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 L2 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.

 

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