Morse decompositions of three-dimensional piecewise constant vector fields: concepts and efficient implementations

Abstract

Discovering and analyzing vector field features, such as stationary points, periodic trajectories or separatrices, is a challenging task. Common approaches to solving this problem use numerical integration to approximate trajectories. The associated inaccuracy brings consistency issues. Accumulated error may permit approximate trajectories to cross, which breaks the assumptions of many algorithms, leading to potentially inconsistent output. Moreover, some vector fields, for example chaotic ones, have infinitely many periodic trajectories and therefore cannot be analyzed using the classical approaches that would attempt to compute all of them. A more robust approach to vector field analysis is based on Morse decomposition, using regions of flow circulation as features. These regions include stationary points and periodic orbits, as well as more complex features, like chaotic dynamics. Even though some Morse decomposition algorithms use numerical integration, they are more robust than the classical methods. One such approach uses Piecewise Constant (PC) vector fields to compute regions of circulation in 2D and 2.5D. This method builds upon mathematical results related to multivalued flows to overcome discontinuities in a PC vector field. Instead of numerical integration, it uses only simple geometric operations to construct trajectories, which facilitates robust handling of numerical error. Features of interest are represented by the strongly connected components of the transition graph, a finite directed graph representing all possible trajectories in a PC vector field. Transition graphs have a natural multi-resolution structure that can be exploited to obtain efficient adaptive refinement algorithms for approximating the circulating regions. In this thesis, we present three new contributions to this approach. First, the PC vector field analysis algorithm is extended to 3D to support a wider range of datasets. We describe related work leading to this modified system, implementation, and results of this approach for a variety of analytical, and simulated data sets. Secondly, we describe a parallel implementation of the PC framework that make it applicable to larger datasets and reduce computation time. The third contribution is a distributed implementation for large scale 3D vector fields and a new representation of the transition graph to reduce memory usage.