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F1308

-  F1308 Iterative

-  F1308 Geometric

-  F1308 New Concepts

-  F1308 Applications

Abstract

Besides Tikhonov regularization, which is probably the 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. The design and analysis of problem adapted iterative regularization methods is one of the major tasks of this project. Additionally, the efficient implementation, coupling with discretization techniques (forward solvers) and the design of preconditioners are fields of ongoing research.

 

Analysis of regularization methods

Many iterative methods have been developed and analysed for well-posed problems. Their application to inverse, especially ill-posed problems, is not straight forward and requires an analysis in the framework of regularization methods. There,



Acceleration of iterative regularization methods

One of the major drawbacks of iterative regularization methods for ill-posed problems is that - due to the ill-posedness, which results in ill-conditioned finite dimensional approximations - usually a high number of iterations is needed in order to reconstruct (order optimal) approximations of a solution. This behavior also appears for Newton-type iterations, if the linearized systems are again solved by iterative regularization methods. In order to reduce the overall numerical effort of the solution process, several strategies are pursued:



Problem adapted regularization strategies and theory

The general theory for the regularization of inverse problems is formulated for very general problems. For special classes of problems, the results can be improved significantly. A problem adapted theory of regularization methods, and even the design of problem adapted regularization algorithms is thus an important task. Topics of ongoing research are, e.g.,



Efficient discretization

Sophisticated discretization strategies become an important factor when it comes to the implementation of regularization algorithms. Especially for large scale problems, the efficient coupling of discretization and iteration process can significantly reduce the overall numerical effort. Topics of recent research are

Discretization issues are investigated in close cooperation with Subproject F1306.

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SpezialForschungsBereich SFB F013 | Special Research Program of the FWF - Austrian Science Fund