Showing papers on "Divide-and-conquer eigenvalue algorithm published in 2001"

The Quadratic Eigenvalue Problem

Abstract: We survey the quadratic eigenvalue problem, treating its many applications, its mathematical properties, and a variety of numerical solution techniques. Emphasis is given to exploiting both the structure of the matrices in the problem (dense, sparse, real, complex, Hermitian, skew-Hermitian) and the spectral properties of the problem. We classify numerical methods and catalogue available software.

A two-grid discretization scheme for eigenvalue problems

Abstract: A two-grid discretization scheme is proposed for solving eigenvalue problems, including both partial differential equations and integral equations. With this new scheme, the solution of an eigenvalue problem on a fine grid is reduced to the solution of an eigenvalue problem on a much coarser grid, and the solution of a linear algebraic system on the fine grid and the resulting solution still maintains an asymptotically optimal accuracy.

Large stable pulse solutions in reaction-diffusion equations

Abstract: In this paper we study the existence and stability of asymp- totically large stationary multi-pulse solutions in a family of singularly perturbed reaction-diVusion equations. This family includes the gen- eralized Gierer-Meinhardt equation. The existence of N-pulse homo- clinic orbits (N 1) is established by the methods of geometric singular perturbation theory. A theory, called the NLEP (=NonLocal Eigenvalue Problem) approach, is developed, by which the stability of these patterns can be studied explicitly. This theory is based on the ideas developed in our earlier work on the Gray-Scott model. It is known that the Evans function of the linear eigenvalue problem associated to the stability of the pattern can be decomposed into the product of a slow and a fast transmission function. The NLEP approach determines explicit leading order approximations of these transmission functions. It is shown that the zero/pole cancellation in the decomposition of the Evans function, called the NLEP paradox, is a phenomenon that occurs naturally in sin- gularly perturbed eigenvalue problems. It follows that the zeroes of the Evans function, and thus the spectrum of the stability problem, can be studied by the slow transmission function. The key ingredient of the analysis of this expression is a transformation of the associated nonlocal eigenvalue problem into an inhomogeneous hypergeometric diVerential equation. By this transformation it is possible to determine both the number and the position of all elements in the discrete spectrum of the linear eigenvalue problem. The method is applied to a special case that corresponds to the classical model proposed by Gierer and Meinhardt. It is shown that the one-pulse pattern can gain (or lose) stability through a Hopf bifurcation at a certain valueHopf of the main parameter. The NLEP approach not only yields a leading order approximation ofHopf , but it also shows that there is another bifurcation value,edge, at which a new (stable) eigenvalue bifurcates from the edge of the essential spec- trum. Finally, it is shown that theN-pulse patterns are always unstable whenN 2.

Eigenvalue Problems for the p-Laplacian with Indefinite Weights

Abstract: We consider the eigenvalue problem pu = V (x)juj p 2 u;u2 W 1;p 0 () where p > 1, p is the p-Laplacian operator, > 0, is a bounded domain in R N and V is a given function in L s () ( s depending on p and N). The weight function V may change sign and has nontrivial positive part. We prove that the least positive eigenvalue is simple, isolated in the spectrum and it is the unique eigenvalue associated to a nonnegative eigenfunction. Further- more, we prove the strict monotonicity of the least positive eigenvalue with respect to the domain and the weight.

Large‐scale eigenvalue calculations for stability analysis of steady flows on massively parallel computers

Abstract: We present an approach for determining the linear stability of steady states of PDEs on massively parallel computers. Linearizing the transient behavior around a steady state leads to a generalized eigenvalue problem. The eigenvalues with largest real part are calculated using Arnoldi's iteration driven by a novel implementation of the Cayley transformation to recast the problem as an ordinary eigenvalue problem. The Cayley transformation requires the solution of a linear system at each Arnoldi iteration, which must be done iteratively for the algorithm to scale with problem size. A representative model problem of 3D incompressible flow and heat transfer in a rotating disk reactor is used to analyze the effect of algorithmic parameters on the performance of the eigenvalue algorithm. Successful calculations of leading eigenvalues for matrix systems of order up to 4 million were performed, identifying the critical Grashof number for a Hopf bifurcation.

