Solution Of Eigenvalue Problem Research Articles

Exceptional Legendre Polynomials and Confluent Darboux Transformations

Exceptional orthogonal polynomials are families of orthogonal polynomials that arise as solutions of Sturm-Liouville eigenvalue problems. They generalize the classical families of Hermite, Laguerre, and Jacobi polynomials by allowing for polynomial sequences that miss a finite number of ''exceptional'' degrees. In this paper we introduce a new construction of multi-parameter exceptional Legendre polynomials by considering the isospectral deformation of the classical Legendre operator. Using confluent Darboux transformations and a technique from inverse scattering theory, we obtain a fully explicit description of the operators and polynomials in question. The main novelty of the paper is the novel construction that allows for exceptional polynomial families with an arbitrary number of real parameters.

A novel high-performance diplexer for mobile communication base stations

In this paper, a novel design of a diplexer for mobile communication base station is proposed, theoretically investigated and successfully tested. The proposed design of the diplexer is implemented on the base of interdigital filters. Successfully fabricated and tested diplexer operates within frequency bands 2.3–2.4/2.49–2.69 GHz and shows high performances: reflection loss within pass bands is less −20 dB, insertion loss is −0.15 dB, isolation is −30 dB and intermodulation is −124 dBm. Computer-aided design of the diplexer is carried out on the basis of combined technique, which includes filter synthesis using coupling matrices, eigenvalue problem solution for single resonator, interresonator coupling calculation and final full wave simulation and optimization of diplexer. The suggested design is a promising candidate for the implementation of diplexers with narrow, moderate and wide bandwidths for standard mobile communication frequency bands.

Rational Spectral Filters with Optimal Convergence Rate

In recent years, contour-based eigensolvers have emerged as a standard approach for the solution of large and sparse eigenvalue problems. Building upon recent performance improvements through non-linear least square optimization of so-called rational filters, we introduce a systematic method to design these filters by minimizing the worst-case convergence ratio and eliminate the parametric dependence on weight functions. Further, we provide an efficient way to deal with the box-constraints which play a central role for the use of iterative linear solvers in contour-based eigensolvers. Indeed, these parameter-free filters consistently minimize the number of iterations and the number of FLOPs to reach convergence in the eigensolver. As a byproduct, our rational filters allow for a simple solution to load balancing when the solution of an interior eigenproblem is approached by the slicing of the sought after spectral interval.

Computation of free boundary minimal surfaces via extremal Steklov eigenvalue problems

Recently Fraser and Schoen showed that the solution of a certain extremal Steklov eigenvalue problem on a compact surface with boundary can be used to generate a free boundary minimal surface, i.e., a surface contained in the ball that has (i) zero mean curvature and (ii) meets the boundary of the ball orthogonally (doi:10.1007/s00222-015-0604-x). In this paper, we develop numerical methods that use this connection to realize free boundary minimal surfaces. Namely, on a compact surface, Σ, with genus γ and b boundary components, we maximize σj(Σ, g) L(∂Σ, g) over a class of smooth metrics, g, where σj(Σ, g) is the jth nonzero Steklov eigenvalue and L(∂Σ, g) is the length of ∂Σ. Our numerical method involves (i) using conformal uniformization of multiply connected domains to avoid explicit parameterization for the class of metrics, (ii) accurately solving a boundary-weighted Steklov eigenvalue problem in multi-connected domains, and (iii) developing gradient-based optimization methods for this non-smooth eigenvalue optimization problem. For genus γ = 0 and b = 2, …, 9, 12, 15, 20 boundary components, we numerically solve the extremal Steklov problem for the first eigenvalue. The corresponding eigenfunctions generate a free boundary minimal surface, which we display in striking images. For higher eigenvalues, numerical evidence suggests that the maximizers are degenerate, but we compute local maximizers for the second and third eigenvalues with b = 2 boundary components and for the third and fifth eigenvalues with b = 3 boundary components.

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.

