The main topic in the basic research of the Industrial Mathematics Institute is the treatment of inverse and ill-posed problems.

Inverse problems are concerned with determining causes for a desired or an observed effect. Inverse problems most often do not fulfill Hadamard's postulates of well-posedness. Thus, they might not have a solution in the strict sense, solutions might not be unique and/or might not depend continuously on the data.

• Computerized tomography: Here, one wants to reconstruct the spatially varying absorption coefficients within the human body from measurements of intensity decays of X-rays sent through the body. Mathematically, this leads to the problem of inverting the Radon transform. Similar problems appear in non-destructive testing
• Inverse heat conduction or diffusion problems: One wants to determine an unknown boundary heat flux which leads to a desired temperature field or to a desired boundary between solid and liquid phase (inverse Stefan problem). This has important industrial applications e.g. in continuous casting of steel
• Inverse scattering problems: From measurements of waves scattered by an obstacle, one wants to determine e.g. the shape or the location of this obstacle. Problems of this type appear also in the exploration of minerals or oil
• Parameter identification: Here, spatially and/or temporally parameters appearing in e.g. partial differential equations have to be determined from measurements of the solution, either in the whole domain, or on the boundary only. The latter case has important applications in medicine and non-destructive testing: an electrical conductivity can be determined from measurements of current and voltage on the boundary (impedance tomography)
• Fluorescence analysis: Here, the aim is to determine the distribution of event-times in time-resolved fluorescence of proteins or polymers. This is of great interest in medical and pharmaceutical research for analysing the structure of and the interaction between proteins

All these problems are ill-posed. A consequence is that arbitrarily small changes in the data may lead to arbitrarily large changes in the solution. As in the numerical treatment of inverse problems data errors are inevitable, one has to use stabilizing procedures for successfully dealing with ill-posed problems, so-called regularization methods.

The Industrial Mathematics Institute has a long tradition in analysing regularization schemes. While in the eighties the emphasis lay on the analysis of regularization methods for linear ill-posed problems, in the last years results have been obtained for regularization methods for nonlinear ill-posed problems, e.g. for Tikhonov regularization, Landweber iteration and the maximum entropy method.

Many of the papers of our group deal with inverse and ill-posed problems, covering both theoretical analysis and practical applications.

To obtain a list of selected papers and proceedings of conferences organized by the group click here.

See also the following scientific book on the subject:
H.W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems, Kluwer, Dordrecht, 1996

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