Stefania Ragni

Splitting and composition methods for explicit time dependence in separable dynamical systems

We consider splitting methods for the numerical integration of separable non-autonomous differential equations. In recent years, splitting methods have been extensively used as geometric numerical integrators showing excellent performances (both qualitatively and quantitatively) when applied on many problems. They are designed for autonomous separable systems, and a substantial number of methods tailored for different structures of the equations have recently appeared. Splitting methods have also been used for separable non-autonomous problems either by solving each non-autonomous part separately or after each vector field is frozen properly. We show that both procedures correspond to introducing the time as two new coordinates. We generalize these results by considering the time as one or more further coordinates which can be integrated following either of the previous two techniques. We show that the performance as well as the order of the final method can strongly depend on the particular choice. We present a simple analysis which, in many relevant cases, allows one to choose the most appropriate split to retain the high performance the methods show on the autonomous problems. This technique is applied to different problems and its performance is illustrated for several numerical examples.

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.

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.

Numerical Solutions to Some Optimal Control Problems Arising fromInnovation Diffusion

In this paper we propose a numerical approach for the solution of some optimalcontrol problems arising in the field of marketing decision models. Inparticular, we account for a specific innovation diffusion model. A numericalapproach may be useful to investigate some features of state variables andparameters of interest. The discrete problem is solved by the SimulatedAnnealing method and the resulting numerical scheme is applied to some testcases.

Numerical Comparison between Different Lie-Group Methods for Solving Linear Oscillatory ODEs

In this paper we deal with high oscillatory systems and numerical methods for the approximation of their solutions. Some classical schemes developed in the literature are recalled and a recent approach based on the expression of the oscillatory solution by means of the exponential map is considered. Moreover we introduce a new method based on the Cayley map and provide some numerical tests in order to compare the different approaches.

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.

The global error of Magnus methods based on the Cayley map for some oscillatory problems

This paper deals with numerical methods for the discretization of highly oscillatory systems. We approach the problem by writing the solution in terms of the Magnus expansion based on the Cayley map. The global error, obtained when the method is applied to the linear oscillator, is investigated. Moreover, we provide numerical experiments in order to validate our theoretical results.

Runge‐Kutta Discretizations of Infinite Horizon Optimal Control Problems with Steady‐State Invariance

Direct numerical approximation of a continuous‐time infinite horizon control problem, requires to recast the model as a discrete‐time, finite‐horizon control model. The quality of the optimization results can be heavily degraded if the discretization process does not take into account features of the original model to be preserved. Restricting their attention to optimal growh problems with a steady state, Mercenier and Michel in [1] and [2], studied the conditions to be imposed for ensuring that discrete first‐order approximation models have the same steady states as the infinite‐horizon continuous‐times counterpart. Here we show that Mercenier and Michel scheme is a first order partitioned Runge‐Kutta method applied to the state‐costate differential system which arises from the Pontryagin maximum principle. The main consequence is that it is possible to consider high order schemes which generalize that algorithm by preserving the steady‐growth invariance of the solutions with respect to the discretizatio...