335 research outputs found
A new approach to nonlinear constrained Tikhonov regularization
We present a novel approach to nonlinear constrained Tikhonov regularization
from the viewpoint of optimization theory. A second-order sufficient optimality
condition is suggested as a nonlinearity condition to handle the nonlinearity
of the forward operator. The approach is exploited to derive convergence rates
results for a priori as well as a posteriori choice rules, e.g., discrepancy
principle and balancing principle, for selecting the regularization parameter.
The idea is further illustrated on a general class of parameter identification
problems, for which (new) source and nonlinearity conditions are derived and
the structural property of the nonlinearity term is revealed. A number of
examples including identifying distributed parameters in elliptic differential
equations are presented.Comment: 21 pages, to appear in Inverse Problem
On a level-set method for ill-posed problems with piecewise non-constant coefficients
We investigate a level-set type method for solving ill-posed problems, with
the assumption that the solutions are piecewise, but not necessarily constant
functions with unknown level sets and unknown level values. In order to get
stable approximate solutions of the inverse problem we propose a Tikhonov-type
regularization approach coupled with a level set framework. We prove the
existence of generalized minimizers for the Tikhonov functional. Moreover, we
prove convergence and stability of the regularized solutions with respect to
the noise level, characterizing the level-set approach as a regularization
method for inverse problems. We also show the applicability of the proposed
level set method in some interesting inverse problems arising in elliptic PDE
models.
Keywords: Level Set Methods, Regularization, Ill-Posed Problems, Piecewise
Non-Constant CoefficientsComment: Accepte
06391 Abstracts Collection -- Algorithms and Complexity for Continuous Problems
From 24.09.06 to 29.09.06, the Dagstuhl Seminar 06391 ``Algorithms and Complexity for Continuous Problems\u27\u27 was held
in the International Conference and Research Center (IBFI),
Schloss Dagstuhl.
During the seminar, participants presented their current
research, and ongoing work and open problems were discussed. Abstracts of
the presentations given during the seminar
are put together in this paper. The first section
describes the seminar topics and goals in general.
Links to extended abstracts or full papers are provided, if available
- …