Algebraic Eigenvalue Research Articles

Eigenvalue asymptotics for differential operators on graphs

We consider the spectral structure for differential equations on graphs. In particular, we show that self-adjointness does not necessarily imply regularity, we also show that the algebraic and geometric eigenvalue multiplicities of formally self-adjoint differential operators on graphs are equal. Asymptotic bounds for the eigenvalues are then found.

Variable-step preconditioned conjugate gradient method for partial symmetric eigenvalue problems

The iterative methods for partial algebraic symmetric eigenvalue problems are considered for sparse positive definite matrices which arise in approximation of 2D and 3D boundary value problems. The approach is based on subspace iterations, Rayleigh–Ritz method, and the variable-step preconditioned conjugate gradient algorithm, including algebraic multigrid and incomplete factorization. Theorems on the properties of convergence rate are presented. The efficiency of the proposed iterative processes is demonstrated by the results of numerical experiments.

Problems and Techniques

We are fortunate to have, in this issue's Problems and Techniques section, three papers that should be of interest to all SIAM Review readers. The first paper, "A Fast and Stable Solution Method for the Radiative Transfer Problem," by P. Edstrom, treats the interaction of radiation with turbid media, an important and longstanding scientific problem. (The first citation is a 1905 paper on radiation through a foggy atmosphere.) For readers unfamiliar with the term, in common parlance something "turbid" contains foreign particles that are stirred up or suspended, as in "I leave a white and turbid wake" (Chapter 37 of Moby Dick); in the present context, a turbid medium is highly scattering. Transmission as well as absorption and multiple scattering must be considered in a turbid medium, and several steps are needed to obtain an appropriate mathematical formulation and efficient numerical solution methods. The paper guides the reader through the motivation for and connections among the resulting multiple procedures: application of Fourier analysis to a fundamental integrodifferential equation, discretization using numerical quadrature, transformation to an algebraic eigenvalue problem, and preconditioning strategies for the equations that define boundary and continuity conditions. Special attention is given to techniques for speeding up the computation, and the solution method described has been implemented in a MATLAB code used in the paper and printing industries. This paper features references that are unusually broad in time (starting with an 1871 paper by Lord Rayleigh) and application (ranging over optics, astrophysics, neutron transport, nuclear engineering, and atmospheric science). The second paper, "Reviving the Method of Particular Solutions," by T. Betcke and L. N. Trefethen, presents a readable and engrossing tour of the authors' successful quest to understand and overcome the difficulties experienced by the method of particular solutions (MPS) defined in a classic 1967 paper from the SIAM Journal on Numerical Analysis by L. Fox, P. Henrici, and C. Moler. As carefully explained in the present paper, the problems are caused by the MPS's inability to ensure that an eigenfunction is nonzero in the interior of the domain; ambiguities and ill-conditioning arise from the existence of linear combinations of functions that are close to zero everywhere, even when the associated parameter is not an eigenvalue. A modified method, based on constraining the problem so that admissible functions are bounded away from zero in the interior, is presented that allows reliable detection of spurious approximate eigenfunctions. (The contrast between Figures 4.1 and 5.4 tells the story.) The authors then go on to describe both a geometric interpretation and the connections between the constrained minimization problem and angles between subspaces. Kac's famous 1966 question "Can one hear the shape of a drum?" is used as an example for the modified MPS on several isospectral examples, including an H-shaped domain. The paper also contains a discussion of error bounds, generalizations, open questions, and very recent related work. Krylov subspace methods for linear systems, featured in last issue's Problems and Techniques section, appear again, this time for eigenvalue problems, in our third paper, "Convergence of Polynomial Restart Krylov Methods for Eigenvalue Computations," by C. A. Beattie, M. Embree, and D. C. Sorensen. Like the other two papers, this paper first presents an illuminating summary of both history and state-of-the-art before moving to its heart: analysis of the angles made by an approximating Krylov subspace of a matrix A with a desired invariant subspace, and convergence of polynomial restarting methods that project A onto a Krylov subspace and modify the starting vector using a polynomial in A. Setting an engaging tone by declaring their interest only in the "good" eigenvalues, the paper provides an explanation of how a Krylov space might converge to an associated good invariant subspace, as well as a discussion of how to assess the proximity of subspaces. After defining the "maximal reachable invariant subspace," the authors consider both the gap between this subspace and the $\ell$th Krylov space as $\ell$ increases, and the dependence of the gap on the polynomial that defines a restart filter. Among the results of this analysis are insights into the design of effective polynomial filters and numerical confirmation of the complications that arise when restarting iterations for nonnormal matrices. The reader is encouraged to examine the pseudospectra in Figures 4.8 and 4.10 for visible evidence of these subtleties.

