Approximate localisation of imperfections in fixed domain

Marta Lipnicka

Mathematica Applicanda, Tome 39 (2011) no. 2, pp. 47-78.

Résumé

In this paper we present a procedure for determining the approximate locationof imperfection in a fixed domain. We define the spectral problem whose solutions areeigenvalues. These values depend on the location and the size of the imperfection. Themain aim in this work is to find the solution of the inverse problem. It means that we findthe location of the imperfection in our domain based on the vector of eigenvalues. For theinverse problem we don’t have the uniqueness of the solutions so we Define a new problemin new domain.For the new problem we obtain the existence of the approximate solution of the inverseproblem. In order to determine the location of imperfection we define a new mapping.This mapping is defined as the conditional expectation of the location of imperfection,provided that we know the finite number of eigenvalues. The mapping is approximated bythe Elman’s neural networks. The networks are built in a dynamic way. Their size dependson the size of the learning set. The approximation method is convergent.Keywords: Neural networks, Eigenvalues, Approximation, Conditonal expectation.

Détail

DOI : 10.14708/ma.v39i2.177

Mots-clés : In this paper we present a procedure for determining the approximate locationof imperfection in a fixed domain. We define the spectral problem whose solutions areeigenvalues. These values depend on the location and the size of the imperfection. Themain aim in this work is to find the solution of the inverse problem. It means that we findthe location of the imperfection in our domain based on the vector of eigenvalues. For theinverse problem we don’t have the uniqueness of the solutions so we Define a new problemin new domain.For the new problem we obtain the existence of the approximate solution of the inverseproblem. In order to determine the location of imperfection we define a new mapping.This mapping is defined as the conditional expectation of the location of imperfection,provided that we know the finite number of eigenvalues. The mapping is approximated bythe Elman’s neural networks. The networks are built in a dynamic way. Their size dependson the size of the learning set. The approximation method is convergent. Neural networks, Eigenvalues, Approximation, Conditonal expectation.