Functional error estimators for the adaptive discretization of inverse problems
This provides a rigorous error estimation framework for adaptive discretization of inverse problems with nonsmooth regularization, benefiting computational scientists solving PDE-constrained inverse problems.
The paper applies functional error estimators to Tikhonov regularization for inverse problems, enabling reliable discretization error estimation for both quadratic and nonsmooth penalties (e.g., sparsity, Ivanov regularization). Numerical results for an elliptic inverse source problem with sparsity regularization demonstrate the approach.
So-called functional error estimators provide a valuable tool for reliably estimating the discretization error for a sum of two convex functions. We apply this concept to Tikhonov regularization for the solution of inverse problems for partial differential equations, not only for quadratic Hilbert space regularization terms but also for nonsmooth Banach space penalties. Examples include the measure-space norm (i.e., sparsity regularization) or the indicator function of an $L^\infty$ ball (i.e., Ivanov regularization). The error estimators can be written in terms of residuals in the optimality system that can then be estimated by conventional techniques, thus leading to explicit estimators. This is illustrated by means of an elliptic inverse source problem with the above-mentioned penalties, and numerical results are provided for the case of sparsity regularization.