Mini Workshop:
Numerical Computing, Optimization and Control of PDE-IV

This minisimposium intends to join researchers in the field of partial differential equations. Both novel applications as well as numerical and theoretical presentations on optimization, control and solution of direct problems on PDE.
Layer Growth Kinetics During Post-Discharge Nitriding

F. Castillo

A. Fraguela

J. Oseguera

J.A. Gómez

Facultad de Ingeniería y Arquitectura
ITESM-CEM, Edo. de México, México
Facultad de Ciencias Físico- Matemáticas
BUAP, Puebla, México
Departamento de Ingeniería Matemática
UFRO, Temuco, Chile

Abstract: Thermo-chemical nitriding treatment produces an important improvement in the mechanical, tribological and chemical properties in steel, thus enhancing the resistance to their wear, corrosion and fatigue. Knowledge about compact layers is an important resource in automating nitriding processes. Moreover, from a practical standpoint, modeling of the process allows description and understanding of the transformation of the Fe-N system and its kinetics.

This work presents a mathematical model of the layer growth kinetics during a postdischarge nitriding process [1]. The model is related to a moving boundary value problem which takes into account the observed qualitative behavior in laboratory. The model assumes several steps: diffusion process, formation of the layers, layer growth and quasi-stabilization of the layer growth. An analytical approximate
solution of Goodman's type [2]-[3] is proposed, which allows us to obtain a representation of the motion of the interfaces and the nitrogen concentration profiles.

[1] J. Bernal-Ponce, A. Fraguela-Collar, J.A. Gómez, J. Oseguera-Peña, F. Castillo Aranguren, Identication of diffusion coefficients during post-discharge nitriding. Proceedings of the 5th International Conference on inverse Problems in Engineering: Theory and Practice, Cambridge, UK, Vol. 1, 2005, B05.

[2] Lifang Xia, Mufu Yan, Mathematical models of nitrogen concentration profile of iron nitrided layers and computer simulation, Acta Metallurgica (English Edition). Series B , 2 (1989), 18-26 .

[3] Crank, J., Free and moving boundary problems. Clarendon Press, Oxford, 1987.

On Solving Parameter Estimation of Simulation Based Optimization Problems

Susana Gómez

IIMAS, Universidad Nacional Autonoma de México
México, D.F.
, México

Abstract: Parameter estimation of simulation based problems, have certain characteristics due to their ill-posednes. These characteristics affect the behaviour of the optimisation methods and can produce convergence to sub-optimal points.

In this work we will discuss these problems and propose solution techniques. An oil reservoir characterization problem using well test historical data, will serve to illustrate these ideas.

Approximation of Discrete Boundary Data by Trigonometric Polynomials

Andrés Fraguela

Facultad de Ciencias Físico Matemáticas,
Benemérita Universidad Autónoma de Puebla

Abstract: This work considers an important problem in applications related to the solution of inverse problems of coefficients identification arising in Biology, Engineering and Natural Sciences. In many of these problems, the use of continuous approximation of discrete data by trigonometric polynomials in the uniform and mean square norms is enough in order to obtain numerical stability of inversion algorithms. As a particular case, the approximation of discrete data obtained with errors from measurements in a grid of nodes that lies on the interval is studied. We want to present numerical algorithms that provide solutions to the above problem. In many problems it is important to consider a discrete version of certain convolution type operators. There are several results that enable us to reduce the discrete case to the non-discrete one (for instance, see [1] and [2]). That is, an inequality of the form where is a sequence of non-discrete linear operators, is an appropriated discretization of and and are independents of n and f. Here we can not use these results because we need a good estimation of the constants involved. It is known that some good quadrature formulae can be obtained by using properties of orthogonal polynomials, but in many cases there are some drawbacks: their nodes cannot be calculated exactly and the coefficients of the fundamentals polynomials are not known. The best constructive results are obtained by discretizing the operators using certain numerical quadrature formulae of appropriate precision. These ideas has been used by different authors (see [5], [4] and the references cited there). In particular, Gonska and Cao considered in [3] and [5] a discretization for the convolution with the Jackson Kernel using the composite trapezoidal rule. In this case the trigonometric background is useful to derive polynomials that can be computed easily using results from Fourier Analysis and standard methods of Numerical Analysis.

[1] D. I. Berman, Some remarks on the problem of the degree of approximation of polynomial operators, Izv. Vyssh. Uchebn. Zaved. Mat., 5 (1961), 3-5.
[2] D. I. Berman, Some inequalities and their applications in the theory of interpolation, in Investigation on Modern Problems of the Constructive Theory of Functions, V. I. Smirnov (ed.), GIZFIL, Moscow, (1961) 116-122 (in russian).
[3] J-D. Cao and H. H. Gonska, Computation of the DeVore- Gopenhauz type approximants, in Approximation Theory VI, C. K. Chui et al eds., Academic Press, New York, (1989), 117-120.
[4] F. Esser and E. Grlich, Diskrete und kontinuier summationsverfahren von orthogonalreihen, in Mathematical Structures- Computational Mathematic- Mathematical Modelling, Bl. Sendov ed., Publishing House of the Bulgar Academy of Sciences, Sofia, (1975), 235-244.
[5] H. H. Gonska and J-D. Cao, On Butzer's problem concerning approximation by algebraic polynomials, Approximation Theory, G. Anastassiuo ed., Marcel Dekker, New York, (1992), 289-313.

Inverse Electroencephalography for Volumetric Sources

María Monserrat Morín Castillo

Andrés Fraguela Collar

José Jacobo Oliveros Oliveros

Facultad de Ciencias de la Electrónica
Puebla, Puebla, México

Abstract: In this work the problem of recovering bioelectrical sources on the cerebral volume, from measurement of the potential generated by these sources on the scalp, is studied. This problem is an ill posed problem, since given a measurement on the scalp, there are different bioelectrical sources that produce this measurement and small variations in the measurement can produce substantial variations in the source localization. The uniqueness is studied through a boundary value problem, which is obtained through a model that describes the head as a system of conductive layers. This model allows relationships between the characteristics of the bioelectrical activity and the EEG to be established. We find conditions under which the inverse solution is unique and we give an algorithm to find this solution. In the case in which the head is modeled through two concentric circles we give a regularization strategy.

A Novel Method to Solve Parameter Identification Problems in a Boundary Value Problem that Arises in the ECT

José Jacobo Oliveros Oliveros

Andrés Fraguela Collar

Susana Gómez Gómez

Facultad de Ciencias Físico Matemáticas
Ciudad Universitaria, Puebla, Puebla, México

Abstract: A method to solve parameter identification problem in a boundary value problem that arises in the inverse problem of image reconstruction of permittivity distribution in the cross section of a pipeline, using Electrical Capacitance Tomography, is described. This method needs the a-priori non-rotationality condition on the electrical displacement vector in the cross flux section and divides the original inverse problem in several simpler problems. Each one of these problems can be solved with a numerically stable procedure, using the theoretically found methods proposed for linear problems. In this work is proved that, under non-rotationality condition, the identification problem has an unique solution and that the boundary value problem is equivalent to a system of integral equations. In the present method it is not necessary to solve the direct problem at each iteration, since the resulting optimization problem obtained in this method, is a linear least-squares subject to quadratic constraints.