Some results on complete Finsler measure spaces

In this paper, we establish a logarithmic Sobolev inequality in complete Finsler measure spaces. As its applications, we prove the existence of solutions of eigenvalue problem for the Finsler Laplacian. Finally, we obtain a Lichnerowicz type theorem which shows that for a complete Finsler measure space with Ric∞≥K>0, the first eigenvalue has lower bound K2.

Eigenfunctions of Ordinary Differential Euler Operators

Asymptotic solutions of the eigenvalue problem for an Euler operator in a neighborhood of a regular singular point are considered. We find a condition under which the asymptotic expansion is free of logarithms. Eigenvalues expressed in terms of elementary functions in the form of a finite sum of quasi-polynomials are obtained for third-order Euler operators and also for commuting Euler operators of sixth and ninth orders. The problem on common eigenfunctions for commuting Euler operators is examined. In the case of operators of rank 2 and 3, it can be reduced to second- and third-order Bessel equations by differential substitutions.

A DRBEM approximation of the Steklov eigenvalue problem

In this study, we propose a novel approach based on the dual reciprocity boundary element method (DRBEM) to approximate the solutions of various Steklov eigenvalue problems. The method consists in weighting the governing differential equation with the fundamental solutions of the Laplace equation where the definition of interior nodes is not necessary for the solution on the boundary. DRBEM constitutes a promising tool to characterize such problems due to the fact that the boundary conditions on part or all of the boundary of the given flow domain depend on the spectral parameter. The matrices resulting from the discretization are partitioned in a novel way to relate the eigenfunction with its flux on the boundary where the spectral parameter resides. The discretization is carried out with the use of constant boundary elements resulting in a generalized eigenvalue problem of moderate size that can be solved at a smaller expense compared to full domain discretization alternatives. We systematically investigate the convergence of the method by several experiments including cases with selfadjoint and non-selfadjoint operators. We present numerical results which demonstrate that the proposed approach is able to efficiently approximate the solutions of various mixed Steklov eigenvalue problems defined on arbitrary domains.

Approximation of PDE eigenvalue problems involving parameter dependent matrices

We discuss the solution of eigenvalue problems associated with partial differential equations that can be written in the generalized form {mathsf {A}}x=lambda {mathsf {B}}x, where the matrices {mathsf {A}} and/or {mathsf {B}} may depend on a scalar parameter. Parameter dependent matrices occur frequently when stabilized formulations are used for the numerical approximation of partial differential equations. With the help of classical numerical examples we show that the presence of one (or both) parameters can produce unexpected results.

Local linearizations of rational matrices with application to rational approximations of nonlinear eigenvalue problems

This paper presents a definition for local linearizations of rational matrices and studies their properties. This definition allows to introduce matrix pencils associated to a rational matrix that preserve its structure of zeros and poles in subsets of any algebraically closed field and also at infinity. This new theory of local linearizations captures and explains rigorously the properties of all the different pencils that have been used from the 1970's until 2020 for computing zeros, poles and eigenvalues of rational matrices. Particular attention is paid to those pencils that have appeared recently in the numerical solution of nonlinear eigenvalue problems through rational approximation.

Flutter characteristics of axially functional graded composite wing system

This paper presents the flutter analysis and optimum design of axially functionally graded box beam cantilever wing section by considering various geometric and material parameters. The coupled dynamic equations of the continuous model of wing system in terms of material and cross-sectional properties are formulated based on extended Hamilton\'s principle. By expressing the lift and pitching moment in terms of plunge and pitch displacements, the resultant two continuous equations are simplified using Galerkin\'s reduced order model. The flutter velocity is predicted from the solution of resultant damped eigenvalue problem. Parametric studies are conducted to know the effects of geometric factors such as taper ratio, thickness, sweep angle as well as material volume fractions and functional grading index on the flutter velocity. A generalized surrogate model is constructed by training the radial basis function network with the parametric data. The optimized material and geometric parameters of the section are predicted by solving the constrained optimal problem using firefly metaheuristics algorithm that employs the developed surrogate model for the function evaluations. The trapezoidal hollow box beam section design with axial functional grading concept is illustrated with combination of aluminium alloy and aluminium with silicon carbide particulates. A good improvement in flutter velocity is noticed by the optimization.

