Abstract

The subject of this project is
the development, analysis and implementation of efficient, mainly iterative, numerical algorithms for solving inverse problems.
Many physically relevant inverse problems are illposed in the sense of Hadamard and have to be solved by regularization methods. Besides Tikhonov regularization, which is the probably most wellknown regularization method for linear as well as nonlinear inverse problems, iterative regularization algorithms have more recently been investigated and applied successfully to the solution of, especially nonlinear and largescale problems.
A sound mathematical theory is the basis for an successful application of the above mentioned regularization methods. The choice of the right method for a specific problem, and optimal choice of parameters in the algorithms, in particular the regularization parameter or stopping index is only possible on behalf of a found theoretical background. A further important issue in applications is the efficient coupling of regularization schemes and discretization strategies. Thus, integration of regularization methods and solvers for the direct problems (partly developed and analysed within other projects of this SFB) is one of the major aims of this project.
Objectives

Design and mathematical analysis of (mainly) iterative regularization methods for
linear and nonlinear inverse and illposed problems. In
many of the problems under consideration, e.g., in
parameter identification, the direct problem is defined
implicitly as the solution of a governing PDE.
 Adaption of methods (e.g. from optimal control) to inverse problems and their analysis in the framework of regularization
 Integration of efficient discretization techniques (direct solvers) and the design an analysis of preconditioning techniques for iterative regularization methods
 Development and analysis of problem adapted regularization strategies, e.g., derivativefree methods.
 Design and mathematical analysis of algorithms for geometric inverse problems. Besides of parametric methods, which can be dealt within a functional analytic framework and hence iterative regularization methods are appropriate, we are concerned with level set methods and phasefield methods to deal with geometric inverse problems.
 Integration of new concepts into regularization theory, e.g., stochastic error concepts, symbolic methods.
 Inverse problems in physical/technical applications.
Internal Cooperations
 F1306: efficient discretization of PDE problems, inverse problems in plasticity
 F1309: optimal design, shape and topology optimization, sqpmethods
 F1322: symbolic methods for operator equations in Hilbert spaces; fundamental solutions to differential equations
 F1305: special function tools in functional analysis
 F1303: polynomial identities in regularization theory; regularization of algorithms for solving over reals
 F1304: differential geometry in inverse problems
 F1315: inverse problems in geometric algorithms; levelset methods and implicit surface representations
People
Prof. Dr. Heinz W. Engl  9219 mail 
Scientific Staff
Dr. Lin He  9229 mail  
Dr. Hanna Katriina Pikkarainen  5233 mail  
Dr. Eva Sincich  5237 mail  
Dr. Mourad Sini  9229 mail  
DI MarieTherese Wolfram 
Publications
SpezialForschungsBereich SFB F013  Special Research Program of the FWF  Austrian Science Fund