Fasma Diele

Gradient flow methods for matrix completion with prescribed eigenvalues

Matrix completion with prescribed eigenvalues is a special type of inverse eigenvalue problem. The goal is to construct a matrix subject to both the structural constraint of prescribed entries and the spectral constraint of prescribed spectrum. The challenge of such a completion problem lies in the intertwining of the cardinality and the location of the prescribed entries so that the inverse problem is solvable. An intriguing question is whether matrices can have arbitrary entries at arbitrary locations with arbitrary eigenvalues and how to complete such a matrix. Constructive proofs exist to a certain point (and those proofs, such as the classical Schur–Horn theorem, are amazingly elegant enough in their own right) beyond which very few theories or numerical algorithms are available. In this paper the completion problem is recast as one of minimizing the distance between the isospectral matrices with the prescribed eigenvalues and the affined matrices with the prescribed entries. The gradient flow is proposed as a numerical means to tackle the construction. This approach is general enough that it can be used to explore the existence question when the prescribed entries are at arbitrary locations with arbitrary cardinalities.

On the inverse problem of constructing symmetric pentadiagonal Toeplitz matrices from their three largest eigenvalues

The inverse problem of constructing a symmetric Toeplitz matrix with prescribed eigenvalues has been a challenge both theoretically and computationally in the literature. It is now known in theory that symmetric Toeplitz matrices can have arbitrary real spectra. This paper addresses a similar problem?can the three largest eigenvalues of symmetric pentadiagonal Toeplitz matrices be arbitrary? Given three real numbers ? ? ? ? ?, this paper finds that the ratio , including infinity if ? = ?, determines whether there is a symmetric pentadiagonal Toeplitz matrix with ?, ? and ? as its three largest eigenvalues. It is shown that such a matrix of size n ? n does not exist if n is even and ? is too large or if n is odd and ? is too close to 1. When such a matrix does exist, a numerical method is proposed for the construction.

On Some Inverse Eigenvalue Problems with Toeplitz-Related Structure

Some inverse eigenvalue problems for matrices with Toeplitz-related structure are considered in this paper. In particular, the solutions of the inverse eigenvalue problems for Toeplitz-plus-Hankel matrices and for Toeplitz matrices having all double eigenvalues are characterized, respectively, in close form. Being centrosymmetric itself, the Toeplitz-plus-Hankel solution can be used as an initial value in a continuation method to solve the more difficult inverse eigenvalue problem for symmetric Toeplitz matrices. Numerical testing results show a clear advantage of such an application.

IMSP schemes for spatially explicit models of cyclic populations and metapopulation dynamics

We examine spatially explicit models described by reaction-diffusion partial differential equations for the study of predator-prey population dynamics. The numerical methods we propose are based on the coupling of a finite difference/element spatial discretization and a suitable partitioned Runge-Kutta scheme for the approximation in time. The RK scheme here implemented uses an implicit scheme for the stiff diffusive term and a partitioned RK symplectic scheme for the reaction term (IMSP schemes). We revisit some results provided in the literature for the classical Lotka-Volterra system and the Rosenzweig-MacArthur model. We then extend the approach to metapopulation dynamics in order to numerically investigate the effect of migration through a corridor connecting two habitat patches. Moreover, we analyze the synchronization properties of subpopulation dynamics, when the migration occurs through corridors of variable sizes.