High performance solution of skew-symmetric eigenvalue problems with applications in solving the Bethe-Salpeter eigenvalue problem

We present a high-performance solver for dense skew-symmetric matrix eigenvalue problems. Our work is motivated by applications in computational quantum physics, where one solution approach to solve the Bethe-Salpeter equation involves the solution of a large, dense, skew-symmetric eigenvalue problem. The computed eigenpairs can be used to compute the optical absorption spectrum of molecules and crystalline systems. One state-of-the art high-performance solver package for symmetric matrices is the ELPA (Eigenvalue SoLvers for Petascale Applications) library. We exploit a link between tridiagonal skew-symmetric and symmetric matrices in order to extend the methods available in ELPA to skew-symmetric matrices. This way, the presented solution method can benefit from the optimizations available in ELPA that make it a well-established, efficient and scalable library. The solution strategy is to reduce a matrix to tridiagonal form, solve the tridiagonal eigenvalue problem and perform a back-transformation for eigenvectors of interest. ELPA employs a one-step or a two-step approach for the tridiagonalization of symmetric matrices. We adapt these to suit the skew-symmetric case. The two-step approach is generally faster as memory locality is exploited better. If all eigenvectors are required, the performance improvement is counteracted by the additional back transformation step. We exploit the symmetry in the spectrum of skew-symmetric matrices, such that only half of the eigenpairs need to be computed, making the two-step approach the favorable method. We compare performance and scalability of our method to the only available high-performance approach for skew-symmetric matrices, an indirect route involving complex arithmetic. In total, we achieve a performance that is up to 3.67 times higher than the reference method using Intel’s ScaLAPACK implementation. Our method is freely available in the current release of the ELPA library.

A Survey of Low-Rank Updates of Preconditioners for Sequences of Symmetric Linear Systems

The aim of this survey is to review some recent developments in devising efficient preconditioners for sequences of symmetric positive definite (SPD) linear systems A k x k = b k , k = 1 , … arising in many scientific applications, such as discretization of transient Partial Differential Equations (PDEs), solution of eigenvalue problems, (Inexact) Newton methods applied to nonlinear systems, rational Krylov methods for computing a function of a matrix. In this paper, we will analyze a number of techniques of updating a given initial preconditioner by a low-rank matrix with the aim of improving the clustering of eigenvalues around 1, in order to speed-up the convergence of the Preconditioned Conjugate Gradient (PCG) method. We will also review some techniques to efficiently approximate the linearly independent vectors which constitute the low-rank corrections and whose choice is crucial for the effectiveness of the approach. Numerical results on real-life applications show that the performance of a given iterative solver can be very much enhanced by the use of low-rank updates.

Beyond the limitations of perturbation methods for real random eigenvalue problems using Exceptional Points and analytic continuation

A numerical method is proposed to approximate the solution of parametric eigenvalue problem when the variability of the parameters exceed the radius of convergence of low order perturbation methods. The radius of convergence of eigenvalue perturbation methods, based on Taylor series, is known to decrease when eigenvalues are getting closer to each other. This phenomenon, known as veering in structural dynamics, is a direct consequence of the existence of branch point singularity in the complex plane of the varying parameters where some eigenvalues are defective. When this degeneracy, referred to as Exceptional Point (EP), is close to the real axis, the veering becomes stronger.The main idea of the proposed approach is to combined a pair of eigenvalues to remove this singularity. To do so, two analytic auxiliary functions are introduced and are computed through high order derivatives of the eigenvalue pair with respect to the parameter. This yields a new robust eigenvalue reconstruction scheme which is compared to Taylor and Puiseux series. In all cases, theoretical bounds are established and all approximations are compared numerically on a three degrees of freedom toy model. This system illustrates the ability of the method to handle the vibrations of a structure with a randomly varying parameter. Computationally efficient, the proposed algorithm could also be relevant for actual numerical models of large size, arising from other applications involving parametric eigenvalue problems, e.g., waveguides, rotating machinery or instability problems such as squeal or flutter.

