Newton’s Method for the Solution of Inverse Spectral Problems

Summary Inverse eigenvalue problems associated with self-adjoint differential equations of order two or higher are considered. The question of what kind of data are necessary and sufficient to insure the existence of a unique solution is examined. A method of solution for well-posed inverse eigenvalue problems is then presented. The implications of this analysis to the inverse problem for the Earth's spheroidal modes of vibration is also discussed. In this talk, I propose to present a partial survey of the mathematical literature on inverse eigenvalue problems, with an eye toward geophysical applications, and to state some new results which I obtained recently. Perhaps the inverse eigenvalue problem which is best known to geophysicists is that related to the normal modes of vibration of the Earth. It can be stated thus: assuming that the Earth is spherically symmetric and given the natural frequencies ,S, and ,T, of the spheroidal and torsional modes of vibration, can we deduce the density p(r) and the speed of propagation Vr Takeuchi & Saito 1972) and very little is known about inverse eigenvalue problems of order higher than two. In order to gain an understanding of inverse eigenvalue problems one should perhaps divorce oneself from a specific problem and investigate rather a sequence of problems of increasing complexity. If you allow me to adopt this attitude, then the first task would be a rough classification of eigenvalue problems. Without pretending to be exhaustive, let me divide eigenvalue problems into two classes depending upon whether they are associated with ordinary or partial differential equations, i.e. depending upon whether they are related to one-dimensional or multi-dimensional problems. The following Sturm-Liouville problem: u+{n-q(x)}u = 0

Left and right inverse eigenvalue problem of (R, S)-symmetric matrices and its optimal approximation problem

This thesis is concerned with presenting convex optimization based tractable solutions for three fundamental problems: 1. Planted subgraph problem: Given two graphs, identifying the subset of vertices of the larger graph corresponding to the smaller one. 2. Graph edit distance problem: Given two graphs, calculating the number of edge/vertex additions and deletions required to transform one graph into the other. 3. Affine inverse eigenvalue problem: Given a subspace e ⊂ 𝕊ⁿ and a vector of eigenvalues λ ∈ ℝⁿ, finding a symmetric matrix with spectrum λ contained in e. These combinatorial and algebraic problems frequently arise in various application domains such as social networks, computational biology, chemoinformatics, and control theory. Nevertheless, exactly solving them in practice is only possible for very small instances due to their complexity. For each of these problems, we introduce convex relaxations which succeed in providing exact or approximate solutions in a computationally tractable manner. Our relaxations for the two graph problems are based on convex graph invariants, which are functions of graphs that do not depend on a particular labeling. One of these convex relaxations, coined the Schur-Horn orbitope, corresponds to the convex hull of all matrices with a given spectrum, and plays a prominent role in this thesis. Specifically, we utilize relaxations based on the Schur-Horn orbitope in the context of the planted subgraph problem and the graph edit distance problem. For both of these problems, we identify conditions under which the Schur-Horn orbitope based relaxations exactly solve the corresponding problem with overwhelming probability. Specifically, we demonstrate that these relaxations turn out to be particularly effective when the underlying graph has a spectrum comprised of few distinct eigenvalues with high multiplicities. In addition to relaxations based on the Schur-Horn orbitope, we also consider outer-approximations based on other convex graph invariants such as the stability number and the maximum-cut value for the graph edit distance problem. On the other hand, for the inverse eigenvalue problem, we investigate two relaxations arising from a sum of squares hierarchy. These relaxations have different approximation qualities, and accordingly induce different computational costs. We utilize our framework to generate solutions for, or certify unsolvability of the underlying inverse eigenvalue problem. We particularly emphasize the computational aspect of our relaxations throughout this thesis. We corroborate the utility of our methods with various numerical experiments.

The inverse interior transmission eigenvalue problem with mixed spectral data

Differential equation models for vibrating systems are associated with matrix eigenvalue problems. Frequently the undamped models lead to problems of the generalized eigenvalue type and damped models lead to problems of the quadratic eigenvalue type. The matrices in these systems are typically real and symmetric and are quite highly structured. The design and stabilisation of systems modelled by these equations (eg., undamped and damped vibrating systems) requires the determination of solutions to the inverse eigenvalue problems which are themselves real, symmetric and possibly have some other structural properties. In this talk we consider some pole assignment problems and inverse spectral problems for generalized and quadratic symmetric pencils, discuss some advances and point to some work that remains to be done.

