Approximate localisation of imperfections in fixed domain
Marta Lipnicka · Roczniki Polskiego Towarzystwa Matematycznego. Seria 3, Matematyka Stosowana/Matematyka Stosowana/Mathematica Applicanda · 2011
In this paper we present a procedure for determining the approximate location of imperfection in a fixed domain. We define the spectral problem whose solutions are eigenvalues. These values depend on the location and the size of the imperfection. The main aim in this work is to find the solution of the inverse problem. It means that we find the location of the imperfection in our domain based on the vector of eigenvalues. For the inverse problem we don’t have the uniqueness of the solutions so we Define a new problem in new domain. For the new problem we obtain the existence of the approximate solution of the inverse problem. 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 by the Elman’s neural networks. The networks are built in a dynamic way. Their size depends on the size of the learning set. The approximation method is convergent. Keywords: Neural networks, Eigenvalues, Approximation, Conditonal expectation.