Representation of solutions for Sturm-Liouville eigenvalue problems with generalized fractional derivative.

We analyze fractional Sturm-Liouville problems with a new generalized fractional derivative in five different forms. We investigate the representation of solutions by means of ρ-Laplace transform for generalized fractional Sturm-Liouville initial value problems. Finally, we examine eigenfunctions and eigenvalues for generalized fractional Sturm-Liouville boundary value problems. All results obtained are compared with simulations in detail under different α fractional orders and real ρ values.

Feature selection based on star coordinates plots associated with eigenvalue problems

Feature selection consists of choosing a smaller number of variables to work with when analyzing high-dimensional data sets. Recently, several visualization tools, techniques, and feature relevance measures have been developed in order to help users carry out the feature selection. Some of these approaches are based on radial axes methods, where analysts perform backward feature elimination by discarding features that have a low impact on the visualizations. Similarly, in this paper, we propose a new feature relevance measure for star coordinates plots associated with the class of linear dimensionality reduction mappings defined through the solutions of eigenvalue problems, such as linear discriminant analysis or principal component analysis. We show that the approach leads to enhanced feature subsets for class separation or variance maximization in the plots for numerous data sets of the UCI repository. Lastly, in practice, the tool allows analysts to decide which features to discard by examining their relevance and by taking into account previous domain knowledge.

Spectrum-based Stability Analysis and Stabilization of Time-periodic Time-delay Systems

We develop an eigenvalue-based approach for the stability assessment and stabilization of linear systems with multiple delays and periodic coefficient matrices. Delays and period are assumed commensurate numbers, such that the Floquet multipliers can be characterized as eigenvalues of the monodromy operator and by the solutions of a finite-dimensional non-linear eigenvalue problem, where the evaluation of the characteristic matrix involves solving an initial value problem. We demonstrate that such a dual interpretation can be exploited in a two-stage approach for computing dominant Floquet multipliers, where global approximation is combined with local corrections. Correspondingly, we also propose two novel characterizations of left eigenvectors. Finally, from the nonlinear eigenvalue problem formulation, we derive computationally tractable expressions for derivatives of Floquet multipliers with respect to parameters, which are beneficial in the context of stability optimization. Several numerical examples show the efficacy and applicability of the presented results.

English

English

The two-grid and multigrid discretizations of the <inline-formula><tex-math id="M1">\begin{document}$ C^0 $\end{document}</tex-math></inline-formula>IPG method for biharmonic eigenvalue problem

In this paper, for the biharmonic eigenvalue problem with clamped boundary condition in \begin{document}$ \mathbb{R}^{2} $\end{document} , we study the two-grid discretization based on shifted-inverse iteration of \begin{document}$ C^0 $\end{document} IPG method. With our scheme, the solution of a biharmonic eigenvalue problem on a fine mesh \begin{document}$ \pi_h $\end{document} can be reduced to the solution of the eigenvalue problem on a coarser mesh \begin{document}$ \pi_H $\end{document} and the solution of a linear algebraic system on the fine mesh \begin{document}$ \pi_h $\end{document} . We prove that the resulting solution still maintains an asymptotically optimal accuracy when \begin{document}$ h\geq O(H^3) $\end{document} . In addition, we also discuss the multigrid discretization and the adaptive \begin{document}$ C^0 $\end{document} IPG algorithm based on Rayleigh quotient iteration. Numerical experiments are provided to validate the theoretical analysis.

Distribution of Stresses Near Angular Notches on an Orthotropic Elastic Plane Under Conditions of Antiplane Deformation

By using a solution of the antiplane eigenvalue problem of the theory of elasticity for an elastic orthotropic wedge, we construct the distribution of singular stresses and displacements in a neighborhood of the tip of the corresponding angular notch under the conditions of antiplane deformation. By the method of singular integral equations, we solve the corresponding problem for an orthotropic plane with semiinfinite angular rounded notch and establish the relationship between the stress intensity and stress concentration factors at the sharp and rounded tips of angular notches.