Certified Computation of planar Morse-Smale Complexes

TL;DR

This paper introduces an algorithm that uses interval arithmetic to reliably compute topologically correct Morse-Smale complexes for smooth functions on planar domains, addressing a gap in existing methods for smooth cases.

Contribution

The paper presents the first certified algorithm for computing Morse-Smale complexes of smooth functions on planar domains, extending previous work limited to piecewise linear cases.

Findings

Algorithm successfully computes certified critical points and separatrices.

Can generate geometrically close Morse-Smale complexes.

Addresses the gap in smooth function analysis.

Abstract

The Morse-Smale complex is an important tool for global topological analysis in various problems of computational geometry and topology. Algorithms for Morse-Smale complexes have been presented in case of piecewise linear manifolds. However, previous research in this field is incomplete in the case of smooth functions. In the current paper we address the following question: Given an arbitrarily complex Morse-Smale system on a planar domain, is it possible to compute its certified (topologically correct) Morse-Smale complex? Towards this, we develop an algorithm using interval arithmetic to compute certified critical points and separatrices forming the Morse-Smale complexes of smooth functions on bounded planar domain. Our algorithm can also compute geometrically close Morse-Smale complexes.