abstracts
Solving inverse problems without using forward operators
Barbara Kaltenbacher (Klagenfurt University)
Inverse problems generally speaking determine causes for desired or observed effects, which has numerous applications ranging from medical imaging via nondestructive testing to seismic prospection.
Computational methods for solving inverse problems usually rely on some kind of inversion of the mentioned cause-to-effect map, which is also called forward operator.
However, this forward operator is often compuationally quite expensive to evaluate or might even not be well-defined. In such cases it can help a lot to take a different viewpoint and consider the inverse problem as a system of model and observation equations, with both the state of the system and the searched for parameter as unknowns. Besides such an all-at-once appraoch, even more generally, reformulation of the inverse problem as an optimization task rather than a (system of) equation(s) allows to avoid the use of a forward operator.
A crucial aspect in the compuational solution of inverse problems is their ill-posedness in the sense that small perturbations in the given observations can lead to large deviations in the reconstructions. To overcome this issue, regularization methods need to be employed and we will discuss the application and adoption of several regularization concepts to all-at-once and minimization based formulations, in contrast to classical reduced ones.
Teaching an old formulation some new tricks: A projection-based approach for the p-center problem
Elisabeth Gaar (JKU Linz)
The p-center problem (PC) is a fundamental problem in location science, where we are given demand points and we want to choose p of these points to locate a facility such that the maximum distance of any demand point to its closest facility is minimized. It is NP-hard and has countless applications including location of fire stations or COVID test-centers. State-of-the-art solution approaches of PC use its connection to the NP-hard set cover problem to solve PC in an iterative fashion by repeatedly solving set cover problems using, e.g., integer programming (IP) techniques. The classical textbook IP formulation of PC is usually dismissed due to its size and bad linear relaxation bounds.
In this talk, we present a novel solution approach that works on an IP formulation that can be obtained by a projection from the classical formulation. The formulation is solved by means of branch-and-cut, where cuts for demand points are iteratively generated. This makes the method suitable for large scale instances. Moreover, the formulation can be strengthened with combinatorial information to obtain a much tighter relaxation. In particular, we present a novel way to use lower bound information to obtain stronger cuts. We show how our formulation is connected to the set cover problem and present a computational comparison of highly sophisticated solution frameworks with our straightforwardly implementable idea.
This is joint work with Markus Sinnl.
A generalization of the medial axis
Elizabeth Stephenson (IST Austria)
In this talk, we re-explore the idea of the medial axis, a subset of an object's symmetry set, which is a valuable image processing tool developed by Blum in the 1960s. Originally introduced as a shape recognition tool in mathematical biology, it has since been widely used in various fields such as computer science and animation. We examine new functionalities of an extension of Blum's medial axis in the context of Morse theory and persistent homology. We hope that our work will facilitate extraction of new and interesting information from images and point cloud data.
Geometry of approximation
Eva Kopecká (Innsbruck University)
We will discuss the geometry behind approximating convex sets by axis-parallel boxes and almost isometries by isometries.
A shape calculus approach for time harmonic solid-fluid interaction problem in stochastic domains
Debopriya Mukherjee (Leoben University)
In this talk, I will introduce an interior solid-fluid interaction problem in harmonic regime with randomly perturbed boundaries. Analysis of the shape derivative and shape Hessian of vector- and tensor-valued functions is provided. Moments of the random solutions are approximated by those of the shape derivative and shape Hessian, and the approximations are of third order accuracy in terms of the size of the boundary perturbation. Our theoretical results are supported by an analytical example on a square domain.
This is a joint work with Thanh Tran, University of New South Wales, Sydney, Australia.
Uncountable trees
Marlene Koelbing (Vienna University)
My research area is set theory. Uncountable trees play an important role in my research. In this talk I will concentrate on one of my projects which is about Suslin trees, Aronszajn trees and Kurepa trees. The main focus will be to define these notions and to discuss their properties and basic results.
Women and mathematics - facts and perspectives
Andrea Blunck (Hamburg University)
In my talk I want to discuss the under-representation of women in mathematics. I will take a look at the history, show some figures about the situation today and present some approaches how to explain and to overcome the under-representation.