On identification of isotropic diffusion coefficient: approximation by Tikhonov-type regularization

Abstract: We consider the inverse problem of identifying an isotropic diffusion coefficient in a parabolic PDE. Since the considered inverse problem is ill-posed, a Tikhonov-type regularization involving a data-dependent linear operator is proposed for obtaining stable approximations. Error estimates are obtained under a general source condition, as it is essential to decide on the quality of convergence of the approximations. We also consider the finite dimensional realization of the proposed
method because of its importance for practical application. In the analysis of finite dimensional realization, we give a procedure to obtain finite dimensional subspaces of L^2 (0, T; H^1(Ω)) by doing double discretization, that is, discretization corresponding to both the space and time domain. Also, we analyze the parameter choice strategy and obtain an aposteriori regularization parameter that is order optimal. This is a joint work with Prof. M. Thamban Nair.