Inverse Eigenvalue Problem Research Articles

A new constrained optimization model for solving the nonsymmetric stochastic inverse eigenvalue problem

ABSTRACT The stochastic inverse eigenvalue problem aims to reconstruct a stochastic matrix from its spectrum. Recently, Zhao et al. [A geometric nonlinear conjugate gradient method for stochastic inverse eigenvalue problems. SIAM J Numer Anal. 2016;54(4):2015–2035] proposed a constrained optimization model on the manifold of so-called isospectral matrices and adapted a modified. Polak-Ribière-Polyak conjugate gradient method to the geometry of this manifold. However, not every stochastic matrix is an isospectral one and the model in Zhao et al. is based on the assumption that for each stochastic matrix there exists a (possibly different) isospectral, stochastic matrix with the same spectrum. We are not aware of such a result in the literature, but prove the claim at least for 3 × 3 matrices. In this paper, we suggest to extend the above model by considering matrices which differ from isospectral ones only by multiplication with a block diagonal matrix with 2 × 2 blocks from the special linear group. First, we show that each stochastic matrix can be written in such a form. We prove that our model has a minimizer and show how the Polak–Ribiére–Polyak conjugate gradient method works on the corresponding more general manifold. We demonstrate by numerical examples that the new, more general method performs similarly as those in Zhao et al.

Matrix inverse eigenvalue problem for stabilization of fractional descriptor discrete-time linear systems by forward and propositional output feedback

In this paper, stabilization of unstable fractional descriptor discrete-time linear system via forward and propositional output feedback is done to obtain satisfactory responses. To gain forward and propositional output feedback matrices, two standard linear systems need to exist. Assigning nonzero arbitrary eigenvalues to the first standard system and inverted the desired eigenvalues for standard descriptor system to the second one, desired eigenvalues are assigned by matrix inverse eigenvalue problem. Numerical examples are also presented to illustrate our method.

Riemannian Newton-CG methods for constructing a positive doubly stochastic matrix from spectral data* *The research of the second author was supported by the National Natural Science Foundation of China (No. 11601112) and the third author was partially supported by the National Natural Science Foundation of China (No. 11671337) and the Fundamental Research Funds for the Central Universities (No. 20720180008).

In this paper, we consider the inverse eigenvalue problem for the positive doubly stochastic matrices, which aims to construct a positive doubly stochastic matrix from the prescribed realizable spectral data. By using the real Schur decomposition, the inverse problem is written as a nonlinear matrix equation on a matrix product manifold. We propose monotone and nonmonotone Riemannian inexact Newton-CG methods for solving the nonlinear matrix equation. The global and quadratic convergence of the proposed methods is established under some assumptions. We also provide invariant subspaces of the constructed solution to the inverse problem based on the computed real Schur decomposition. Finally, we report some numerical tests, including an application in digraph, to illustrate the effectiveness of the proposed methods.

An inverse eigenvalue problem for a nonlocal Sturm-Liouville operator

This paper is concerned with the inverse eigenvalue problem for a nonlocal Sturm-Liouville operator generated by its rank one perturbations. We give a necessary and sufficient condition that a sequence of real numbers is the spectrum of a nonlocal Sturm-Liouville operator, provided that the potential is known a priori. An explicit algorithm for constructing all such operators is also provided.

On a regularization approach to the inverse transmission eigenvalue problem

We consider the irregular (in the Birkhoff and even the Stone sense) transmission eigenvalue problem of the form −y″ + q(x)y = ρ 2 y, y(0) = y(1) cos ρa − y′(1)ρ −1 sin ρa = 0. The main focus is on the ‘most’ irregular case a = 1, which is important for applications. The uniqueness questions of recovering the potential q(x) from transmission eigenvalues were studied comprehensively. Here we investigate the solvability and stability of this inverse problem. For this purpose, we suggest the so-called regularization approach, under which there should first be chosen some regular subclass of eigenvalue problems under consideration, which actually determines the course of the study and even the precise statement of the inverse problem. For definiteness, by assuming q(x) to be a complex-valued function in possessing the zero mean value and q(1) ≠ 0, we study properties of transmission eigenvalues and prove the local solvability and stability of recovering q(x) from the spectrum along with the value q(1). In the appendices, we provide some illustrative examples of regular and irregular transmission eigenvalue problems, and we also obtain necessary and sufficient conditions in terms of the characteristic function for the solvability of the inverse problem of recovering an arbitrary real-valued square-integrable potential q(x) from the spectrum for any fixed .

An Inverse Spectral Problem of Sturm–Liouville Problems with Transmission Conditions

In this paper, a class of inverse Sturm–Liouville problems of Atkinson type with transmission conditions is investigated. Based on the method of well-known theory for inverse matrix eigenvalue problems and the spectral properties of Sturm–Liouville problems with transmission conditions of Atkinson type, the corresponding Sturm–Liouville problems can be constructed using two sets of interlacing real numbers and some other conditions. We solve the problems not only under the separated boundary conditions, but also under the real coupled boundary conditions and the general real self-adjoint transmission conditions.

