Mini-workshop in mathematics

Schedule:

• 14:00 Stefan Sommer
• 14:40 Karen Habermann
• 15:20 Fika
• 15:40 Helmut Harbrecht
• 16:20 Tomasz Tyranowski

Titles and abstracts:

Karen Habermann: Geodesic and stochastic completeness for landmark space

Abstract: In computational anatomy and, more generally, shape analysis, the Large Deformation Diffeomorphic Metric Mapping framework models shape variations as diffeomorphic deformations. An important shape space within this framework is the space consisting of shapes characterised by n ≥ 2 distinct landmark points in R^d. In diffeomorphic landmark matching, two landmark configurations are compared by solving an optimisation problem which minimises a suitable energy functional associated with flows of compactly supported diffeomorphisms transforming one landmark configuration into the other one. The landmark manifold Q of n distinct landmark points in R^d can be endowed with a Riemannian metric g such that the above optimisation problem is equivalent to the geodesic boundary value problem for g on Q. Despite its importance for modelling stochastic shape evolutions, no general result concerning long-time existence of Brownian motion on the Riemannian manifold (Q,g) is known. I will present joint work with Philipp Harms and Stefan Sommer on first progress in this direction which provides a full characterisation of long-time existence of Brownian motion for configurations of exactly two landmarks, governed by a radial kernel. I will further discuss joint work with Stephen C. Preston and Stefan Sommer which, for any number of landmarks in R^d and again with respect to a radial kernel, provides a sharp criterion guaranteeing geodesic completeness or geodesic incompleteness, respectively, of (Q,g).

Helmut Harbrecht: Shape optimization under constraints on the probability of a quadratic functional to exceed a given threshold

Abstract: This talk is dedicated to shape optimization of elastic materials under random loadings where the particular focus is on the minimization of failure probabilities. Our approach relies on the fact that the area of integration is an ellipsoid in the high-dimensional parameter space when the shape functional of interest is quadratic. We derive the respective expressions for the shape functional and the related shape gradient. As showcase for the numerical implementation, we assume that the random loading is a Gaussian random field. By exploiting the specialties of this setting, we derive an efficient shape optimization algorithm. Numerical results in three spatial dimensions validate the feasibility of our approach.

Stefan Sommer: Kunita flows, shape stochastics, and phylogenetic inference

Abstract: I will discuss constructions of stochastic processes on shape spaces in different settings and which properties of such processes are relevant in applications in morphological analysis in evolutionary biology. Particularly, Kunita flows of diffeomorphisms and their actions on shapes can be used as a shape equivalents of the standard Brownian motion model often used in evolutionary studies. I will detail this application, including how statistical inference of properties of the stochastic flow can be performed from observed shapes, in both finite and infinite dimensional settings.

Tomasz Tyranowski: Stochastic variational principles for the collisional Vlasov–Maxwell and Vlasov–Poisson equations

Abstract: In this talk I will recast the collisional Vlasov-Maxwell and Vlasov-Poisson equations as systems of coupled stochastic and partial differential equations, and I will derive stochastic variational principles which underlie such reformulations. I will also propose a stochastic particle method for the collisional Vlasov-Maxwell equations and provide a variational characterization of it, which can be used as a basis for a further development of stochastic structure-preserving particle-in-cell integrators.