Oberseminar Analysis Summer 2025

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.

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.