Well-posed Inverse Eigenvalue Problems

The inverse matrix eigenvalue problem is the problem of reconstruction of a matrix (determination of the unknown parameters of a certain matrix) for a given finite set of spectral data (points of the spectrum). The spectral data used in this case may contain complete or only partial information on the eigenvalues or eigenvectors. The aim of an inverse eigenvalue problem is to construct a matrix that preserves a certain specific structure and the prescribed spectral properties. There are two basic questions connected with any inverse eigenvalue problem: the theoretical question on the possibility of construction of its solutions and the practical question of its determination. The main investigations connected with solvability deal with establishing necessary or sufficient conditions under which the inverse eigenvalue problem is solvable. On the other hand, the main difficulty in calculations is connected with the development of a numerical procedure which would enable us to construct the matrix itself according to the a-priori specified spectral data. Both questions are important and complicated. Numerous works are devoted to the conditions of existence and uniqueness of solutions for different formulations of inverse matrix spectral problems and the methods used for their solution (see, e.g., [1–3, 7, 9–13, 15] and the literature therein). In the present work, we consider a numerical algorithm used for the solution of inverse eigenvalue problems under the assumption that this solution exists. Hence, we consider inverse algebraic eigenvalue problems.

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.

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.

Inverse eigenvalue problems for extended Hessenberg and extended tridiagonal matrices

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.

In this paper, the symmetric positive semidefinite Procrustes problem is considered. By using matrix inner product and matrix decomposition theory, an explicit expression of the solution is given. Based on the explicit expression given in this paper, it is easy to derive the explicit expression of the solution given by Woodgate [K.G. Woodgate. Least-squares solution of F = PG over positive semidefinite symmetric P . Linear Algebra Appl., 245:171–190, 1996.] and by Liao [A.P. Liao. On the least squares problem of a matrix equation. J. Comput. Math., 17:589–594, 1999.] for the Procrustes problem in some special cases. The explicit expression given in this paper also shows that the solution of the special inverse eigenvalue problem considered by Zhang [L. Zhang. A class of inverse eigenvalue problem for symmetric nonnegative definite matrices. J. Hunan Educational Inst., 2:11–17, 1995 (in Chinese).] can be computed exactly. Examples to illustrate the correctness of the theory results are given.

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

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.

We present two versions of the same inequality, relating the maximal diagonal entry of a nonnegative matrix to its eigenvalues. We demonstrate a matrix factorization of a companion matrix, which leads to a solution of the nonnegative inverse eigenvalue problem (denoted the nniep) for 4×4 matrices of trace zero, and we give some sufficient conditions for a solution to the nniep for 5×5 matrices of trace zero. We also give a necessary condition on the eigenvalues of a 5×5 trace zero nonnegative matrix in lower Hessenberg form. Finally, we give a brief discussion of the nniep in restricted cases. Keywords: Nonnegative matricesMaximal diagonal entrynniepCompanion matricesAMS Subject Classification: 15A1815A2315A42

A structured method for the design-for-frequency of a brace-soundboard system using a scalloped brace

Over the last two decades, anomalous diffusion processes in which the mean squares variance grows slower or faster than that in a Gaussian process have found many applications. At a macroscopic level, these processes are adequately described by fractional differential equations, which involves fractional derivatives in time or/and space. The fractional derivatives describe either history mechanism or long range interactions of particle motions at a microscopic level. The new physics can change dramatically the behavior of the forward problems. For example, the solution operator of the time fractional diffusion diffusion equation has only limited smoothing property, whereas the solution for the space fractional diffusion equation may contain weak singularity. Naturally one expects that the new physics will impact related inverse problems in terms of uniqueness, stability, and degree of ill-posedness. The last aspect is especially important from a practical point of view, i.e., stably reconstructing the quantities of interest. In this paper, we employ a formal analytic and numerical way, especially the two-parameter Mittag-Leffler function and singular value decomposition, to examine the degree of ill-posedness of several ‘classical’ inverse problems for fractional differential equations involving a Djrbashian–Caputo fractional derivative in either time or space, which represent the fractional analogues of that for classical integral order differential equations. We discuss four inverse problems, i.e., backward fractional diffusion, sideways problem, inverse source problem and inverse potential problem for time fractional diffusion, and inverse Sturm–Liouville problem, Cauchy problem, backward fractional diffusion and sideways problem for space fractional diffusion. It is found that contrary to the wide belief, the influence of anomalous diffusion on the degree of ill-posedness is not definitive: it can either significantly improve or worsen the conditioning of related inverse problems, depending crucially on the specific type of given data and quantity of interest. Further, the study exhibits distinct new features of ‘fractional’ inverse problems, and a partial list of surprising observations is given below. (a) Classical backward diffusion is exponentially ill-posed, whereas time fractional backward diffusion is only mildly ill-posed in the sense of norms on the domain and range spaces. However, this does not imply that the latter always allows a more effective reconstruction. (b) Theoretically, the time fractional sideways problem is severely ill-posed like its classical counterpart, but numerically can be nearly well-posed. (c) The classical Sturm–Liouville problem requires two pieces of spectral data to uniquely determine a general potential, but in the fractional case, one single Dirichlet spectrum may suffice. (d) The space fractional sideways problem can be far more or far less ill-posed than the classical counterpart, depending on the location of the lateral Cauchy data. In many cases, the precise mechanism of these surprising observations is unclear, and awaits further analytical and numerical exploration, which requires new mathematical tools and ingenuities. Further, our findings indicate fractional diffusion inverse problems also provide an excellent case study in the differences between theoretical ill-conditioning involving domain and range norms and the numerical analysis of a finite-dimensional reconstruction procedure. Throughout we will also describe known analytical and numerical results in the literature.

This report demonstrates the use of a genetic algorithm search in the solution of an inverse problem. The genetic algorithm is used to solve the one-dimensional inverse heat conduction problem using numerical data generated by solution of the corresponding direct problem. Both "pure" and noisy data are considered. If used with regularization, the method is shown to yield reasonable results even in the face of noisy data. Although only a simple one-dimensional case is presented to demonstrate the method, the genetic algorithm approach is expected to excel in the solution of complicated multidimensional inverse problems.

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.