Juan Carlos Jimenez

Juan Carlos Jimenez, Alina Sotolongo, Jose Miguel Sanchez-Bornot

In this paper, the effect that produces the local linearization of the embedded Runge-Kutta formulas of Dormand and Prince for initial value problems is studied. For this, embedded Locally Linearized Runge-Kutta formulas are defined and their performance is analyzed by means of exhaustive numerical simulations. For a variety of well-known physical equations with different dynamics, the simulation results show that the locally linearized formulas exhibit significant higher accuracy than the original ones, which implies a substantial reduction of the number of time steps and, consequently, a sensitive reduction of the overall computation cost of their adaptive implementation.

Juan Carlos Jimenez

Explicit formulas for the mean and variance of linear stochastic differential equations are derived in terms of an exponential matrix. This result improved a previous one by means of which the mean and variance are expressed in terms of a linear combination of higher dimensional exponential matrices. The important role of the new formulas for the system identification as well as numerical algorithms for their practical implementation are pointed out.

Juan Carlos Jimenez

In this paper, approximate Linear Minimum Variance (LMV) filters for continuous-discrete state space models are introduced. The filters are obtained by means of a recursive approximation to the predictions for the first two moments of the state equation. It is shown that the approximate filters converge to the exact LMV filter when the error between the predictions and their approximations decreases. As particular instance, the order-$β$ Local Linearization filters are presented and expounded in detail. Practical algorithms are also provided and their performance in simulation is illustrated with various examples. The proposed filters are intended for the recurrent practical situation where a nonlinear stochastic system should be identified from a reduced number of partial and noisy observations distant in time.

Zochil González Arenas, Juan Carlos Jimenez, Li-Vang Lozada-Chang et al.

Estimation of Distribution Algorithms (EDAs) and Innovation Method are recognized methods for solving global optimization problems and for the estimation of parameters in diffusion processes, respectively. Well known is also that the quality of the Innovation Estimator strongly depends on an adequate selection of the initial value for the parameters when a local optimization algorithm is used in its computation. Alternatively, in this paper, we study the feasibility of a specific EDA - a continuous version of the Univariate Marginal Distribution Algorithm (UMDAc) - for the computation of the Innovation Estimators. Numerical experiments are performed for two different models with a high level of complexity. The numerical simulations show that the considered global optimization algorithms substantially improves the effectiveness of the Innovation Estimators for different types of diffusion processes with complex nonlinear and stochastic dynamics.