On the property of the curl-curl matrix in finite element analysis with edge elements

Abstract: This paper discusses properties of the curl-curl matrix in the finite element formulation with edge elements. Moreover, the observed deceleration in convergence of the CG and ICCG methods applied to magnetostatic problems through the tree-cotree gauging is explained on the basis of the eigenvalue separation property. From the eigenvalue separation property it follows that neither minimum nonzero eigenvalue of the curl-curl matrix nor maximum one increase through the tree-cotree gauging. Hence it is concluded that the condition number of the curl-curl matrix tends to grow by its definition. Moreover, the maximum eigenvalue tends to keep constant whereas the minimum nonzero eigenvalue reduces. This property also makes the condition number worse.

Optimal Stability and Eigenvalue Multiplicity

Abstract: We consider the problem of minimizing over an affine set of square matrices the maximum of the real parts of the eigenvalues. Such problems are prototypical in robust control and stability analysis. Under nondegeneracy conditions, we show that the multiplicities of the active eigenvalues at a critical matrix remain unchanged under small perturbations of the problem. Furthermore, each distinct active eigenvalue corresponds to a single Jordan block. This behavior is crucial for optimality conditions and numerical methods. Our techniques blend nonsmooth optimization and matrix analysis.

Variational formulas and approximation theorems for the first eigenvalue in dimension one

Abstract: Some complete variational formulas and approximation theorems for the first eigenvalue of elliptic operators in dimension one or a class of Markov chains are presented.

Riesz basis of root vectors of a non-symmetric system of first-order ordinary differential operators and application to inverse eigenvalue problems

Abstract: We consider a nonsymmetric first order differential operator where P is a 2 × 2 matrix whose compnents are of C1 class on [0,1]. We study an eigenvalue problem for A with boundary conditions at x=0,1. We establish an asympotic form of the eigenvalues and prove that the set of the root vectors forms a Riesz basis in {l2(0,1)}2. Next we apply the Riesz basis for showing the uniqueness in inverse eigenvalue problems for nonsymmetric systems. The key is the transformation formula.

Inverse Spectral problem for the Sturm-Liouville Operator with Eigenvalue Parameter Dependent Boundary Conditions

Abstract: To solve the inverse problem for the Sturm-Liouville operator with eigenvalue parameter dependent boundary conditions we reconstruct the spectral distribution function from two spectra of the boundary-value problems with equal Θ(λ) and different real constants in the boundary conditions. The well-known results of A.V. Strauss [5] concerning the connection between the eigenvalue problems with the spectral parameter in the boundary conditions and the theory of generalized resolvents is used.

"Single Ring Theorem" and the Disk-Annulus Phase Transition

Abstract: Recently, an analytic method was developed to study in the large $N$ limit non-hermitean random matrices that are drawn from a large class of circularly symmetric non-Gaussian probability distributions, thus extending the existing Gaussian non-hermitean literature. One obtains an explicit algebraic equation for the integrated density of eigenvalues from which the Green's function and averaged density of eigenvalues could be calculated in a simple manner. Thus, that formalism may be thought of as the non-hermitean analog of the method due to Brezin, Itzykson, Parisi and Zuber for analyzing hermitean non-Gaussian random matrices. A somewhat surprising result is the so called "Single Ring" theorem, namely, that the domain of the eigenvalue distribution in the complex plane is either a disk or an annulus. In this paper we extend previous results and provide simple new explicit expressions for the radii of the eigenvalue distiobution and for the value of the eigenvalue density at the edges of the eigenvalue distribution of the non-hermitean matrix in terms of moments of the eigenvalue distribution of the associated hermitean matrix. We then present several numerical verifications of the previously obtained analytic results for the quartic ensemble and its phase transition from a disk shaped eigenvalue distribution to an annular distribution. Finally, we demonstrate numerically the "Single Ring" theorem for the sextic potential, namely, the potential of lowest degree for which the "Single Ring" theorem has non-trivial consequences.

