||his talk presents research which was in various combinations partly done
in collaboration with Sergei Pereverzev (RICAM Linz), Peter Mathe (WIAS
Berlin), Thorsten Hohage (Univ. Göttingen), Axel Munk (Univ. Göttingen),
Nicolai Bissantz (Univ. Göttingen) and Olha Ivanyshyn (Univ. Göttingen).
Inverse problems occur in very many different situations ranging from
determining not directly accessible physical quantities out of indirect
measurements to approximation/regression to financial models and learning
theory with kernel based methods.
Due to their instability, one of the major tasks for efficient
regularization methods are reliable algorithms for choosing the
regularization parameter. This topic became pressing because with new
noise models and areas of applications well-known methods like Morozov's
balancing principle proved to be not appropriate any more.
In this talk a new strategy, the Lepskij type balancing principle is
presented. We were able to prove that it yields as good results as the
former methods in all classical situations and gives significant
improvements in stochastical noise models. Numerics supporting these
theoretical considerations will be shown.
Parameter choice methods like the balancing principle (but also any other
proven one known to me) need the size of the error in one way or another.
We will present a way out using a more Baysian approach which could give
rise to new methods with much less restrictive requirements on the data;
again we will present supporting numerics.