Carlos E. Mejía

Discrete mollification and automatic numerical differentiation

Abstract An automatic method for numerical differentiation, based on discrete mollification and the principle of generalized cross validation is presented. With data measured at a discrete set of points of a given interval, the method allows for the approximate recovery of the derivative function on the entire domain. No information about the noise is assumed. Error estimates are included together with several numerical examples of interest.

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.

Mollified Hyperbolic Method for Coefficient Identification Problems

Abstract We introduce a stable numerical method for the identification of a transmissivity coefficient in a one-dimensional parabolic equation. It is a combination of the Mollification Method and a well-known space marching implementation of the Hyperbolic Regularization procedure. The new method succesfully restores a certain type of continuity with respect to the initial condition and the boundary data. The accuracy of the algorithm is demonstrated by means of several examples where exact and perturbed data are considered.

New stable numerical inversion of Abel's integral equation

Abstract The 3-D image reconstruction from cone-beam projections in computerized tomography leads naturally, in the case of radial symmetry, to the study of Abels type of integral equations. A new method for the numerical solution of such equations when the experimental information is obtained through measured data, on a discrete set of points, is presented and rigorously analyzed.

Numerical solution of generalized IHCP by discrete mollification

A numerical space marching algorithm based on discrete mollification and automatic iterative filtering by Generalized Cross Validation is applied to the solution of a generalized one-dimensional inverse heat conduction problem. No information about the noise is assumed. With data temperature and heat flux functions measured at a discrete set of points on the boundary x = 1, 0 @? t @? 1, the temperature and heat flux solution functions are approximately recovered in the unit square of the (x, t) plane, including its boundaries. Error bounds and numerical examples are provided

Generalized time fractional IHCP with Caputo fractional derivatives

The numerical solution of the generalized time fractional inverse heat conduction problem (GTFIHCP) on a finite slab is investigated in the presence of measured (noisy) data when the time fractional derivative is interpreted in the sense of Caputo. The GTFIHCP involves the simultaneous identification of the heat flux and temperature transient functions at one of the boundaries of the finite slab together with the initial condition of the original direct problem from noisy Cauchy data at a discrete set of points on the opposite (active) boundary. A finite difference space marching scheme with adaptive regularization, using trigonometric mollification techniques and generalized cross validation is introduced. Error estimates for the numerical solution of the mollified problem and numerical examples are provided.

Numerical identification of diffusivity coefficient and initial condition by discrete mollification

Abstract We discuss the simultaneous identification of the initial condition and the space-time depending diffusivity coefficient for general linear one-dimensional parabolic equations when the measured information is obtained only at the active boundary. We solve these problems by introducing stable space marching implementations of the Mollification Method which restore continuity with respect to the data. Several numerical examples show the properties of the methods.

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.

Some Applications of the Mollification Method

The Mollification Method is a filtering procedure that is appropriate for the regularization of a variety of ill-posed problems. In this review, we briefly introduce the method, including its main feature, which is its ability to automatically select regularization parameters. After this introduction, we present several applications of the method, illustrated with numerical examples. Most of these applications are the subject of our current research.

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.