Graph Theoretic Methods

This chapter is written from a different viewpoint relative to the two that preceded it. Rather than seeking predictive models, we adopt a graph theoretic perspective for analysis of large scale, high-fidelity solutions to a few example problems in continuum physics. As before, our interest remains in problems that can be described by partial differential equations, with solutions being obtained by a direct numerical simulation. The motivation is to gain insight to high-dimensional solutions by using graphs constructed on low-dimensional states defined on the systems. The states will typically be functionals of the high-dimensional solutions and therefore retain certain high-fidelity information that may be present in the original, computed solutions. We regard each state as a vertex on a graph and identify edges as being induced either by numerical solution strategies or by the physics. We identify correspondences between the physics that determines the relative arrangement of stationary states, or the time evolution of dynamic phenomena, and the analytic machinery of graph theory. We use this framework on a collection of computations to gain insights to them by analyzing the corresponding graphs. As in the previous chapters, the treatment is deterministic. This chapter builds upon work that was presented in Banerjee et al (Comput Methods Appl Mech Eng 351:501–530, 2019).