Model Variational Inverse Problems Governed by Partial Differential Equations

We discuss solution methods for inverse problems, in which the unknown parameters are connected to the measurements through a partial differential equation (PDE). Various features that commonly arise in these problems, such as inversions for a coefficient field, for the initial condition in a time-dependent problem, and for source terms are being studied in the context of three model problems. These problems cover distributed, boundary, as well as point measurements, different types of regularizations, linear and nonlinear PDEs, and bound constraints on the parameter field. The derivations of the optimality conditions are shown and efficient solution algorithms are presented. Short implementations of these algorithms in a generic finite element toolkit demonstrate practical strategies for solving inverse problems with PDEs. The complete implementations are made available to allow the reader to experiment with the model problems and to extend them as needed.