On identification of matrix diffusion coefficient in a parabolic PDE

Abstract: In this talk, we shall consider an inverse problem of identifying the diffusion coefficient in matrix form in a parabolic PDE. Using the idea of natural linearisation, considered in [H. Cao and S.V. Pereverzev, Inverse Problems, 22 (2006), no.6, 2311-2330] for some scalar coefficient identification problem, the non-linear inverse problem is transformed into a problem of solving an operator equation where the operator involved is linear. Solving the linear operator equation turns out to be an ill-posed problem. The Tikhonov-type regularization method is employed for obtaining stable approximations and the finite dimensional analysis is done based on the Galerkin method, for which a sequence of finite rank orthogonal projection on the space of matrices with entries from L2(Ω) is introduced. Since the error estimates in Tikhonov regularization method rely heavily on the adjoint operator, an explicit representation of the adjoint of the linear operator involved is obtained. Finally, we shall present a numerical experiment. It is worth mentioning that, although there a few existing results about existence and uniqueness related to this type of problem, not much is available about the regularization and error analysis. So, this work intends to contribute in that direction.