Oberseminar Analysis Summer 2025

We consider the heat equation on finite cubes and all of $R^d$. We give criteria for null-controllability from two different classes of sensor sets. The first consist of well distributed deterministic balls, the second of balls around a Poisson point process. We focus on the later case and discuss the dependence of the control cost on the intensity of the process and the size of the balls.

In this talk I will give a basic introduction to inverse problems, highlighting commonalities and differences between deterministic and statistical ones, specific difficulties, and important examples. Two major topics will be covered: regularization (i.e. the construction of suitable estimators from noisy data) and inference (i.e. statistically rigorous statements about properties of the unknown quantity of interest based on noisy data). For both topics, I will explain a general approach to design and analyze corresponding methods, and I will also discuss applications e.g. from medical imaging and super-resolution microscopy.

In this talk I will describe a class of minimization problems arising in modelling shape memory alloys. I will start with a shape memory material illustration, followed by a simple one dimensional model of it. Its extensions in 2D and 3D will help us understand the energetic mechanism behind the formation of twin patterns in physical experiments. By means of sharp upper and lower bounds, we show that certain experimentally observed structures provide optimal energy scaling law, and - under certain assumptions - are true energy minimizers. I will also discuss the extension of the linear elastic models for two twins to model multiscale patterns, which appear in recent physical experiments and involve mixing of four twins.

In this talk, I present a linearization result for Griffith functionals of brittle fracture. By means of Gammma-convergence we derive effective linearized functionals in the limit of vanishing strains, where the resulting free discontinuity problem in linear elasticity is formulated in the space of generalized special functions of bounded deformation. We also discuss a result in the evolutionary setting, namely the convergence of energetic solutions describing quasistatic crack growth. Finally, I will address the possibility of including a non-interpenetration constraint in the nonlinear and linearized models. Based on joint works with S. Almi, E. Davoli, P. Steinke, and K. Stinson.

Compressed sensing allows for the recovery of sparse signals from a small number of measurements, which are proportional (up to logarithmic factors) to the sparsity of the unknown signal. Classical theory primarily considers randomly subsampled isometries in the finite-dimensional setting. In this talk, I will show how the theory of compressed sensing can also be rigorously applied to a variety of ill-posed inverse problems, including X-ray and photoacoustic tomography. Specifically, we will investigate, within a general framework, the relationship between the sparsity level of a signal, the ill-posedness of the problem, and the number of measurements required for stable reconstruction.