Gauss–Seidel-type methods for energy states of a multi-component Bose–Einstein condensate

In this paper, we propose two iterative methods, a Jacobi-type iteration (JI) and a Gauss–Seidel-type iteration (GSI), for the computation of energy states of the time-independent vector Gross–Pitaevskii equation (VGPE) which describes a multi-component Bose–Einstein condensate (BEC). A discretization of the VGPE leads to a nonlinear algebraic eigenvalue problem (NAEP). We prove that the GSI method converges locally and linearly to a solution of the NAEP if and only if the associated minimized energy functional problem has a strictly local minimum. The GSI method can thus be used to compute ground states and positive bound states, as well as the corresponding energies of a multi-component BEC. Numerical experience shows that the GSI converges much faster than JI and converges globally within 10–20 steps.

Spatiospectral Concentration on a Sphere

We pose and solve the analogue of Slepian's time-frequency concentration problem on the surface of the unit sphere to determine an orthogonal family of strictly bandlimited functions that are optimally concentrated within a closed region of the sphere, or, alternatively, of strictly spacelimited functions that are optimally concentrated within the spherical harmonic domain. Such a basis of simultaneously spatially and spectrally concentrated functions should be a useful data analysis and representation tool in a variety of geophysical and planetary applications, as well as in medical imaging, computer science, cosmology and numerical analysis. The spherical Slepian functions can be found either by solving an algebraic eigenvalue problem in the spectral domain or by solving a Fredholm integral equation in the spatial domain. The associated eigenvalues are a measure of the spatiospectral concentration. When the concentration region is an axisymmetric polar cap the spatiospectral projection operator commutes with a Sturm-Liouville operator; this enables the eigenfunctions to be computed extremely accurately and efficiently, even when their area-bandwidth product, or Shannon number, is large. In the asymptotic limit of a small concentration region and a large spherical harmonic bandwidth the spherical concentration problem approaches its planar equivalent, which exhibits self-similarity when the Shannon number is kept invariant.

Twin-characteristic-parameters analysis for elastic dynamic buckling of thin cylindrical shells under axial step loading

The relations of the critical stress and transverse inertia effect to the loading duration are investigated for the dynamic buckling of thin cylindrical shells under axial step loading. The critical stress and the inertial exponent are treated as the two characteristic parameters. The criterion of energy conservation is used to derive the supplementary restraint condition for buckling deformations at compression wave front. By use of the Galerkin method, an algebraic eigenvalue problem for the two characteristic parameters is derived from the governing equations and boundary conditions. The solution of the eigenvalue problem, which satisfies the supplementary restraint condition, gives the values of the critical stress and the inertial exponent for the dynamic buckling. The relation of critical stress to loading duration, predicted by the theoretical analysis, is in reasonable agreement with the experimental results.