Domain Identification for Inverse Problem via Conformal Mapping and Fixed Point Methods in Two Dimensions

In this paper, we present a survey of the inverse eigenvalue problem for a Laplacian equation based on available Cauchy data on a known part Γ0 and a homogeneous Dirichlet condition on an unknown part Γ0 of the boundary of a bounded domain, Ω⊂ℝN. We consider variations in the eigenvalues and propose a conformal mapping tool to reconstruct a part of the boundary curve of the two-dimensional bounded domain based on the Cauchy data of a holomorphic function that maps the unit disk onto the unknown domain. The boundary values of this holomorphic function are obtained by solving a nonlocal differential Bessel equation. Then, the unknown boundary is obtained as the image of the boundary of the unit disk by solving an ill-posed Cauchy problem for holomorphic functions via a regularized power expansion. The Cauchy data were restricted to a nonvanishing function and to the normal derivatives without zeros. We prove the existence and uniqueness of the holomorphic function being considered and use the fixed-point method to numerically analyze the results of convergence. We’ll calculate the eigenvalues and compare the result with the shape obtained via minimization functional method, as developed in a previous study. Further, we’ll observe via simulations the shape of Γ and if it preserves its properties with varying the eigenvalues.

An extension of the Cayley transform method for a parameterized generalized inverse eigenvalue problem

SummarySince recent studies have shown that the Cayley transform method can be an effective iterative method for solving the inverse eigenvalue problem, in this work, we consider using an extension of it for solving a type of parameterized generalized inverse eigenvalue problem and prove its locally quadratic convergence. This type of inverse eigenvalue problem, which includes multiplicative and additive inverse eigenvalue problems, appears in many applications. Also, we consider the case where the given eigenvalues are multiple. In this case, we describe a modified problem that is not overdetermined and discuss the extension of the Cayley transform method for this modified problem. Finally, to demonstrate the effectiveness of these algorithms, we present some numerical examples to show that the proposed methods are practical and efficient.

Simultaneous assignment of resonances and antiresonances in vibrating systems through inverse dynamic structural modification

This paper proposes a method for the concurrent assignment of resonance and antiresonance frequencies in cross-receptances of undamped vibrating systems, by exploiting inverse dynamic structural modification. The technique also allows partially assigning the mode shapes of interest, which in turn makes it possible to perform pole-zero flipping in cross receptances. This is useful to shape receptances in the frequency range of interest. The method uses the inverse eigenvalue problem of the actual system and the adjunct one, by formulating it through the system matrices, to overcome the numerical issues of using the characteristic polynomials. Besides the theoretical formulation and the numerical application of the method, experimental validation is proposed through a laboratory cantilever beam, that shows the method effectiveness and ease of implementation.

Left and right inverse eigenpairs problem with a submatrix constraint for the generalized centrosymmetric matrix

Abstract Left and right inverse eigenpairs problem is a special inverse eigenvalue problem. There are many meaningful results about this problem. However, few authors have considered the left and right inverse eigenpairs problem with a submatrix constraint. In this article, we will consider the left and right inverse eigenpairs problem with the leading principal submatrix constraint for the generalized centrosymmetric matrix and its optimal approximation problem. Combining the special properties of left and right eigenpairs and the generalized singular value decomposition, we derive the solvability conditions of the problem and its general solutions. With the invariance of the Frobenius norm under orthogonal transformations, we obtain the unique solution of optimal approximation problem. We present an algorithm and numerical experiment to give the optimal approximation solution. Our results extend and unify many results for left and right inverse eigenpairs problem and the inverse eigenvalue problem of centrosymmetric matrices with a submatrix constraint.

Solving generalized inverse eigenvalue problems via L-BFGS-B method

ABSTRACT The parameterized generalized inverse eigenvalue problems containing multiplicative and additive inverse eigenvalue problems appear in vibrating systems design, structural design, and inverse Sturm–Liouville problems. In this article, by using the Cholesky factorization and the Jacobi method, we propose two efficient algorithms based on Newton's method and the L-BFGS-B method for solving these problems. To demonstrate the effectiveness of the algorithms, we present three numerical examples.

Inverse eigenvalue problems for skew-Hermitian reflexive and anti-reflexive matrices and their optimal approximations

In this paper, the inverse eigenvalue problems for skew-Hermitian reflexive and anti-reflexive matrices and their associated optimal approximation problems which are constrained by their partially prescribed eigenpairs are considered, respectively. First, the necessary and sufficient conditions of the solvability for the inverse eigenvalue problems of skew-Hermitian reflexive and anti-reflexive matrices are both derived, and the general solutions are also presented. Then the solutions of the corresponding optimal approximation problems in the Frobenius norm to a given matrix are also given, respectively. Furthermore, we give the algorithms to compute the optimal approximate skew-Hermitian reflexive and anti-reflexive solutions and present some illustrative numerical examples.

