Regularization of Inverse Problems by Spline-Approximation Method and Applications
by
Alexandre Ivanovich Grebennikov
Benemerita Universidad Autonoma de Puebla, Facultad de Ciencias Fisico Matematicas, Av. San Claudio y 18 sur,72570 Puebla, Pue. Mexico

It is presented the generalization of the theoretical results of the Spline-Approximation Method [1], [2] for resolving wide class of the operator equations. The application to the Inverse Problem in Electroencephalography (IPEEG) and Coefficient Inverse Problems for Differential Equations (CIPDE) of the parabolic type is consided. IPEEG consists in to determine the spatial distribution of the cortical current produced by electric activity of the brain. The input data presents the electroencephalographic measurement of the electric potential on the scalp. In proposed model electric potential satisfies the Laplace's equation and original suitable boundary conditions. Using the potential theory, we obtain the system of two-dimensional slow singular integral equations [3], [4]. CIPDE consists in a restoration of the coefficients of an equation of the parabolic type, using as the input data the measurements of the solution of the equation on nonregular net. In [5] some results are obtaned para restoration of depending only of time coefficient at the derivative on the time. Here is consided the problem for restoring of the coefficientes at the derivatives in Laplacian, written in the divergent form. This model corresponding to the applide problems of heat-conduction so as of identifiyng the characteristics of the porous media of confined aquifers.

On the base of the extended theory of Spline-Approximation Method the stable and fast algorithms of numerical solution of formulated mathematical problem are constructed and justified. The results of numerical calculations for model examples are presented.

1.Grebennikov, A.I. Spline approximation method for restoring functions. Sov. J. Numer. Anal. Mathem. Modelling, Vol.4, N 4, pp.1-15(1989).

2. Grebennikov, A.I. Solving integral equations with a singularity in the kernel by spline approximation method. Sov.J.Numer.Anal. Vol.5, N 3, pp.199-208(1990).

3.Grebennikov A. I. On solving some inverse problems of electroencephalographia on spline - approximation method. Collection of the Papers of the International Simposium on Optimization of Calculating, Kiev, 1999.

4. Grebennikov A. I., Fraguela A. Statemen and numerical analysis of some inverse problems of electroencephalography // Numerical Analysis: theory, applications, programms. Moscow, MSU, 1999, pp. 28-46.

5. Grebennikov A. I. Fast spline-algorithms for multidimentional data processing and solving coefficient inverse problems for heat-conduction equation // Numerical analysis: methods and algorithms. Moscow, MSU, 1998, pp. 62-71.

Date received: February 1, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caeb-82.