Asymptotic eigenvalue moments for linear multiuser detection

Abstract: Using the theory of lattices of non-crossing partitions, an explicit expression is obtained for the asymptotic moments of certain infinite random matrices that arise in multiuser detection in DS-CDMA systems with random spreading. By using this explicit expression, we obtain a self-contained proof of the Tse-Hanly formula for the output signal-to-interference-plus-noise ratio of a MMSE multiuser detector. The asymptotic moment results are used to design a low-complexity polynomial approximation of a linear multiuser detector. Analytical and numerical results of their performance analysis are also given.

Nonlinear eigenvalue problems for quasilinear operators on unbounded domains

Abstract: We prove several existence results for eigenvalue problems involving the p-Laplacian and a nonlinear boundary condition on unbounded domains. We treat the non-degenerate subcritical case and the solutions are found in an appropriate weighted Sobolev space.

Changing poles in the rational Lanczos method for the Hermitian eigenvalue problem

Abstract: Applications such as the modal analysis of structures and acoustic cavities require a number of eigenvalues and eigenvectors of large-scale Hermitian eigenvalue problems. The most popular method is probably the spectral transformation Lanczos method. An important disadvantage of this method is that a change of pole requires a complete restart. In this paper, we investigate the use of the rational Krylov method for this application. This method does not require a complete restart after a change of pole. It is shown that the change of pole can be considered as a change of Lanczos basis. The major conclusion of this paper is that the method is numerically stable when the poles are chosen in between clusters of the approximate eigenvalues. Copyright (C) 2001 John Wiley & Sons, Ltd.

An eigenvalue problem for a quasilinear elliptic field equation

Abstract: We study the field equation −∆u + V (x)u + ε(−∆pu + W ′(u)) = μu on Rn, with ε positive parameter. The function W is singular in a point and so the configurations are characterized by a topological invariant: the topological charge. By a min-max method, for ε sufficiently small, there exists a finite number of solutions (μ(ε), u(ε)) of the eigenvalue problem for any given charge q ∈ Z \\ {0}.

Finite Volume Methods for Eigenvalue Problems

Abstract: Based on a linear finite element space, in this paper, two symmetric finite volume schemes are proposed for self-adjoint elliptic boundary eigenvalue problems. Both convergence and superconvergence are discussed.

Convergence of numerical methods and parameter dependence of min-plus eigenvalue problems, Frenkel-Kontorova models and homogenization of Hamilton-Jacobi equations

Abstract: Using the min-plus version of the spectral radius formula, one proves: 1) that the unique eigenvalue of a min-plus eigenvalue problem depends continuously on parameters involved in the kernel defining the problem; 2) that the numerical method introduced by Chou and Griffiths to compute this eigenvalue converges. A toolbox recently developed at I.n.r.i.a. helps to illustrate these results. Frenkel-Kontorova models serve as example. The analogy with homogenization of Hamilton-Jacobi equations is emphasized.

Necessary and sufficient conditions for global optimality of eigenvalue optimization problems

Abstract: The necessary and sufficient conditions for global optimality are derived for an eigenvalue optimization problem. We consider the generalized eigenvalue problem where real symmetric matrices on both sides are linear functions of design variables. In this case, a minimization problem with eigenvalue constraints can be formulated as Semi-Definite Programming (SDP). From the Karush-Kuhn-Tucker conditions of SDP, the necessary and sufficient conditions are derived for arbitrary multiplicity of the lowest eigenvalues for the case where important lower bound constraints are considered for the design variables.

Successive eigenvalue relaxation: a new method for the generalized eigenvalue problem and convergence estimates

Abstract: We present a new subspace iteration method for the efficient computation of several smallest eigenvalues of the generalized eigenvalue problem Au = lambda Bu for symmetric positive definite operators A and B. We call this method successive eigenvalue relaxation, or the SER method (homoechon of the classical successive over-relaxation, or SOR method for linear systems). In particular, there are two significant features of SER which render it computationally attractive: (i) it can effectively deal with preconditioned large-scale eigenvalue problems, and (ii) its practical implementation does not require any information about the preconditioner used: it can routinely accommodate sophisticated preconditioners designed to meet more exacting requirements (e.g. three-dimensional elasticity problems with small thickness parameters). We endow SER with theoretical convergence estimates allowing for multiple and clusters of eigenvalues and illustrate their usefulness in a numerical example for a discretized partial differential equation exhibiting clusters of eigenvalues.

