DFG Project

Adaptive Discretization Methods for the Regularization of Inverse Problems

Project leaders
Prof. Dr. Barbara Kaltenbacher Pfeil, Universität Klagenfurt, Austria
Prof. Dr. Boris Vexler, Technische Universität München, Germany

Project staff
Dipl.-Math. Alana Kirchner, Technische Universität München, Germany
Dr. Slobodan Veljovic, ehemals Universität Graz, Austria (until 2011)

Project description
Many complex processes in the eld of natural sciences, medicine and engineering are described by mathematical models with partial diff erential equations (PDEs). The mentioned systems of PDEs mostly contain unknown data, e.g. space-dependent coefficient functions, source terms, initial and boundary data, whose determination leads to high-dimensional inverse problems. The numerical e ort for solving inverse problems with PDEs is usually much higher than for the numerical simulation of the underlying process with a given data set. Moreover, the inherent instability of inverse problems requires the use of appropriate regularization techniques. Great potential for the construction of ecient algorithms for the solution of such inverse problems lies in adaptive discretizations. While the use of adaptive concepts for the choice of the discretizations for numerical simulation has become state of the art in the last years, adaptivity in the context of inverse problems presents a new and highly relevant topic. The goal of the project consists in fi nding as generally applicable and analytically established methods as possible for the adaptive discretization of inverse problems. In this process, the main focus is on the efficiency of the constructed algorithms on the one hand and on the rigorous convergence analysis on the other hand.

Key words
Inverse Problems, Adaptivity

Field of work
Applied Mathematics, Numerical Mathematics und inverse problems

Publications
Barbara Kaltenbacher, Alana Kirchner and Boris Vexler:
Adaptive discretizations for the choice of a Tikhonov regularization parameter in nonlinear inverse problems
Inverse Problems 27, 2011.
Alana Kirchner, Dominik Meidner and Boris Vexler:
Trust region methods with hierarchical finite element models for PDE-constrained optimization
Control and Cybernetics 40(4), 2011.

Related Publications
Anke Griesbaum, Barbara Kaltenbacher and Boris Vexler:
Efficient computation of the Tikhonov regularization parameter by goal oriented adaptive discretization
Inverse Problems, Vol. 24(2), 2008.
Barbara Kaltenbacher and Jonas O fftermatt:
A re finement and coarsening indicator algorithm for finding sparse solutions of inverse problems
Inverse Problems and Imaging (IPI), 5, pp. 391 - 406, 2011.
Barbara Kaltenbacher and Jonas Of ftermatt:
A convergence analysis of regularization by discretization in preimage space
Mathematics of Computation 81, pp. 2049 - 2069, 2012.

Software
Gascoigne, RoDoBo
 
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