Welcome to the research group Inverse Problems
Inverse problems consist in reconstructing causes from observed effects. For example, in magnetic resonance imaging (MRI) one has to reconstruct the proton density in a human or animal body which gave rise to a measured magnetic signal. In coherent x-ray imaging, one tries to reconstruct the refractive index of a sample which causes a measured diffraction pattern.
The basic difficulty in the numerical solution of inverse problems consists in the ill-posedness of most of these problems in the sense that arbitrarily small measurement errors may cause large errors in the reconstruction. Therefore, naive approaches fail. A remedy against ill-posedness is regularization. Here a-priori information about the solution are incorporated into the reconstruction procedure. In more mathematical terms, instead of the discontinuous inverse of the forward operator, which maps the unknown to the data, a continuous approximation of this inverse is applied to the data to obtain a stable reconstruction.
We work, among others within the CRC 1456 Mathematics of Experiment: The challenge of indirect observations in the natural sciences on inverse problems in fields like
Further examples and other groups working on similar problems can be found on the web pages of the German Speaking Inverse Problems Society (GIP).
Inverse problems consist in reconstructing causes from observed effects. For example, in magnetic resonance imaging (MRI) one has to reconstruct the proton density in a human or animal body which gave rise to a measured magnetic signal. In coherent x-ray imaging, one tries to reconstruct the refractive index of a sample which causes a measured diffraction pattern.
The basic difficulty in the numerical solution of inverse problems consists in the ill-posedness of most of these problems in the sense that arbitrarily small measurement errors may cause large errors in the reconstruction. Therefore, naive approaches fail. A remedy against ill-posedness is regularization. Here a-priori information about the solution are incorporated into the reconstruction procedure. In more mathematical terms, instead of the discontinuous inverse of the forward operator, which maps the unknown to the data, a continuous approximation of this inverse is applied to the data to obtain a stable reconstruction.
We work, among others within the CRC 1456 Mathematics of Experiment: The challenge of indirect observations in the natural sciences on inverse problems in fields like
Further examples and other groups working on similar problems can be found on the web pages of the German Speaking Inverse Problems Society (GIP).
members of the research group