On singularities in the solution of three‐dimensional Stokes flow and incompressible elasticity problems with corners

AbstractIn this paper, a numerical procedure is presented for the computation of corner singularities in the solution of three‐dimensional Stokes flow and incompressible elasticity problems near corners of various shape. For obtaining the order and mode of singularity, a neighbourhood of the singular point is considered with only local boundary conditions. The weak formulation of this problem is approximated using a mixed u, p Galerkin–Petrov finite element method. Additionally, a separation of variables is used to reduce the dimension of the original problem. As a result, the quadratic eigenvalue problem (P+λQ+λ2R)d=0 is obtained, where the saddle‐point‐type matrices P, Q, R are defined explicitly. For a numerical solution of the algebraic eigenvalue problem an iterative technique based on the Arnoldi method in combination with an Uzawa‐like scheme is used. This technique needs only one direct matrix factorization as well as few matrix–vector products for finding all eigenvalues in the interval 𝓈(λ) ∈ (−0.5, 1.0), as well as the corresponding eigenvectors. Some benchmark tests show that this technique is robust and very accurate. Problems from practical importance are also analysed, for instance the surface‐breaking crack in an incompressible elastic material and the three‐dimensional viscous flow of a Newtonian fluid past a trihedral corner. Copyright © 2004 John Wiley & Sons, Ltd.

Wave propagation and damping in linear viscoelastic laminates

This analysis is concerned with wave propagation and damping in linear viscoelastic laminates. A spectral finite-element method is developed and used to calculate the dispersion properties of the first few wave types of a given laminate; the proposed approach provides a robust and numerically efficient alternative to the transfer matrix method in certain applications. The proposed approach is also well suited to the calculation of the wave types of sections whose material properties vary continuously throughout the thickness of the section. A one-dimensional finite-element mesh is used to describe the through-thickness deformation of the laminate, and the dispersion equation for plane-wave propagation is formulated as a linear algebraic eigenvalue problem in wave number at each frequency of interest. The resulting eigenvectors and eigenvalues can be computed using standard numerical routines and used to investigate the dispersion characteristics of the propagating wave types of the section. The damping loss factor of each wave type is estimated from the cross-sectional strain energy distribution of the laminate. The proposed approach is well suited to modeling the structural-acoustic response of sandwich panels, constrained layer damping treatments, and general viscoelastic laminate sections in statistical energy analysis (SEA) codes.

Thermoviscous stability of ice-sheet flows

The shallow-ice approximation is used to analyse the normal-mode stability of long-wavelength, thermally coupled, free-surface flows of glaciers and ice sheets along infinite flow sections. The viscosity of the ice changes with temperature, and the ice is also thermally coupled with the bedrock. An assumption of quasi-uniform flow, where the downstream flux of mass and energy is assumed to be constant over relatively short wavelengths, permits vertical advection due to accumulation to be considered. The assumption allows realistic representation of the vertical velocity distribution and temperature distributions within the modelled flow. Nevertheless, the linearized equations are formally independent of the assumption of quasi-uniform flow. The linearized equations are Fourier-transformed over the horizontal plane and an algebraic eigenvalue problem constructed which considers the Fourier-transforms of the vertically varying temperature and thickness. The dependence of the eigenvalues on wavenumber, both parallel and orthogonal to flow, accumulation rate, geothermal heat flux, surface temperature and slope is computed. Extensive linearly unstable regimes where the base is well below melting point are found, which correspond only approximately to negative slopes in the flux/thickness relationship (supercritical zones). The most unstable parts of these regimes have finite along-flow wavelength, and have very long transverse wavelength. These modes are found to be more unstable in two senses: (i) they bifurcate for lesser thicknesses; (ii) they have greater growth rates. All computed wave velocities of unstable modes were downslope. Less unstable modes with short and medium transverse wavelengths are found which excite the temperature field only. Time constants range between ten and one thousand years. There is no strong evidence for stream-pattern formation through a thermoviscous instability mechanism.

