Comparative analysis of inverse coefficient problems for parabolic equations. Part II: Coarse-fine grid algorithm

Abstract

This article presents a computational analysis of the adjoint problem approach for parabolic inverse coefficient problems based on boundary measured data. The proposed coarse-fine grid algorithm constructed on the basis of this approach is an effective computational tool for the numerical solution of inverse coefficient problems with various Neumann or/and Dirichlet type measured output data. In the previous Part I paper it was shown that the ill-posedness also depends on where Neumann and Dirichlet conditions are given: in the direct problem or as an output data. Based on integral identities relating solutions of direct problems to appropriate adjoint problems solutions, a coarse-fine grid algorithm for parabolic coefficient identification problems is constructed. It is shown that use of a coarse grid for the interpolation of the unknown coefficient and a fine grid for the numerical solution of the well-posed forward and backward parabolic problems guarantees an optimal compromise between the accuracy and stability in numerically solving the inverse problems. The efficiency and applicability of this method is demonstrated on various numerical examples with noisy free and noisy data.