Parameter identification problems are formulated in a probabilistic language, where the randomness reflects the uncertainty about the knowledge of the true values. This setting allows conceptually easily incorporating new information, e. g. through a measurement, by connecting it to Bayes's theorem. The unknown quantity is modelled as a (may be high-dimensional) random variable. Such a description has two constituents, the measurable function and the measure. One group of methods is identified as updating the measure, the other group changes the measurable function. We connect both groups with the relatively recent methods of functional approximation of stochastic problems, and introduce especially in combination with the second group of methods a new procedure which does not need any sampling, hence works completely deterministically. It also seems to be the fastest and more reliable when compared with other methods. We show by example that it also works for highly nonlinear non-smooth problems with non-Gaussian measures.
Hermann Matthies studied mathematics, computer science, physics, and engineering at Technische Universität Berlin, Germany, where he received his “Diploma” degree in mathematics in 1976. Thereafter he studied at the Massachusetts Institute of Technology (MIT), USA, where he received his PhD in 1978 in the field of mathematics. In 1979 he joined Germanischer Lloyd in Hamburg, Germany – an engineering company -- and subsequently there held various positions in the Research-, Wind-, and Offshore Departments. In 1995 he returnd to academia as a professor and Head of the Institute of Scientific Computing at Technische Universität Braunschweig, Germany; a position he maintains until today. From 1996 to 2006 he also served as the Director of the University Computing Centre. His research interests are primarily in numerical algorithms for partial differential equations, coupled and multi-physics problems, multi-scale problems, with a current emphasis on computational methods for stochastic partial differential equations and parametric identification problems.