A gentle introduction to interpolation on the Grassmann manifold

03/2024

A gentle introduction to interpolation on the Grassmann manifold

Tuesday 16th January 2024

Ciaramella, G.; Gander, M.J.; Vanzan, T.

These notes originated from the authors’ effort while studying interpolation techniques on the Grassmann manifold. This has been a hot topic recently since it is an important tool in parametric reduced order modelling. Fortunately, there is an extensive literature available with seminal contributions both from the engineering and mathematical communities. More generally, the development of numerical methods involving manifolds is a very active research area. Given all literature about interpolation on Grassmann manifold, the reader may immediately ask the following question: is there any need for an additional introductory manuscript? It is the authors’ belief that this is actually the case. The aim of these notes is to precisely fill a gap in the literature, by providing a reference which gently introduces numerical analysts to the very interesting research topic of interpolation on the Grassmann manifold. Indeed, on the one hand, the engineering literature often does not provide the necessary mathematical details needed by a numerical analyst to understand the subject and to solidly build new computational algorithms. On the other hand, manuscripts from the mathematical community, despite being seminal references, tend to be overwhelming in terms of details, and mathematically concepts that are often not familiar to numerical analysts approaching the topic for the first time. These notes are meant to be a first very gentle introduction to these numerical methods, before approaching the more organic references. Further, the notes are self-contained concerning the derivation of geodesics, the algorithms to compute the exponential and logarithmic maps, and interpolation algorithms on the Grassmann manifold. These mathematical results are all well-known, but the original proofs are scattered across several manu scripts, often using different notations and level of detail, so that their study may not be immediate.