Numerical Solution of the two‐dimensional time independent Schrödinger Equation by symplectic schemes

AbstractThe solution of the two‐dimensional time‐independent Schrödinger equation is considered by partial discretization. The discretized problem is treated as an ordinary differential equation problem and solved numerically by asymptotically symplectic methods. The problem is then transformed into an algebraic eigenvalue problem involving real, symmetric matrices. The eigenvalues of the two‐dimensional harmonic oscillator and the twodimensional Henon‐Heils potential are computed by the application of the methods developed. The results are compared with the results produced by full discretization. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Linear time-varying eigenstructure assignment with flight control application

This work is concerned with the assignment of a desired PD-eigenstructure for linear time-varying systems. Despite its well-known limitations, gain scheduling control appeared to be a focus of the research efforts. Scheduling of frozen-time, frozen-state controllers for fast time-varying dynamics is known to be mathematically fallacious, and practically hazardous. Therefore, recent research efforts are being directed towards applying time-varying controllers. In this paper we: 1) introduce a differential algebraic eigenvalue theory for linear time-varying systems; and 2) propose a PD-eigenstructure assignment scheme via a differential Sylvester equation and a command generator tracker (CGT) for linear time-varying systems. The PD-eigenstructure assignment is utilized as a regulator. A feedforward gain for tracking control is computed by using the command generator tracker. The whole design procedures of the proposed PD-eigenstructure assignment scheme are systematic in nature. The scheme could be used to determine the stability of linear time-varying systems easily as well as to provide a new horizon of designing controllers for the linear time-varying systems. A missile flight control application is presented to validate the proposed schemes.

Computing Cavity Resonances Using Eigenvalues Displacement

The numerical discretization of the field inside a cavity by means of edge elements results in a generalized algebraic eigenvalues problem that contains several undesired null eigenvalues. This occurrence prevents the effective use of iterative eigensolvers. To overcome this difficulty, a complementary eigenproblem has been proposed in the literature. This paper extends this method by introducing a family of algebraically built complementary eigenproblems, and determines, by numerical experiments and heuristics, which complementary eigenproblems are best suited for the preconditioned inverse iteration eigensolver and the Lanczos method.

Spurious roots in the algebraic Dirac equation

It is shown that the spurious roots appearing among solutions of the algebraic Dirac Hamiltonian eigenvalue problem represent the positive spectrum states of the Dirac Hamiltonian and that each of them has its variational non-relativistic counterpart.

Adjoint high-order vectorial finite elements for nonsymmetric transversally anisotropic waveguides

In this paper, a vector finite-element method (FEM) for the investigation of microwave and optical waveguides with arbitrary cross section is presented. In particular, the FEM solves the vector wave equation in the frequency domain for waveguides, containing nonsymmetric transversally anisotropic (passive and active) materials. Therefore, a bilinear form has been derived that is solved for the original and adjoint wave equation. Biorthogonality, relations of original and adjoint solutions for simpler materials, the role of nonphysical solutions for both problems, and their proper approximations are discussed in detail. Special focus has been set on shifting the propagation constants of the nonphysical modes for nonzero frequencies to infinity. Therefore, an excellent rate of convergence for iterative solvers is observed. The occurrence of nonphysical solutions is shown. Different nth-order basis functions are systematically derived, which are identical to the well-known tangential continuous finite elements. Both algebraic eigenvalue problems are solved via a generalized Lanczos algorithm, which solves both eigenvalue problems efficiently. The capability of the FEM is illustrated by critical examples.

An approximate solution scheme for the algebraic random eigenvalue problem

A reduced basis formulation is presented for the efficient solution of large-scale algebraic random eigenvalue problems. This formulation aims to improve the accuracy of the first order perturbation method, and also allow the efficient computation of higher order statistical moments of the eigenparameters. In the present method, the two terms of the first order perturbation approximation for the eigenvector are used as basis vectors for Ritz analysis of the governing random eigenvalue problem. This leads to a sequence of reduced order random eigenvalue problems to be solved for each eigenmode of interest. Since, only two basis vectors are used to represent each eigenvector, explicit expressions for the random eigenvalues and eigenvectors can readily be derived. This enables the statistics of the random eigenparameters and the forced response to be efficiently computed. Numerical studies are presented for free and forced vibration analysis of a linear stochastic structural system. It is demonstrated that the reduced basis method gives better results as compared to the first order perturbation method.

