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Solving coefficient inverse problems of mathematical physics using radial basis functions networks

Keywords:

M.V. Zhukov – Post-graduate Student, Penza State University. E-mail: maxim.zh@gmail.com


The inverse problems are marked out among mathematical physics problems. These problems have the known mathematical model structure of a researched process but have some unknown the model’s components. To solve them not only the solution should be defined, but also the model should be identified. The inverse problems are often ill-posed problems in the classical sense (ill-posed by Hadamard). The breach of the solution’s continuous dependence on input is a typical situation. To make ill-posed problem conditionally well-posed (well-posed by Tikhonov) additional information and regularization are used. The mess-free method is used to solve coefficient inverse problems here. The unknown solution and the unknown coefficients of the mathematical model are approximated with radial basis functions networks. The network’s parameters are determined by a condition of residuals minimization between the left and the right parts of an equation, initial boundary and additional conditions. The iteration method is used to regularize a solution. A quantity of iterations acts as a regularization factor. It is defended by a residual between the right and the left parts of additional conditions. The approach was considered by example of coefficient inverse problems for elliptic and non-linear parabolic equations. This choice allowed to reflect peculiarities of stationary and non-stationary problems solution. The results of four experiments are presented in this article. They helped to come to the conclusion that the introduced approach is effective from the point of view of computational complexity and precision as well as from the point of view of realization complexity.
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