One-dimensional approximations of the eigenvalue problem of curved rods

Abstract: In this work we analyse the asymptotic behaviour of eigenvalues and eigenfunctions of the linearized elasticity eigenvalue problem of curved rod-like bodies with respect to the small thickness ϵ of the rod We show that the eigenfunctions and scaled eigenvalues converge, as ϵ tends to zero, toward eigenpairs of the eigenvalue problem associated to the one-dimensional curved rod model which is posed on the middle curve of the rod Because of the auxiliary function appearing in the model, describing the rotation angle of the cross-sections, the limit eigenvalue problem is non-classical This problem is transformed into a classical eigenvalue problem with eigenfunctions being inextensible displacements, but the corresponding linear operator is not a differential operator Copyright © 2001 John Wiley & Sons, Ltd

A Characterization of the Eigenvalue of a General (Min,Max, +)-System

Abstract: In this paper we consider general (min,max,+)-systems for which the eigenvalue exists. The major contribution of this paper is a characterization of the eigenvalue of such systems in terms of the convergence of a specific iteration. Conditions for the eigenvalue to exist are also given.

A low-complexity divide-and-conquer method for computing eigenvalues and eigenvectors of symmetric band matrices ∗

Abstract: A framework for an efficient low-complexity divide-and-conquer algorithm for computing eigenvalues and eigenvectors of an n × n symmetric band matrix with semibandwidth b ≪ n is proposed and its arithmetic complexity analyzed The distinctive feature of the algorithm—after subdivision of the original problem into p subproblems and their solution—is a separation of the eigenvalue and eigenvector computations in the central synthesis problem The eigenvalues are computed recursively by representing the corresponding symmetric rank b(p−1) modification of a diagonal matrix as a series of rank-one modifications Each rank-one modifications problem can be solved using techniques developed for the tridiagonal divide-and-conquer algorithm Once the eigenvalues are known, the corresponding eigenvectors can be computed efficiently using modified QR factorizations with restricted column pivoting It is shown that the complexity of the resulting divide-and-conquer algorithm is O (n2b2) (in exact arithmetic)

Accurate modal perturbation in non-self-adjoint eigenvalue problem

Abstract: This paper presents an algorithm for the accurate modal perturbation analysis in the non-self-adjoint eigenvalue problem. Complete perturbation items are obtained from the given straightforward process, satisfying two conditions in a modal analysis: the eigenvalue equations and normality condition. The zeroth-order perturbation, solved from equations in a form of Rayleigh quotient, is employed in the later perturbations, which helps to improve the accuracy of analysis. Two examples are given to show the modal perturbation with distinct eigenvalues and with close eigenvalues. It is confirmed that the algorithm is applicable to any mode with a distinct eigenvalue, repeated eigenvalues, or close eigenvalues, and can give an improved accuracy. Copyright © 2001 John Wiley & Sons, Ltd.

New Methods for Calculations of the Lowest Eigenvalues of the Real Symmetric Generalized Eigenvalue Problem

Abstract: A new iterative method based on a Newton correction vector for extension of the Krylov subspace, its diagonal, and band versions are proposed for calculation of selected lowest eigenvalues and corresponding eigenvectors of the generalized symmetric eigenvalue problem. Additionally, diagonal and band Jacobi?Davidson methods are introduced. Test calculations show that the new iterative method usually converges faster than quadratic near a solution. The new iterative method along with its band version uses a smaller number of iterative steps to obtain a solution compared to the Jacobi?Davidson, band Jacobi?Davidson method, and generalized Davidson method correspondingly. The diagonal version of the new method preserves an advantage over the diagonal Jacobi?Davidson and the Davidson method.