Inverse eigenvalue problems for extended Hessenberg and extended tridiagonal matrices

The inverse eigenvalue problem consisting of the differential equation u(2n)-(p1u(n-1))(n-1)+…+(-1)npnu = λu together with suitable boundary conditions is examined. It is shown that n + 1 spectra associated with n + 1 distinct sets of boundary conditions are required in order to reconstruct the unknown coefficients p1, …, pn. The sixth order case is analogous to the eigenvalue problem for the spheroidal modes of vibrations of earth which have been used to infer the density, the bulk modulus and shear modulus.

We present a method for automatic elasticity tuning of industrial robot manipulators. The main contributions of the work are: a) The parameters of a mechanical mass-spring-damper equivalent of any order are solved given, only partial state information (motor encoder position and motor torque). b) The method is fully automatic with no operator input and can easily be applied in the field to update the dynamic model parameters. The ability to automatically update the elasticity parameters is particularly useful when the robot operators mount flexible tooling or equipment on the robot arms. c) The method separates friction and elasticity identification. d) The method is demonstrated on an industrial ABB robot. e) We combine an important result from the vibration literature (Yam and Elhay, 1996), with the solution of inverse eigenvalue problems (Gladwell, 1986). To our knowledge, this is the first time that these methods have been combined and applied to the identification of flexible robot manipulators. The main advantage of the method compared to other identification methods is the fact that only motor encoder position and motor torque are required to identify the springs, masses and dampers of an Nth order system.

We establish the relationship between the inverse eigenvalue problem and Jacobi matrices within the framework of the theory of singular perturbations of unbounded self-adjoint operators.

Firstly we study necessary and sufficient conditions for the constant row sums symmetric inverse eigenvalue problem to have a solution and sufficient conditions for the symmetric stochastic inverse eigenvalue problem to have a solution. Then we introduce the concept of general solutions for the symmetric stochastic inverse eigenvalue problem and the concept of totally general solutions for the $3\times3$ symmetric stochastic inverse eigenvalue problem. Finally we study the necessary and sufficient conditions for the symmetric stochastic inverse eigenvalue problems of order 3 to have general solutions, and the necessary and sufficient conditions for the symmetric stochastic inverse eigenvalue problems of order 3 to have a totally general solution.

The inverse eigenvalue problem for a Hermitian reflexive matrix and the optimization problem

The reconstruction of a symmetric matrix from the spectral data

A Jacobi matrix inverse eigenvalue problem with mixed data

We investigate the capacity of finite-input finite-output discrete memoryless channels (DMC) whose channel matrix is n × n square and furthermore is assumed to be nonsingular with n linearly independent real eigenvectors. For any given DMC with such a channel matrix, we characterize the mutual information in terms of its eigenvalues. Our main result, obtained by using the method of Lagrange multipliers, is to derive an analytic expression for the capacity, depending on the eigenvectors and the eigenvalues of the invertible channel matrix. In particular, by using the inverse eigenvalue problem, we characterize the capacity of (2, 2) channels, with invertible channel matrices, in terms of lower and upper bounds that exist in the literature. In addition, numerical examples are provided, and probability of error is discussed.

In this paper, we establish the Newton‐like and inexact Newton‐like based methods for solving a type of parameterized generalized inverse eigenvalue problem. This type of parameterized generalized inverse eigenvalue problem, including multiplicative and additive inverse eigenvalue problems, appears in many applications. We show that the direction produced by the Newton‐like method does not depend explicitly on the eigenvalues. Also, the inexact version can minimize the oversolving problem of Newton‐like methods and hence improve efficiency. We discuss the convergence properties of the presented methods. Finally, the performance and effectiveness of the algorithms are tested on three numerical examples and compared to the Newton algorithm.