Convergence of inexact inverse iteration with application to preconditioned iterative solves

In this paper we study inexact inverse iteration for solving the generalised eigenvalue problem A x=λM x. We show that inexact inverse iteration is a modified Newton method and hence obtain convergence rates for various versions of inexact inverse iteration for the calculation of an algebraically simple eigenvalue. In particular, if the inexact solves are carried out with a tolerance chosen proportional to the eigenvalue residual then quadratic convergence is achieved. We also show how modifying the right hand side in inverse iteration still provides a convergent method, but the rate of convergence will be quadratic only under certain conditions on the right hand side. We discuss the implications of this for the preconditioned iterative solution of the linear systems. Finally we introduce a new ILU preconditioner which is a simple modification to the usual preconditioner, but which has advantages both for the standard form of inverse iteration and for the version with a modified right hand side. Numerical examples are given to illustrate the theoretical results. AMS subject classification (2000)  65F15, 65F10     

An Ulm-Like Cayley Transform Method for Inverse Eigenvalue Problems with Multiple Eigenvalues

We study the convergence of an Ulm-like Cayley transform method for solving inverse eigenvalue problems which avoids solving approximate Jacobian equations. Under the nonsingularity assumption of the relative generalized Jacobian matrices at the solution, a convergence analysis covering both the distinct and multiple eigenvalues cases is provided and the quadratical convergence is proved. Moreover, numerical experiments are given in the last section to illustrate our results.     

Inexact inverse subspace iteration for generalized eigenvalue problems

n this paper, we present an inexact inverse subspace iteration method for computing a few eigenpairs of the generalized eigenvalue problem Ax=λBx. We first formulate a version of inexact inverse subspace iteration in which the approximation from one step is used as an initial approximation for the next step. We then analyze the convergence property, which relates the accuracy in the inner iteration to the convergence rate of the outer iteration. In particular, the linear convergence property of the inverse subspace iteration is preserved. Numerical examples are given to demonstrate the theoretical results.     

Analyzing the convergence factor of residual inverse iteration

We will establish here a formula for the convergence factor of the method called residual inverse iteration, which is a method for nonlinear eigenvalue problems and a generalization of the well-known inverse iteration. The formula for the convergence factor is explicit and involves quantities associated with the eigenvalue to which the iteration converges, in particular the eigenvalue and eigenvector. Residual inverse iteration allows for some freedom in the choice of a vector w _k and we can use the formula for the convergence factor to analyze how it depends on the choice of w _k . We also use the formula to illustrate the convergence when the shift is close to the eigenvalue. Finally, we explain the slow convergence for double eigenvalues by showing that under generic conditions, the convergence factor is one, unless the eigenvalue is semisimple. If the eigenvalue is semisimple, it turns out that we can expect convergence similar to the simple case.     

AN ADAPTIVE INVERSE ITERATION FEM FOR THE INHOMOGENEOUS DIELECTRIC WAVEGUIDES

We introduce an adaptive finite element method for computing electromagnetic guided waves in a closed, inhomogeneous, pillared three-dimensional waveguide at a given frequency based on the inverse iteration method. The problem is formulated as a generalized eigenvalue problems. By modifying the exact inverse iteration algorithm for the eigenvalue problem, we design a new adaptive inverse iteration finite element algorithm. Adaptive finite element methods based on a posteriori error estimate are known to be successful in resolving singularities of eigenfunctions which deteriorate the finite element convergence. We construct a posteriori error estimator for the electromagnetic guided waves problem. Numerical results are reported to illustrate the quasi-optimal performance of our adaptive inverse iteration finite element method.     

Convergence of the modified discrete ordinates method for the anisotropic scattering transport equation

Criticality problem of nuclear tractors generally refers to an eigenvalue problem for the transport equations. In this paper, we deal with the eigenvalue of the anisotropic scattering transport equation in slab geometry. We propose a new discrete method which was called modified discrete ordinates method. It is constructed by redeveloping and improving discrete ordinates method in the space of L¹(X). Different from traditional methods, norm convergence of operator approximation is proved theoretically. Furthermore, convergence of eigenvalue approximation and the corresponding error estimation are obtained by analytical tools.     