Spurious Roots in the Algebraic Dirac Equation

The nature of spurious roots discovered by Drake and Goldman (G W F Drake and S P Goldman 1981 Phys. Rev. A 23 2093) among solutions of the algebraic Dirac Hamiltonian eigenvalue problem is discussed. It is shown that the spurious roots represent the positive spectrum states of the Dirac Hamiltonian and that each of them has its variational non-relativistic counterpart. Sufficient conditions to avoid the appearance of the spuriouses in the forbidden gap of Dirac energies are formulated. Numerical examples for κ = 1(P1/2) states of an electron in a spherical Coulomb potential (in Slater-type bases) are presented.

Analysis of Bi-anisotropic PBG Structure Using Plane Wave Expansion Method

An algebraic eigenvalue problem for analyzing the propagation characteristics of electromagnetic waves inside the PBG Structure consisting of complex medium is established by using Bloch theorem and plane wave expansion. Two eigen-solvers are employed. One is matrix-based and another is iterative. Calculated results show that both methods are effective for 2-D PBG structure, but the iterative eigen-solver is more attractive in both CPU-time and memory requirement. A sample of 2-D PBG structure, with Chiral medium as host and air cylinders arranged in triangular lattice as inclusion, is analyzed using both methods. It is found that the introduction of chirality increases the band gap width significantly.

ANALYSIS OF BI-ANISOTROPIC PBG STRUCTURE USING PLANE WAVE EXPANSION METHOD - ABSTRACT

An algebraic eigenvalue problem for analyzing the propagation characteristics of electromagnetic waves inside the PBG Structure consisting of complex medium is established by using Bloch theorem and plane wave expansion. Two eigen-solvers are employed. One is matrix-based and another is iterative. Calculated results show that both methods are effective for 2-D PBG structure, but the iterative eigen-solver is more attractive in both CPU-time and memory requirement. A sample of 2-D PBG structure, with Chiral medium as host and air cylinders arranged in triangular lattice as inclusion, is analyzed using both methods. It is found that the introduction of chirality increases the band gap width significantly.

Numerical methods for solving the eigenvalue problem for a positive branch confocal unstable resonator

The resonator problem for a positive branch confocal unstable resonator reduces to a Fredholm homogeneous integral equation of the second kind, whose numerical solution here is based on a sequence of algebraic eigenvalue problems. We compare two algorithms for the solution of an optical resonator problem. These are obtained by (i) successive degenerate kernel approximation by Taylor polynomials of the Fredholm kernel and (ii) Nyström’s method with Simpson’s rule as the subordinate numerical integration method. The numerical results arising from these routines compare well with other published results, and have the added advantage of simplicity and easy adaptability to other resonator problems.

Analytical and hybrid solutions of diffusion problems within arbitrarily shaped regions via integral transforms

Linear diffusion problems defined within irregular multidimensional regions are analytically solved through integral transforms, requiring numerical routines only for integration purposes, when a general functional boundary representation is considered. Auxiliary one-dimensional eigenvalue problems mapping the irregular region are applied with an integral transformation procedure so that the original differential Sturm–Liouville system gives place to an algebraic eigenvalue problem. The exact analytical inversion formula is then employed to yield the desired potential, explicitly, at any point within the domain. To allow for improved flexibility and further applicability, the related integration is simplified through an approximate boundary representation using lines connecting user provided points instead of the former exact representation of the irregular bounds, which is particularly advantageous when a functional description of the boundaries is not available. A cylindrical region test case with known exact solution is considered, and treated as an irregular region in the Cartesian coordinates system. Convergence behavior and error analysis are carefully undertaken and illustrated.