Criticality is often accompanied by the onset of ill-posedness, for example in the form of instability. Settings in which one is particularly confronted with this challenge are physical systems which can only be accessed by indirect measurements. In these situations, only limited, often noise-corrupted information on the system is available, from which one, nevertheless, seeks to efficiently reconstruct the relevant physical properties. Here, the strong competition between the only limited availability of data and the necessity to, yet, extract information on the system at hand often leads to severe instabilities or even non-uniqueness. As a consequence, numerical and analytical approaches are notoriously difficult and expensive, and necessitate efficient solution representations.

The projects in this research area are

• C01 - Inverse problems for genuinely nonlocal elliptic operators: uniqueness, stability and reconstruction (Rüland)
• C02 - Reconstructing cardiac electrical activation with learned spatiotemporal regularization on unstructured grids (Effland)
• C04 - Multilevel operator sampling in natural function spaces (Dölz)
• C05 - Inverse problems and nonlinear PDEs (Koch, Rüland)
• C06 - Variational estimates in multi- and nonlinear harmonic analysis (Thiele)