The NIEP and the positive realization problem

The nonnegative inverse eigenvalue problem is the problem of determining necessary and sufficient conditions for a multiset of complex numbers to be the spectrum of a nonnegative real matrix of size equal to the cardinality of the multiset itself. The problem is longstanding and proved to be very difficult so that several variations have been defined by considering particular classes of multisets and nonnegative real matrices. In this paper, a novel variation of the problem is proposed. This variation is motivated by a practical application in the positive realization problem, that is the problem of characterizing existence and minimality of a positive state--space representation of a given transfer function.

Supervised learning of time-independent Hamiltonians for gate design

We present a general framework to tackle the problem of finding time-independent dynamics generating target unitary evolutions. We show that this problem is equivalently stated as a set of conditions over the spectrum of the time-independent gate generator, thus translating the task into an inverse eigenvalue problem. We illustrate our methodology by identifying suitable time-independent generators implementing Toffoli and Fredkin gates without the need for ancillae or effective evolutions. We show how the same conditions can be used to solve the problem numerically, via supervised learning techniques. In turn, this allows us to solve problems that are not amenable, in general, to direct analytical solution, providing at the same time a high degree of flexibility over the types of gate-design problems that can be approached. As a significant example, we find generators for the Toffoli gate using only diagonal pairwise interactions, which are easier to implement in some experimental architectures. To showcase the flexibility of the supervised learning approach, we give an example of a non-trivial four-qubit gate that is implementable using only diagonal, pairwise interactions.

An inverse eigenvalue problem for Hamiltonian matrices

Let J=0In−In0∈R2n×2n. A matrix H∈R2n×2n is called Hamiltonian if (HJ)⊤=HJ. In this paper, the inverse eigenvalue problem for Hamiltonian matrices is considered. The solvability condition for the inverse problem is derived and the representation of the general solution is presented by the generalized singular value decomposition of a matrix pair. Furthermore, the associated optimal approximation problem for this inverse eigenvalue problem is discussed and the expression of the solution for the optimal approximation problem is presented.

On the direct and inverse transmission eigenvalue problems for the Schrödinger operator on the half line

For the direct problem, we give the asymptotic distribution of the (real and nonreal) transmission eigenvalues for the Schrödinger operator on the half line. For the inverse problem, we prove that the potential can be uniquely determined by all transmission eigenvalues, the parameter in the boundary condition, and some nonzero value related to the potential (whereas without the certain constant γ). The reconstruction algorithms are also revealed.

Inverse eigenvalue problems for a damped Stieltjes string with mixed data

In this paper, we consider the inverse eigenvalue problem for a Stieltjes string subject to the Robin condition at the left end and a damping condition at the right end with mixed data, which consists of the values of a part of masses and lengths and a subset of its spectrum. We use Krein-Nudelman's interpolation formula for a rational S-function to deal with our problem. We present a decomposition of the S0-function associated with the Stieltjes string, which makes it possible to use Krein-Nudelman's interpolation formula. Necessary and sufficient conditions are given for existence and uniqueness of solution to the inverse problem such that a finite number of simple not purely imaginary complex numbers lying in the open upper half-plane are a subset of the spectrum of a damped Stieltjes string.

Solvability and stability of the inverse Sturm–Liouville problem with analytical functions in the boundary condition

The paper deals with the Sturm–Liouville eigenvalue problem with the Dirichlet boundary condition at one end of the interval and with the boundary condition containing entire functions of the spectral parameter at the other end. We study the inverse problem, which consists in recovering the potential from a part of the spectrum. This inverse problem generalizes partial inverse problems on finite intervals and on graphs and also the inverse transmission eigenvalue problem. We obtain sufficient conditions for global solvability of the studied inverse problem, which prove its local solvability and stability. In addition, application of our main results to the partial inverse Sturm–Liouville problem on the star‐shaped graph is provided.

A Note on NIEP for Leslie and Doubly Leslie Matrices

The nonnegative inverse eigenvalue problem (NIEP) consists of finding necessary and sufficient conditions for the existence of a nonnegative matrix with a given list of complex numbers as its spectrum. If the matrix is required to be Leslie (doubly Leslie), the problem is called the Leslie (doubly Leslie) nonnegative eigenvalue inverse problem. In this paper, necessary and/or sufficient conditions for the existence and construction of Leslie and doubly Leslie matrices with a given spectrum are considered.

On the universal realizability problem

We say that a list Λ={λ1,λ2,…,λn} of complex numbers is realizable if it is the spectrum of an entrywise nonnegative matrix A. We say that Λ is universally realizable (UR) if Λ is realizable for each possible Jordan canonical form allowed by Λ. The universal realizability problem of spectra (URP) has been solved for certain kind of real and complex spectra, mainly spectra in the left half-plane. In this paper we look for extending the kind of spectra that are UR by given new sufficient conditions for the URP to have a solution. In particular, we show that some realizability criteria for the real and symmetric nonnegative inverse eigenvalue problem are realizability criteria for the URP as well.