Compare conforming Morse–Smale complex methods with the steepest descent approach

Investigate the similarities and differences between conforming Morse–Smale complex constructions that use an auxiliary map to enforce equal-valued pairings (e.g., Gyulassy et al., 2014; 2019) and the steepest descent discrete gradient construction of Robins et al. (2011), focusing on both the resulting geometric embeddings of separatrices and the topological connectivity between critical points.

Background

The paper reviews methods for computing Morse–Smale complexes, emphasizing that probabilistic approaches can improve geometric embedding but may alter topology. Gyulassy et al. proposed conforming Morse–Smale complexes where additional information via a map L can influence pairings to achieve application-specific outcomes.

The authors explicitly leave for future work a systematic investigation comparing these conforming approaches to the standard steepest descent method by Robins et al., which is widely used and has established topological guarantees.