Equivalence of the suggested triangle-grid approach to the uniform-grid Morse–Smale complex after ε-simplification

Determine whether applying the steepest descent method of Robins et al. (2011) on the suggested triangle grid derived from a uniform grid—constructed by adding edge-midpoint and random interior vertices per cell with bilinear interpolation and Delaunay triangulation, followed by an ε-simplification—produces a Morse–Smale complex that is topologically equivalent to the Morse–Smale complex obtained from the original uniform grid after ε-simplification.

Background

To improve geometric accuracy while preserving topological consistency, the authors propose converting a uniform grid into a triangle mesh by adding auxiliary vertices (edge midpoints and random interior points) and triangulating each cell, then applying a small ε-simplification. They empirically observe improved geometric embeddings when using the steepest descent method on this mesh.

They explicitly state that it remains to be determined whether the resulting Morse–Smale complex is equivalent to that from the uniform grid (after ε-simplification), noting reasons to expect equivalence due to using the same steepest descent method and value bounds from bilinear interpolation.