Regularization Methods for the Identification of Parameters in Partial Differential Equations

In physical or technical applications, one often has the situation that the physical laws governing the process are known, but quantitative information about physical parameters is not available. One can then try to determine these parameters (which might be functions of space, time or of the unknown solution) from (noisy) measurements of the solution; this is called "parameter identification". In this proposal we deal with numerical methods for parameter identification for partial differential equations, which is known to be an ill-posed problem in the sense that the parameters depend in a discontinuous way on the measurements. Therefore, stable numerical methods, so-called "regularization methods", have to be developed. Based on our prior work in the theory of regularization methods and on a new approach to Tikhonov regularization which uses the special structure of parameter identification problems, especially the weak formulation of the direct problem, we plan to develop a theory for iterative regularization methods using this special structure, hoping that this yields better results than the application of the general theory available. Moreover, we will try to develop (in a theory-based way) iterative regularization methods for parameter identification problems in complicated (and practically relevant) systems of partial differential equations like the ones modelling the crystallization of polymers.