A linear regularization method for a nonlinear parameter identification problem

In order to obtain regularized approximations for the solution q of the parameter identification problem -del.(q del u) = f in Omega along with the Neumann boundary condition q partial derivative u/partial derivative v = g on partial derivative Omega, which is an ill-posed problem, we consider its weak formulation as a linear operator equation with operator as a function of the data u is an element of W-1,W-infinity (Omega), and then apply the Tikhonov regularization and a finite-dimensional approximation procedurewhen the data is noisy. Here, Omega is a bounded domain in R-d with Lipschitz boundary, f is an element of L-2 (Omega) and g is an element of H-1/2 (partial derivative Omega). This approach is akin to the equation error method of Al- Jamal and Gockenback (2012) wherein error estimates are obtained in terms of a quotient norm, whereas our procedure facilitates to obtain error estimates in terms of the regularization parameters and data errors with respect to the norms of the spaces under consideration. In order to obtain error estimates when the noisy data belongs to L-2 (Omega) instead of W-1,W-infinity (Omega), we shall make use of a smoothing procedure using the Clement operator under additional assumptions of Omega and u.