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 ill-posed in the sense of Hadamard and have to be solved by regularization methods. Besides Tikhonov regularization, which is the probably most well-known 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 large-scale 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.

• Design and mathematical analysis of (mainly) iterative regularization methods for linear and nonlinear inverse and ill-posed 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., derivative-free 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 phase-field 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.

• F1306: efficient discretization of PDE problems, inverse problems in plasticity
• F1309: optimal design, shape- and topology optimization, sqp-methods
• 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; level-set methods and implicit surface representations

List of Publications.