led by Otmar Scherzer

Inverse problems are concerned with determining (usually numerically) causes for desired or observed effects, which is a type of problems frequently arising in science and engineering. Application areas where the Linz group has been active include inverse scattering, parameter identification in partial differential equations, imaging science, deconvolution problems in physics, inverse problems in refractive optics, inverse heat conduction problems with applications in steel processing, and non-destructive testing.

The property that makes inverse problems mathematically challenging is their "ill-posedness", which renders traditional numerical methods inherently unstable. This means that a solution to an inverse problem might neither exist nor be unique, and even if some generalised concept of solution is introduced, then this solution depends in a discontinuous way on the data. Since in applications, data always contain noise, which is highly amplified due to the ill-posedness of the inverse problem, the (in)stability issue is of high practical relevance. Special "regularisation" methods have to be developed to stabilise the solution process of inverse problems. Until about a decade ago, mathematical research concentrated on linear ill-posed problems. For non-linear problems, a variational method carried over from the linear theory, namely Tikhonov regularisation, had long been used and mathematically analysed. The main emphasis now, however, seems to be on iterative regularisation methods. We aim at developing iterative regularisation methods in a "theory-based" way, i.e., at analysing their stability, convergence and convergence rates and, on this basis, develop a-posteriori stopping rules for optimal convergence rates: a peculiar feature of iterative methods for inverse problems is that, while the error initially decreases with progressing iterations, in the presence of noise, after some time, the error starts to increase again, finally without bound. This is due to noise propagation; for practical success of an iterative regularisation method, it is therefore of high importance to stop the iteration at the right time, which is possible only based on a mathematical analysis as indicated. Methods to be used come from non-linear functional analysis. When solving a complex inverse problems, e.g., a parameter identification problem for a transient and spatially three-dimensional partial differential equation, the efficient coupling of the "inverse iteration" with the "forward solver" (in this case, the method used for solving the pde with a given parameter) is crucial. The final aim is to obtain an accurate solution in as little overall computing time as possible, and in a way which is stable with respect to data noise. To reach this aim, one has to analyse the direct and the inverse problems as an entity; there are various ways to do this, which we are both currently investigating: either, one can control the necessary accuracy with which the direct solver has to work via its a posteriori error estimates from the inverse iteration, or one can use an all-at-once approach, where the inverse problem is formulated as an optimisation problem with the forward equation as a constraint. This issue is the topic of an ongoing co-operation, which we will further intensify in the Radon Institute, between the groups of Profs. Langer and Engl.

---

The Institute is named after the famous Austrian mathematician Johann Radon (1887-1956)

Medieninhaber:
Österreichische Akademie der Wissenschaften
Juristische Person öffentlichen Rechts (BGBl 569/1921 idF BGBl I 130/2003)
Dr. Ignaz Seipel-Platz 2, 1010 Wien
Diese Website dient zur Information über die wissenschaftlichen Aktivitäten der Österreichischen Akademie der Wissenschaften und setzt somit den gesetzlichen Auftrag um, die Wissenschaft in jeder Hinsicht zu fördern.

This RICAM page was made with 100% valid HTML & CSS - Send comments to Webmaster
Today's date and time is 02/09/12 - 06:08 CET and this file ( /research/invprob/index.html ) was last modified on 02/02/12 - 13:33 CET