Mathematical aspects of registration methods in bounded domains

TL;DR

This paper reviews mathematical techniques for registration methods in bounded domains, focusing on vector flows and compositional maps, with analysis of bijectivity and approximation properties in parametric model reduction.

Contribution

It provides a comprehensive overview and analysis of registration methods in bounded domains, emphasizing their mathematical properties and applications in model reduction.

Findings

Analysis of bijectivity constraints in registration methods

Comparison of vector flows and compositional maps

Insights into approximation capabilities in Lipschitz domains

Abstract

Registration methods in bounded domains have received significant attention in the model reduction literature, as a valuable tool for nonlinear approximation. The aim of this work is to provide a concise yet complete overview of relevant results for registration methods in $n$-dimensional domains, from the perspective of parametric model reduction. We present a thorough analysis of two classes of methods, vector flows and compositional maps: we discuss the enforcement of the bijectivity constraint and we comment on the approximation properties of the two methods, for Lipschitz $n$-dimensional domains.