Efficient Computation of a Tikhonov Regularization Parameter for Nonlinear Inverse Problems with Adaptive Discretization Methods

Alana Kirchner^*

Parameter and coefficient identification problems for PDEs usually lead to nonlinear inverse problems, which require regularization techniques due to their instability. We will present a combination of Tikhonov regularization, Morozov's discrepancy principle, and adaptive finite element discretizations as a Tikhonov parameter choice rule. The discrepancy principle is implemented via an inexact Newton method, where we control the accuracy by means of mesh refinement based on a posteriori goal oriented error estimators. In order to further reduce the computational costs, we apply a generalized Gauss-Newton approach for the optimal control problem, where the stopping index for this iteration plays the part of an additional regularization parameter, also determined by the discrepancy principle. The obtained theoretical convergence results (optimal rates under usual source conditions) will be illustrated by several numerical experiments. This is joint work with Barbara Kaltenbacher and Boris Vexler.

Mathematics Subject Classification: 49N45 65N50 65N30 65J20 49J20 35J60

Keywords: inverse problems; regularization; finite elements; mesh refinement; a posteriori error estimation; optimal control;

Minisymposion: Adaptivity and Model Order Reduction in PDE Constrained Optimization