Convergence and adaptive discretization of the IRGNM Tikhonov and the IRGNM Ivanov method under a tangential cone condition in Banach space
For researchers in inverse problems, this provides a theoretical foundation for IRGNM in Banach spaces under weaker nonlinearity assumptions, though the results are incremental as they extend existing theory.
The paper proves convergence of the IRGNM Tikhonov and Ivanov methods in Banach space under a tangential cone condition, without requiring source conditions, and extends convergence to discretized problems with error control via goal-oriented weighted dual residual estimators. Numerical results are shown for an inverse source problem with L∞ sources.
In this paper we consider the Iteratively Regularized Gauss-Newton Method (IRGNM) in its classical Tikhonov version and in an Ivanov type version, where regularization is achieved by imposing bounds on the solution. We do so in a general Banach space setting and under a tangential cone condition, while convergence (without source conditions, thus without rates) has so far only been proven under stronger restrictions on the nonlinearity of the operator and/or on the spaces. Moreover, we provide a convergence result for the discretized problem with an appropriate control on the error and show how to provide the required error bounds by goal oriented weighted dual residual estimators. The results are illustrated for an inverse source problem for a nonlinear elliptic boundary value problem, for the cases of a measure valued and of an $L^\infty$ source. For the latter, we also provide numerical results with the Ivanov type IRGNM.