara. We generalize the results and develop analytical solutions to certain generalized matrix eigenvalue problems (GEVPs)Ax=λBxwhich arise from the finite element method (FEM) and isogeometric analysis (IGA). The FEM matrices are corner-overlapped block-diagonal while the IGA matrices are almost Toeplitz-plus-Hankel. In fact, IGA with a correction that results in Toeplitz-plus-Hankel matrices gives a better numerical method. In this paper, we focus on finding the analytical eigenpairs to the GEVPs while developing better numerical methods is our motivation. Analytical solutions are also obtained for some polynomial eigenvalue problems (PEVPs). Lastly, we generalize the eigenvector-eigenvalue identity (rediscovered and coined recently for EVPs) for GEVPs and derive some trigonometric identities.

A Rational Even-IRA Algorithm for the Solution of $T$-Even Polynomial Eigenvalue Problems

In this work we present a rational Krylov subspace method for solving real large-scale polynomial eigenvalue problems with $T$-even (that is, symmetric/skew-symmetric) structure. Our method is base...

Multiparameter Eigenvalue Problems and Shift-invariance

We discuss four eigenvalue problems of increasing generality and complexity: rooting a univariate polynomial, solving the polynomial eigenvalue problem, rooting a set of multivariate polynomials and solving multi-parameter eigenvalue problems. In doing so, we provide a unifying framework for solving these eigenvalue problems, where we exploit properties of (block-) (multi-) shift-invariant subspaces and use multi-dimensional realization algorithms.

Nonlinearizing Two-parameter Eigenvalue Problems

We investigate a technique to transform a linear two-parameter eigenvalue problem, into a nonlinear eigenvalue problem (NEP). The transformation stems from an elimination of one of the equations in the two-parameter eigenvalue problem, by considering it as a (standard) generalized eigenvalue problem. We characterize the equivalence between the original and the nonlinearized problem theoretically and show how to use the transformation computationally. Special cases of the transformation can be interpreted as a reversed companion linearization for polynomial eigenvalue problems, as well as a reversed (less known) linearization technique for certain algebraic eigenvalue problems with square-root terms. Moreover, by exploiting the structure of the NEP we present algorithm specializations for NEP methods, although the technique also allows general solution methods for NEPs to be directly applied. The nonlinearization is illustrated in examples and simulations, with focus on problems where the eliminated equation is of much smaller size than the other two-parameter eigenvalue equation. This situation arises naturally in domain decomposition techniques. A general error analysis is also carried out under the assumption that a backward stable eigenvalue solver method is used to solve the eliminated problem, leading to the conclusion that the error is benign in this situation.

Linearizations for Interpolatory Bases - a Comparison: New Families of Linearizations

One strategy to solve a nonlinear eigenvalue problem $T(\lambda)x=0$ is to solve a polynomial eigenvalue problem (PEP) $P(\lambda)x=0$ that approximates the original problem through interpolation. Then, this PEP is usually solved by linearization. Because of the polynomial approximation techniques, in this context, $P(\lambda)$ is expressed in a non-monomial basis. The bases used with most frequency are the Chebyshev basis, the Newton basis and the Lagrange basis. Although, there exist already a number of linearizations available in the literature for matrix polynomials expressed in these bases, new families of linearizations are introduced because they present the following advantages: 1) they are easy to construct from the matrix coefficients of $P(\lambda)$ when this polynomial is expressed in any of those three bases; 2) their block-structure is given explicitly; 3) it is possible to provide equivalent formulations for all three bases which allows a natural framework for comparison. Also, recovery formulas of eigenvectors (when $P(\lambda)$ is regular) and recovery formulas of minimal bases and minimal indices (when $P(\lambda)$ is singular) are provided. The ultimate goal is to use these families to compare the numerical behavior of the linearizations associated to the same basis (to select the best one) and with the linearizations associated to the other two bases, to provide recommendations on what basis to use in each context. This comparison will appear in a subsequent paper.

Min-max elementwise backward error for roots of polynomials and a corresponding backward stable root finder

A new measure called min-max elementwise backward error is introduced for approximate roots of scalar polynomials p(z). Compared with the elementwise relative backward error, this new measure allows for larger relative perturbations on the coefficients of p(z) that do not participate much in the overall backward error. By how much these coefficients can be perturbed is determined via an associated max-times polynomial and its tropical roots. An algorithm is designed for computing the roots of p(z). It uses a companion linearization C(z)=A−zB of p(z) to which we added an extra zero leading coefficient, and an appropriate two-sided diagonal scaling that balances A and makes B graded in particular when there is variation in the magnitude of the coefficients of p(z). An implementation of the QZ algorithm with a strict deflation criterion for eigenvalues at infinity is then used to obtain approximations to the roots of p(z). Under the assumption that this implementation of the QZ algorithm exhibits a graded backward error when B is graded, we prove that our new algorithm is min-max elementwise backward stable. Several numerical experiments show the superior performance of the new algorithm compared with the MATLAB roots function. Extending the algorithm to polynomial eigenvalue problems leads to a new polynomial eigensolver that exhibits excellent numerical behaviour compared with other existing polynomial eigensolvers, as illustrated by many numerical tests.