On Computing the Spectral Decomposition of Symmetric Arrowhead Matrices

The computation of the spectral decomposition of a symmetric arrowhead matrix is an important problem in applied mathematics [10]. It is also the kernel of divide and conquer algorithms for computing the Schur decomposition of symmetric tridiagonal matrices [2,7,8] and diagonal–plus–semiseparable matrices [3,9]. The eigenvalues of symmetric arrowhead matrices are the zeros of a secular equation [5] and some iterative algorithms have been proposed for their computation [2,7,8]. An important issue of these algorithms is the choice of the initial guess. Let α 1 ≤ α 2 ≤... ≤ α n − 1 be the entries of the main diagonal of a symmetric arrowhead matrix of order n. Denoted by λ i , i=1, ..., n, the corresponding eigenvalues, it is well know that α i ≤ λ i + 1 ≤ α i + 1, i=1,..., n-2. An algorithm for computing each eigenvalue λ i , i=1, ..., n, of a symmetric arrowhead matrix with monotonic quadratic convergence, independent of the choice of the initial guess in the interval ]α i − 1,α i [ is proposed in this paper. Although the eigenvalues of a symmetric arrowhead matrix can be computed efficiently, a loss of orthogonality can occur in the computed matrix of eigenvectors [2,7,8].In this paper we propose also a simple, stable and efficient way to compute the eigenvectors of arrowhead matrices.

Isospectral flows and the inverse eigenvalue problem for Toeplitz matrices

Abstract The inverse eigenvalue problem for Toeplitz matrices (ITEP), concerning the reconstruction of a symmetric Toeplitz matrix from prescribed spectral data, is considered. To numerically construct such a matrix the approach introduced by Chu in (SIAM Rev. 40(1) (1998) 1–39) is followed. He proposed to solve the ITEP by using an isospectral flow whose equilibria are symmetric Toeplitz matrices. In this paper we study the previous isospectral flow for reversed times and we obtain some formal properties of the solution. The case n =3 for ITEP is analytically investigated by following an approach different from the one in (Chu, SIAM Rev. 40(1) (1998) 1–39). We prove that the flow globally converges to a regular Toeplitz matrix starting from a tridiagonal symmetric and centro-symmetric matrix. Numerical experiments confirm the above results and suggest their extension in higher dimension.

Exponential Lawson integration for nearly Hamiltonian systems arising in optimal control

Abstract: We are concerned with the discretization of optimal control problems when a Runge-Kutta scheme is selected for the related Hamiltonian system. It is known that Lagrangians first order conditions on the discrete model, require a symplectic partitioned Runge-Kutta scheme for state-costate equations. In the present paper this result is extended to growth models, widely used in Economics studies, where the system is described by a current Hamiltonian. We prove that a correct numerical treatment of the state-current costate system needs Lawson exponential schemes for the costate approximation. In the numerical tests a shooting strategy is employed in order to verify the accuracy, up to the fourth order, of the innovative procedure we propose.

Numerical Methods Based on Gaussian Quadrature and Continuous Runge-Kutta Integration for Optimal Control Problems

This paper provides a numerical approach for solving optimal control problems governed by ordinary differential equations. Continuous extension of an explicit, fixed step-size Runge-Kutta scheme is used in order to approximate state variables; moreover, the objective function is discretized by means of Gaussian quadrature rules. The resulting scheme represents a nonlinear programming problem, which can be solved by optimization algorithms. With the aim to test the proposed method, it is applied to different problems.

Direct Optimization Using Gaussian Quadrature and Continuous Runge-Kutta Methods: Application to an Innovation Diffusion Model

In the present paper the discretization of a particular model arising in the economic field of innovation diffusion is developed. It consists of an optimal control problem governed by an ordinary differential equation. We propose a direct optimization approach characterized by an explicit, fixed step-size continuous Runge-Kutta integration for the state variable approximation. Moreover, high-order Gaussian quadrature rules are used to discretize the objective function. In this way, the optimal control problem is converted into a nonlinear programming one which is solved by means of classical algorithms.

SB3A splitting for approximation of invariants in time-dependent Hamiltonian systems ☆

Abstract Most physical phenomena are described by time-dependent Hamiltonian systems with qualitative features that should be preserved by numerical integrators used for approximating their dynamics. The initial energy of the system together with the energy added or subtracted by the outside forces, represent a conserved quantity of the motion. For a class of time-dependent Hamiltonian systems [8] this invariant can be defined by means of an auxiliary function whose dynamics has to be integrated simultaneously with the system’s equations. We propose splitting procedures featured by a SB 3 A property that allows to construct composition methods with a reduced number of determining order equations and to provide the same high accuracy for both the dynamics and the preservation of the invariant quantity.