Steepest descent paths on simplicial meshes of arbitrary dimensions

Abstract

This paper introduces an algorithm to compute steepest descent paths on multivariate piecewise-linear functions on Euclidean domains of arbitrary dimensions and topology. The domain of the function is required to be a finite PL-manifold modeled by a simplicial complex. Given a starting point in such a domain, the resulting steepest descent path is represented by a sequence of segments terminating at a local minimum. Existing approaches for two and three dimensions define few ad hoc procedures to calculate these segments within simplices of dimensions one, two and three. Unfortunately, in a dimension-independent setting this case-by-case approach is no longer applicable, and a generalized theory and a corresponding algorithm must be designed. In this paper, the calculation is based on the derivation of the analytical form of the hyperplane containing the simplex, independent of its dimension. Our prototype implementation demonstrates that the algorithm is efficient even for significantly complex domains.