The Effectiveness of the Passive Damping System Combining the Viscoelastic Dampers and Inerters

Passive damping systems are energy dissipating devices that are used in building constructions to reduce excessive vibration caused by wind or earthquakes. Recently, the use of inerters, devices using the inertia of the rotating mass, as a passive damping system has become more and more popular. Inerter, regardless of the construction, generates a resistance force that is proportional to the relative acceleration at the terminal ends of the device. The purpose of this study is to evaluate the effectiveness of the viscoelastic (VE) dampers in combination with the inerter. For the derived equations of motion of structures with VE dampers and inerters, the Laplace transformation is applied, which leads to a nonlinear eigenproblem. The solution to the eigenproblem is obtained using the continuation method, also known as the homotopy or as the path following method. It was found that for some cases of inerter connection with ve damper, there are some additional solutions in relation to the degrees of freedom of the structure. Furthermore, the dynamic characteristics of the structure associated with the above-mentioned additional eigenvalue are strongly dependent on the equivalent inerter mass. Damping efficiency is tested by determining changes in the system’s dynamic characteristics: natural frequencies, non-dimensional damping ratios and displacement transfer functions. The presented approach allows determining the appropriate parameters of the inerter and VE damper and their appropriate location on the structures that correspond to the most effective damping. The obtained numerical results confirm the effectiveness of the proposed approach.

A New Approach to Solve Perturbed Symmetric Eigenvalue Problems

The objective of this study is to ef-ciently resolve a perturbed symmetric eigen-value problem, without resolving a completelynew eigenvalue problem. When the size of aninitial eigenvalue problem is large, its multipletimes solving for each set of perturbations can becomputationally expensive and undesired. Thistype of problems is frequently encountered inthe dynamic analysis of mechanical structures.This study deals with a perturbed symmetriceigenvalue problem. It propose to develop atechnique that transforms the perturbed sym-metric eigenvalue problem, of a large size, toa symmetric polynomial eigenvalue problem ofa much reduced size. To accomplish this, weonly need the introduced perturbations, the sym-metric positive-de nite matrices representing theunperturbed system and its rst eigensolutions.The originality lies in the structure of the ob-tained formulation, where the contribution of theunknown eignsolutions of the unperturbed sys-tem is included. The e ectiveness of the pro-posed method is illustrated with numerical tests.High quality results, compared to other existingmethods that use exact reanalysis, can be ob-tained in a reduced calculation time, even if theintroduced perturbations are very signi cant.

Sampling the feasible sets of SDPs and volume approximation

We present algorithmic, complexity, and implementation results on the problem of sampling points in the interior and the boundary of a spectrahedron, that is the feasible region of a semidefinite program. Our main tool is random walks. We define and analyze a set of primitive geometric operations that exploits the algebraic properties of spectrahedra and the polynomial eigenvalue problem and leads to the realization of a broad collection of efficient random walks. We demonstrate random walks that experimentally show faster mixing time than the ones used previously for sampling from spectrahedra in theory or applications, for example Hit and Run. Consecutively, the variety of random walks allows us to sample from general probability distributions, for example the family of log-concave distributions which arise frequently in numerous applications. We apply our tools to compute (i) the volume of a spectrahedron and (ii) the expectation of functions coming from robust optimal control. We provide a C++ open source implementation of our methods that scales efficiently up to dimension 200. We illustrate its efficiency on various data sets.

Observer-based sliding mode control for piezoelectric wing bending-torsion coupling flutter involving delayed output

An observer-based sliding mode control scheme is proposed for suppressing bending-torsion coupling flutter motions of a wing aeroelastic system with delayed output by using the piezoelectric patch actuators. The wing structure is modeled as a thin-walled beam, and the aerodynamics on the wing are computed by the strip theory. For the implementation of the control algorithm, the piezoelectric patch is bonded on the top surface of the beam to act as the actuator. Ignoring the effect of piezoelectric actuators on structural dynamics, only considering the bending moments induced by piezoelectric effects, the corresponding dynamic motion equation is established by using the Lagrange method with the assumed mode method. The flutter speed and frequency of the closed-loop system with time delay are obtained by solving a polynomial eigenvalue problem. An observer-based controller that does not dependent on time delay is developed for suppressing the flutter, and the corresponding gain matrices are obtained by solving linear matrix inequalities. The sufficient condition for the asymptotic stability of the closed-loop system is derived in terms of linear matrix inequalities. The simulation results demonstrate that the proposed control strategy based on the piezoelectric actuator is effective in wing bending-torsion coupling flutter system with a delayed output.

Non-linear eigenvalue problems with GetDP and SLEPc: Eigenmode computations of frequency-dispersive photonic open structures

We present a framework to solve non-linear eigenvalue problems suitable for a Finite Element discretization. The implementation is based on the open-source finite element software GetDP and the open-source library SLEPc. As template examples, we propose and compare in detail different ways to address the numerical computation of the electromagnetic modes of frequency-dispersive objects. This is a non-linear eigenvalue problem involving a non-Hermitian operator. A classical finite element formulation is derived for five different solutions and solved using algorithms adapted to the large size of the resulting discrete problem. The proposed solutions are applied to the computation of the dispersion relation of a diffraction grating made of a Drude material. The important numerical consequences linked to the presence of sharp corners and sign-changing coefficients are carefully examined. For each method, the convergence of the eigenvalues with respect to the mesh refinement and the shape function order, as well as computation time and memory requirements are investigated. The open-source template model used to obtain the numerical results is provided. Details of the implementation of polynomial and rational eigenvalue problems in GetDP are given in the appendix. Program summaryProgram title: NonLinearEVP.proCPC Library link to program files:http://dx.doi.org/10.17632/r57nxxtc62.1Licensing provisions: GNU General Public License 3Programming language: Gmsh (http://gmsh.info), GetDP (http://getdp.info)Nature of problem: Computing the eigenvalues and eigenvectors of electromagnetic wave problems involving frequency-dispersive materials. The resulting eigenvalue problem is non-linear and non-hermitian.Solution method: Finite element method coupled to efficient non-linear eigenvalue solvers: Relevant SLEPc solvers were interfaced to the Finite Element software GetDP. Several linearization schemes are benchmarked.