R1: Inverse problems and parameter identification for partial differential equations
Overview
Rapidly and accurately identifying parameters from measurements of physical quantities modelled by partial differential equations (PDEs) is a universal task in many industrial applications and most natural sciences. Technological innovation causes such inversion problems to permanently re-emerge in various facets, which makes parameter identification for PDEs a crucial field of research in inverse problems.
Parameter identification problems for PDEs feature two intricate properties that make their analytical and numerical solution challenging: First, the searched-for parameters do not depend continuously on the measured data: Small measurement errors lead to instability and huge reconstruction errors. Indeed, ill-posedness of the operator equation for the parameter-to-data map that mathematically defines the parameter identification problem crucially requires to stabilize the inversion process by regularisation.
For parameter identification in the context of PDEs an additional second difficulty arises: As the solution to a differential equation depends on the parameters of that equation, the identification of such a parameter from (partial) measurements is non-linear. Algorithms tackling such problems numerically thus need to cope with high complexity and high dimensionality.
The PhD projects of the 1st cohort focused on:
- Parameter identification for scattering problems
- Regularization properties of Tikhonov functionals with tolerances
The PhD projects of the 2nd cohort are addressing:
- Mathematical models and regularization concepts for magnetic particle imaging
- Deep Learning concepts for inverse problems (Collaboration with R3)
Dynamic Inverse Wave Problems
As an example of the approach for scattering problems we highlight the thesis of Thies Gerken. The classical approach for scattering problems usually rely on the assumption that all quantities (parameters, incident waves) are time-harmonic. The project R1-1, however, was concerned with scenarios in which the time-dependence of the parameters is unknown. In this case the underlying hyperbolic PDEs cannot simply be reformulated as a Helmholtz equation. Thies Gerken in particular analysed the acoustic wave equation, and the related inverse problems of reconstructing a time- and space-dependent wave speed and mass density from the solution of this equation. The time-dependence of the principal part of the differential equation makes it especially complicated to obtain a Fréchet-differentiable forward operator to this problem. This difficulty was solved through the development of suitable regularity results for such evolution equations. The numerical inversion also had to be carried out in a computationally intensive four-dimensional setting. Nevertheless, with the regularization method CG-REGINN it was possible to obtain satisfying reconstructions. One of the papers resulting from this thesis was listed as Highlight Paper of the year by the most important journal of the field, "Inverse Problems" (T. Gerken, A. Lechleiter: Reconstruction of a Time-dependent Potential from Wave Measurements, 2017).
Magnetic Particle Imaging
Magnetic particle imaging (MPI) is another highlight of Research Area R1. This field is strongly influenced by our former PostDoc, Tobias Kluth, who developed a mathematical theory for inverse problems of MPI. His results include a mathematical hierarchy of model refinement from linear approximations to non-linear models based on the Landau-Lifshitz-Gilbert equation (see T. Kluth: Mathematical models for magnetic particle imaging, 2018; T. Kluth, B. Jin, G. Li: On the Degree of Ill-Posedness of Multi-Dimensional Magnetic Particle Imaging, 2018). MPI is now a benchmark application for the 2nd cohort and also serves as a prototypical application for deep learning concepts to inverse problems.
