Efficient computation of the Tikhonov regularization parameter by goal oriented adaptive discretization. Inverse Problems

Parameter identification problems for partial differential equations (PDEs) often lead to large-scale inverse problems. For their numerical solution it is necessary to repeatedly solve the forward and even the inverse problem, as it is required for determining the regularization parameter, e.g., according to the discrepancy principle in Tikhonov regularization. To reduce the computational effort, we use adaptive finite-element discretizations based on goal-oriented error estimators. This concept provides an estimate of the error in a so-called quantity of interest, which is a functional of the searched for parameter q and the PDE solution u. Based on this error estimate, the discretizations of q and u are locally refined. The crucial question for parameter identification problems is the choice of an appropriate quantity of interest. A convergence analysis of the Tikhonov regularization with the discrepancy principle on discretized spaces for q and u provides a possible answer: it shows, that in order to determine the correct regularization parameter, one has to guarantee sufficiently high accurac