Variational Second-Order Interpolation on the Group of Diffeomorphisms with a Right-Invariant Metric

TL;DR

This paper introduces a variational approach for second-order interpolation on the diffeomorphism group with a right-invariant metric, ensuring well-posedness and existence of minimizers for numerical applications.

Contribution

It establishes a new variational framework that guarantees the existence of minimizers for acceleration-based interpolation on diffeomorphism groups.

Findings

Proves well-posedness of the variational problem

Ensures existence of minimizers for numerical simulations

Provides a theoretical foundation for second-order interpolation

Abstract

In this note, we propose a variational framework in which the minimization of the acceleration on the group of diffeomorphisms endowed with a right-invariant metric is well-posed. It relies on constraining the acceleration to belong to a Sobolev space of higher-order than the order of the metric in order to gain compactness. It provides the theoretical guarantee of existence of minimizers which is compulsory for numerical simulations.