Nonlinear Markov semigroups and refinement schemes on metric spaces

TL;DR

This paper proves convergence of barycentric subdivision schemes on Hadamard spaces by linking them to nonlinear Markov semigroups, extending linear theory results to nonlinear metric space settings.

Contribution

It establishes a convergence criterion for multivariate barycentric subdivision schemes on Hadamard spaces, generalizing linear convergence results to nonlinear metric spaces.

Findings

Convergence of schemes is equivalent to convergence on real data.

Characterization of convergence via nonlinear Markov semigroups.

Analysis of approximation qualities and relation to characteristic Markov chains.

Abstract

This article settles the convergence question for multivariate barycentric subdivision schemes with nonnegative masks on complete metric spaces of nonpositive Alexandrov curvature, also known as Hadamard spaces. We establish a link between these types of refinement algorithms and the theory of Markov chains by characterizing barycentric subdivision schemes as nonlinear Markov semigroups. Exploiting this connection, we subsequently prove that any such scheme converges on arbitrary Hadamard spaces if and only if it converges for real valued input data. Moreover, we generalize a characterization of convergence from the linear theory, and consider approximation qualities of barycentric subdivision schemes. A concluding section addresses the relationship between the convergence properties of a scheme and its so-called characteristic Markov chain.