Variational imbedding approach to coefficient identification in an elliptic partial differential equation

Abstract

We consider the inverse problem for identification of the coefficient in an elliptic partial differential equation inside the unit square D, with overposed boundary data. This problem is not investigated enough in the literature due to the lack of results about the uniqueness of the inverse problem. Following the main idea of the so-called Method of Variational Imbedding, we imbed the solution of the inverse problem into the elliptic Boundary Value Problem (BVP) stemming from the necessary conditions for minimisation of the quadratic functional of the original equation. The system contains a well-posed fourth-order BVP for the sought function and an explicit equation for the unknown coefficient. We solve imbedding BVP numerically by making use of operator-splitting for the fourth-order BVP. The convergence and stability of the numerical method are established. We introduce an effective way for finding the best approximation for the coefficient. Featuring examples are elaborated numerically.