## Invited Session Thu.1.H 2051

#### Thursday, 10:30 - 12:00 h, Room: H 2051

**Cluster 24: Variational analysis** [...]

### Optimization methods for nonsmooth inverse problems in PDEs

**Chair: Akhtar A. Khan and Christian Clason**

**Thursday, 10:30 - 10:55 h, Room: H 2051, Talk 1**

**Barbara Kaltenbacher**

Iterative regularization of parameter identification in PDEs in a Banach space framework

**Coauthors: Bernd Hofmann, Frank Schoepfer, Thomas Schuster**

**Abstract:**

Natural formulations of inverse problems for PDEs often lead to a Banach space setting, so that the well-established Hilbert space theory of regularization methods does not apply.

The talk will start with an illustration of this fact by some parameter identification problems in partial differential equations. Then, after a short detour to variational regularization, we will mainly focus on iterative regularization methods in Banach spaces. We will dwell on gradient and Newton type methods as well as on their extension from the original Hilbert space setting to smooth and convex Banach spaces. Therewith, convexity of the Newton step subproblems is preserved while often nondifferentiability might be introduced, which results in the requirement of solving a PDE with nonsmooth nonlinearity for evaluating the duality mapping. Convergence results for iterative methods in a Banach space framework will be discussed and illustrated by numerical experiments for one of the above mentioned parameter identification problems.

**Thursday, 11:00 - 11:25 h, Room: H 2051, Talk 2**

**Bernd Hofmann**

On smoothness concepts in regularization

**Abstract:**

A couple of new results on the role of smoothness

and source conditions in Tikhonov type regularization

in Hilbert and Banach spaces are presented. Some aspect

refers to the the role of appropriate choice rules for

the regularization parameter. The study is motivated

by examples of nonlinear inverse problems from inverse

option pricing and laser optics.

**Thursday, 11:30 - 11:55 h, Room: H 2051, Talk 3**

**Christian Clason**

Inverse problems for PDEs with uniform noise

**Abstract:**

For inverse problems where the data is corrupted by uniform noise, it is well-known that the *L*^{\}infty norm is a more robust data fitting term than the standard *L*^{2} norm.

Such noise can be used as a statistical model of quantization errors appearing in digital data acquisition and processing. Although there has been considerable progress in the regularization theory in Banach spaces, the numerical solution of inverse problems in *L*^{\}infty has received rather little attention in the mathematical literature so far, possibly due to the nondifferentiability of the Tikhonov functional.

However, using an equivalent formulation,

it is possible to derive optimality conditions that are amenable to numerical solution by a superlinearly convergent semi-smooth Newton method.

The automatic choice of the regularization parameter *α* using a simple fixed-point iteration is also addressed. Numerical examples illustrate the performance of the proposed approach as well as the qualitative behavior of *L*^{\}infty fitting.