Title

Discrete Geodesic Calculus for Complete Riemannian Metrics

Abstract

Geodesic calculus on Riemannian manifolds refers to the exponential and logarithmic mappings, the parallel transport, and the Riemannian connection, whose evaluation involves solving complex systems of second-order ordinary differential equations. Closed expressions for solving such problems can be obtained only in exceptional cases. This fact is important while dealing with optimisation problems on manifolds since we require the evaluation of such mappings multiple times per iteration.

In this talk, we will develop a discrete geodesic calculus based on the definition of an inexpensive dissimilarity measure which approximates the Riemannian distance. We will focus on complete Riemannian metrics as the ones described in [Gordon, 1973], for which a simple dissimilarity measure can be identified by exploiting their specific structure. We will present numerical experiments in three different manifolds showing the advantage of working within this framework.

Karen Estefania Loayza Romero is a Chapman Fellow in the Mathematics Department at Imperial College London. Before joining Imperial, she was a postdoctoral researcher at the Cluster of Excellence “Mathematics Münster” in the group of mathematical optimisation led by Prof. Benedikt Wirth. She completed her PhD at the University of Heidelberg under the supervision of Prof. Roland Herzog. She finished her undergraduate and MSc studies at Escuela Politécnica Nacional del Ecuador, in the Research Center on Mathematical Modelling (MODEMAT) supervised by Prof. Juan Carlos De lo Reyes and Prof. Pedro Merino.

Karen’s research focuses on computational PDE-constrained optimization. Among others, She is interested in non-smooth optimization algorithms and their applications, data assimilation, shape optimization, and optimization on manifolds.