06.06.2023, 15 Uhr
Phenomena in High Dimensions
Pierre Youssef (New York University, Abu Dhabi)
Bernd Hofmann (TU Chemnitz), Elena Resmerita (Alpen-Adria Universität Klagenfurt)
14:00 Bernd Hofmann: An introduction to regularization theory for inverse problems
15:00 Tea and Coffee break
15:30 Elena Resmerita: Old and new facets of sparsity promoting regularization
Bernd Hofmann: An introduction to regularization theory for inverse problems
Inverse problems play a prominent role in modern applied mathematics, strongly motivated by applications in natural sciences, technology and finance. Such problems tend to be ill-posed, which means that small perturbations in the input data may lead to arbitrarily large errors in the solution. For constructing stable approximate solutions to inverse problems, regularization approaches are required. We start this talk with four examples that outline the wide field of inverse problems formulated in form of linear and nonlinear operator equations in infinite dimensional Hilbert and Banach spaces. Then we discuss the phenomenon of ill-posedness and how to overcome the resulting instability by regularization techniques. Our focus is on variational regularization and its interplay with conditional stability estimates. In this context, the distinguished role of smoothness must be emphasized, occurring for example in form of source conditions for obtaining error estimates and convergence rates of the regularized solutions.
Elena Resmerita: Old and new facets of sparsity promoting regularization
A broad range of real-life problems can be modeled by operator equations that are ill-posed. Approximating sparse solutions for such problems in a stable way has been a challenging task during the past two decades. In this talk, we will review qualitative and quantitative aspects of typical sparsity-promoting regularization in the infinite dimensional settings of Lebesgue spaces of sequences. We will also discuss recent non-standard approaches in more general spaces, which are better suitable for more complex problems (arising, e.g., in image reconstruction).