
Mini
Workshop:
Numerical Computing, Optimization and Control of PDEIV
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 PostDischarge Nitriding
F. Castillo
francast@itesm.mx
A. Fraguela
fraguela@fcfm.buap.mx
J. Oseguera
joseguer@itesm.mx
J.A. Gómez
jagomez@ufro.cl
Facultad de Ingeniería y Arquitectura
ITESMCEM, Edo. de México, México
Facultad de Ciencias Físico Matemáticas
BUAP, Puebla, México
Departamento de Ingeniería Matemática
UFRO, Temuco, Chile
Abstract: Thermochemical
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 FeN 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 quasistabilization
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. BernalPonce, A. FraguelaCollar, J.A. Gómez,
J. OsegueraPeña, F. Castillo Aranguren, Identication of diffusion
coefficients during postdischarge 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), 1826 .
[3] Crank, J., Free and moving boundary problems. Clarendon
Press, Oxford, 1987.
On
Solving Parameter Estimation of Simulation Based Optimization Problems
Susana Gómez
susanag@servidor.unam.mx
IIMAS, Universidad Nacional Autonoma de
México
México, D.F., México
Abstract: Parameter
estimation of simulation based problems, have certain characteristics
due to their illposednes. These characteristics affect the behaviour
of the optimisation methods and can produce convergence to suboptimal
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
fraguela@fcfm.buap.mx
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
nondiscrete one (for instance, see [1] and [2]). That is, an inequality
of the form
where
is a sequence of nondiscrete 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.
Bibliography
[1] D. I. Berman, Some remarks on the problem of the degree of approximation
of polynomial operators, Izv. Vyssh. Uchebn. Zaved. Mat., 5 (1961),
35.
[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) 116122 (in russian).
[3] JD. 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), 117120.
[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), 235244.
[5] H. H. Gonska and JD. Cao, On Butzer's problem concerning approximation
by algebraic polynomials, Approximation Theory, G. Anastassiuo ed.,
Marcel Dekker, New York, (1992), 289313.
Inverse
Electroencephalography for Volumetric Sources
María Monserrat Morín
Castillo
mmorin@ece.buap.mx
Andrés Fraguela
Collar
fraguela@fcfm.buap.mx
José Jacobo Oliveros Oliveros
oliveros@fcfm.buap.mx
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
oliveros@fcfm.buap.mx
Andrés Fraguela Collar
fraguela@fcfm.buap.mx
Susana Gómez Gómez
susanag@servidor.unam.mx
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
apriori nonrotationality 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 nonrotationality
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 leastsquares subject to quadratic constraints.