A Riemannian variant of the Fletcher–Reeves conjugate gradient method for stochastic inverse eigenvalue problems with partial eigendata

In this paper, we focus on the stochastic inverse eigenvalue problem with partial eigendata of constructing a stochastic matrix from the prescribed partial eigendata. A Riemannian variant of the Fletcher–Reeves conjugate gradient method is proposed for solving a general unconstrained minimization problem on a Riemannian manifold, and the corresponding global convergence is established under some assumptions. Then, we reformulate the inverse problem as a nonlinear least squares problem over a matrix oblique manifold, and the application of the proposed geometric method to the nonlinear least squares problem is investigated. The proposed geometric method is also applied to the case of prescribed entries and the case of column stochastic matrix. Finally, some numerical tests are reported to illustrate that the proposed geometric method is effective for solving the inverse problem.     

Perturbed preconditioned inverse iteration for operator eigenvalue problems with applications to adaptive wavelet discretization

In this paper we discuss an abstract iteration scheme for the calculation of the smallest eigenvalue of an elliptic operator eigenvalue problem. A short and geometric proof based on the preconditioned inverse iteration (PINVIT) for matrices (Knyazev and Neymeyr, SIAM J Matrix Anal 31:621–628, 2009) is extended to the case of operators. We show that convergence is retained up to any tolerance if one only uses approximate applications of operators which leads to the perturbed preconditioned inverse iteration (PPINVIT). We then analyze the Besov regularity of the eigenfunctions of the Poisson eigenvalue problem on a polygonal domain, showing the advantage of an adaptive solver to uniform refinement when using a stable wavelet base. A numerical example for PPINVIT, applied to the model problem on the L-shaped domain, is shown to reproduce the predicted behaviour.     

Convergence of a Ulm-like method for square inverse singular value problems with multiple and zero singular values

An interesting problem was raised in Vong et al. (SIAM J. Matrix Anal. Appl. 32:412–429, 2011): whether the Ulm-like method and its convergence result can be extended to the cases of multiple and zero singular values. In this paper, we study the convergence of a Ulm-like method for solving the square inverse singular value problem with multiple and zero singular values. Under the nonsingularity assumption in terms of the relative generalized Jacobian matrices, a convergence analysis for the multiple and zero case is provided and the quadratical convergence property is proved. Moreover, numerical experiments are given in the last section to demonstrate our theoretic results.     

The Inexact Newton-Like Method for Inverse Eigenvalue Problem

In this paper, we consider using the inexact Newton-like method for solving inverse eigenvalue problem. This method can minimize the oversolving problem of Newton-like methods and hence improve the efficiency. We give the convergence analysis of the method, and provide numerical tests to illustrate the improvement over Newton-like methods.     

Local convergence analysis of several inexact Newton-type algorithms for general nonlinear eigenvalue problems

We study the local convergence of several inexact numerical algorithms closely related to Newton’s method for the solution of a simple eigenpair of the general nonlinear eigenvalue problem $T(\lambda )v=0$ . We investigate inverse iteration, Rayleigh quotient iteration, residual inverse iteration, and the single-vector Jacobi–Davidson method, analyzing the impact of the tolerances chosen for the approximate solution of the linear systems arising in these algorithms on the order of the local convergence rates. We show that the inexact algorithms can achieve the same order of convergence as the exact methods if appropriate sequences of tolerances are applied to the inner solves. We discuss the connections and emphasize the differences between the standard inexact Newton’s method and these inexact algorithms. When the local symmetry of $T(\lambda )$ is present, the use of a nonlinear Rayleigh functional is shown to be fundamental in achieving higher order of convergence rates. The convergence results are illustrated by numerical experiments.     

A Newton iteration process for inverse eigenvalue problems

Summary We suppose an inverse eigenvalue problem which includes the classical additive and multiplicative inverse eigenvalue problems as special cases. For the numerical solution of this problem we propose a Newton iteration process and compare it with a known method. Finally we apply it to a numerical example.     

