Carlos D. Acosta

Stabilization of explicit methods for convection diffusion equations by discrete mollification

The main goal of this paper is to show that discrete mollification is a simple and effective way to speed up explicit time-stepping schemes for partial differential equations. The second objective is to enhance the mollification method with a variety of alternatives for the treatment of boundary conditions. The numerical experiments indicate that stabilization by mollification is a technique that works well for a variety of explicit schemes applied to linear and nonlinear differential equations.

Numerical identification of a nonlinear diffusion coefficient by discrete mollification

The discrete mollification method is a convolution-based filtering procedure suitable for the regularization of ill-posed problems. Combined with explicit space-marching finite difference schemes, it provides stability and convergence for a variety of coefficient identification problems in linear parabolic equations. In this paper, we extend such a technique to identify some nonlinear diffusion coefficients depending on an unknown space dependent function in one dimensional parabolic models. For the coefficient recovery process, we present detailed error estimates and to illustrate the performance of the algorithms, several numerical examples are included.

Efficient parameter estimation in a macroscopic traffic flow model by discrete mollification

Our concern is the numerical identification of traffic flow parameters in a macroscopic one-dimensional model whose governing equation is strongly degenerate parabolic. The unknown parameters determine the flux and the diffusion terms. The parameters are estimated by repeatedly solving the corresponding direct problem under variation of the parameter values, starting from an initial guess, with the aim of minimizing the distance between a time-dependent observation and the corresponding numerical solution. The direct problem is solved by a modification of a well-known monotone finite difference scheme obtained by discretizing the nonlinear diffusive term by a formula that involves a discrete mollification operator. The mollified scheme occupies a larger stencil but converges under a less restrictive Courant-Friedrichs-Lewy (CFL) condition, which allows one to employ a larger time step. The ability of the proposed procedure for the identification of traffic flow parameters is illustrated by a numerical experiment.

A mollification based operator splitting method for convection diffusion equations

The main goal of this paper is to show that discrete mollification is a suitable ingredient in operator splitting methods for the numerical solution of nonlinear convection-diffusion equations. In order to achieve this goal, we substitute the second step of the operator splitting method of Karlsen and Risebro (1997) [1] for a mollification step and prove that the convergence features are fairly well preserved. We end the paper with illustrative numerical experiments.

Stability Analysis of a Finite Difference Scheme for a Nonlinear Time Fractional Convection Diffusion Equation

The nonlinear time fractional convection diffusion equation (TFCDE) is obtained from a standard nonlinear convection diffusion equation by replacing the first-order time derivative with a fractional derivative (in Caputo sense) of order \(\alpha \in (0,1)\). Developing numerical methods for solving fractional partial differential equations is of increasing interest in many areas of science and engineering. In this chapter, an explicit conservative finite difference scheme for TFCDE is introduced. We find its Courant–Friedrichs–Lewy (CFL) condition and prove encouraging results regarding stability, namely, monotonicity, the total variation diminishing (TVD) property and several bounds. Illustrative numerical examples are included in order to evaluate potential uses of the new method.

Comparación de Métodos de Reducción de Dimensión Basados en Análisis por Localidades

In this paper, a comparison of methods for nonlinear dimensionality reduction is proposed in order to determine which technique preserves better the local properties, without losing the overall structure of the original data. We seek to establish which of these methods is the most appropriate for visualization tasks. The embeddings obtained with each technique are evaluated by two criteria Preservation Neighborhood Error and Preserved Neighbors Average. The methodologies were tested on artificial and real-world data sets which allow us to visually confirm the quality of the embedding. The results obtained show that Maximum variance unfolding computes high quality embeddings, because the optimization problem pretends to preserve exactly the local pair-wise distance between neighbors and conserve the global manifold structure.

Estimación Dinámica Neuronal a partir de Señales Electroencefalográficas sobre un Modelo Realista de la Cabeza

In this paper is presented a method for neural activity estimation over the brain that take into account, for the solution of the inverse problem, a dynamic model for the neural activity in a realistic head model calculated with bounded elements method, according to a physiologically based model that describes the real interaction between neurons. The solution of the inverse problem is calculated using high performance computing. This analysis is performed for simulated EEG signals for SNR of 25 dB, 15 dB and 5 dB. The obtained results show the robustness of the estimation method that includes the dynamic model in comparison with the static model for several levels of noise.