Alexander Schmeding

Elena Celledoni, Helge Glöckner, Jørgen Riseth et al.

One of the fundamental problems in shape analysis is to align curves or surfaces before computing geodesic distances between their shapes. Finding the optimal reparametrization realizing this alignment is a computationally demanding task, typically done by solving an optimization problem on the diffeomorphism group. In this paper, we propose an algorithm for constructing approximations of orientation-preserving diffeomorphisms by composition of elementary diffeomorphisms. The algorithm is implemented using PyTorch, and is applicable for both unparametrized curves and surfaces. Moreover, we show universal approximation properties for the constructed architectures, and obtain bounds for the Lipschitz constants of the resulting diffeomorphisms.

Geir Bogfjellmo, Alexander Schmeding

The Butcher group is a powerful tool to analyse integration methods for ordinary differential equations, in particular Runge--Kutta methods. Recently, a natural Lie group structure has been constructed for this group. Unfortunately, the associated topology is too coarse for some applications in numerical analysis. In the present paper, we propose to remedy this problem by replacing the Butcher group with the subgroup of all exponentially bounded elements. This "tame Butcher group" turns out to be an infinite-dimensional Lie group with respect to a finer topology. As a first application we then show that the correspondence of elements in the tame Butcher group with their associated B-series induces certain Lie group (anti)morphisms.