特征值反问题的迭代解法

§1.引言 近年来,由于许多应用科学,如地球物理、海洋、地质、声学、光学、量子力学和识别等问题的需要,提出了特征值反问题和广义特征值反问题.这些问题形成一类区别于经典代数特征值问题的复杂非线性问题.这类问题中只有少量在理论上、数值上有一些求解的方法,前人的工作主要集中于sturm-Liouville反问题,见[1,2,3,4].本文讨论下列各种特征值反问题:     

A geometric theory for preconditioned inverse iteration I: Extrema of the Rayleigh quotient

The discretization of eigenvalue problems for partial differential operators is a major source of matrix eigenvalue problems having very large dimensions, but only some of the smallest eigenvalues together with the eigenvectors are to be determined. Preconditioned inverse iteration (a “matrix-free” method) derives from the well-known inverse iteration procedure in such a way that the associated system of linear equations is solved approximately by using a (multigrid) preconditioner. A new convergence analysis for preconditioned inverse iteration is presented. The preconditioner is assumed to satisfy some bound for the spectral radius of the error propagation matrix resulting in a simple geometric setup. In this first part the case of poorest convergence depending on the choice of the preconditioner is analyzed. In the second part the dependence on all initial vectors having a fixed Rayleigh quotient is considered. The given theory provides sharp convergence estimates for the eigenvalue approximations showing that multigrid eigenvalue/vector computations can be done with comparable efficiency as known from multigrid methods for boundary value problems.     

The calculation of multi-soliton solutions of the Vakhnenko equation by the inverse scattering method

A Bäcklund transformation both in bilinear form and in ordinary form for the transformed Vakhnenko equation is derived. An inverse scattering problem is formulated. The inverse scattering method has a third-order eigenvalue problem. A procedure for finding the exact N-soliton solution of the Vakhnenko equation via the inverse scattering method is described. The procedure is illustrated by considering the cases N=1 and N=2.     

A block inverse-free preconditioned Krylov subspace method for symmetric generalized eigenvalue problems

The inverse-free preconditioned Krylov subspace method of Golub and Ye [G.H. Golub, Q. Ye, An inverse free preconditioned Krylov subspace method for symmetric generalized eigenvalue problems, SIAM J. Sci. Comp. 24 (2002) 312-334] is an efficient algorithm for computing a few extreme eigenvalues of the symmetric generalized eigenvalue problem. In this paper, we first present an analysis of the preconditioning strategy based on incomplete factorizations. We then extend the method by developing a block generalization for computing multiple or severely clustered eigenvalues and develop a robust black-box implementation. Numerical examples are given to illustrate the analysis and the efficiency of the block algorithm.     

An inexact inverse iteration for large sparse eigenvalue problems

In this paper, we propose an inverse inexact iteration method for the computation of the eigenvalue with the smallest modulus and its associated eigenvector for a large sparse matrix. The linear systems of the traditional inverse iteration are solved with accuracy that depends on the eigenvalue with the second smallest modulus and iteration numbers. We prove that this approach preserves the linear convergence of inverse iteration. We also propose two practical formulas for the accuracy bound which are used in actual implementation. © 1997 John Wiley & Sons, Ltd.     

A modified Leverrier-Faddeev algorithm for matrices with multiple eigenvalues

The Leverrier algorithm as modified by Faddeev gives the characteristic equation of a matrix A, its inverse, and the eigenvector corresponding to a simple eigenvalue λ of A. These results are extended (1) to give a generalized inverse when A is not of full rank and (2) to examine the modification required when λ is a multiple eigenvalue.     

Contingent derivatives and regularization for noncoercive inverse problems

Abstract

We study the inverse problem of parameter identification in noncoercive variational problems that commonly appear in applied models. We examine the differentiability of the set-valued parameter-to-solution map using the first-order and the second-order contingent derivatives. We explore the inverse problem using the output least-squares and the modified output least-squares objectives. By regularizing the noncoercive variational problem, we obtain a single-valued regularized parameter-to-solution map and investigate its smoothness and boundedness. We also consider optimization problems using the output least-squares and the modified output least-squares objectives for the regularized variational problem. We give a complete convergence analysis showing that for the output least-squares and the modified output least-squares, the regularized minimization problems approximate the original optimization problems suitably. We also provide the first-order and the second-order adjoint method for the computation of the first-order and the second-order derivatives of the output least-squares objective. We provide discrete formulas for the gradient and the Hessian calculation and present numerical results.     

Greedy Algorithms for High-Dimensional Eigenvalue Problems

In this article, we present two new greedy algorithms for the computation of the lowest eigenvalue (and an associated eigenvector) of a high-dimensional eigenvalue problem and prove some convergence results for these algorithms and their orthogonalized versions. The performance of our algorithms is illustrated on numerical test cases (including the computation of the buckling modes of a microstructured plate) and compared with that of another greedy algorithm for eigenvalue problems introduced by Ammar and Chinesta.