Mathematical aspects of registration methods in bounded domains

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.