Method of variational imbedding for identification of heat-conduction coefficient from overposed boundary data

Abstract

We consider the inverse problem of identifying a spatially varying coefficient in diffusion equation from overspecified boundary conditions. We make use of a technique called Method of Variational Imbedding (MVI) which consists in replacing the original inverse problem by the boundary value problem for the Euler-Lagrange equations presenting the necessary conditions for minimization of the quadratic functional of the original equations. The latter is well-posed for redundant data at boundaries. The equivalence of the two problems is demonstrated. In the recent authors' works difference scheme and algorithm have been created to apply MVI to the problem under consideration. In the present work we show that the number of boundary conditions can be decreased, replacing them with the so-called "natural conditions" for minimization of a functional. A difference scheme of splitting type is employed and featuring examples are elaborated numerically.