A graph theoretic view is taken for a range of phenomena in continuum physics in order to develop representations that will allow analysis of large scale, high-fidelity solutions to these problems. Of interest are phenomena whose description leads to partial differential equations, with solutions being obtained by computation. The motivation is to gain insight that may otherwise be difficult to attain because of the high dimensionality of computed solutions.
We consider graph theoretic representations that are made possible by low-dimensional states defined on the systems. These states are typically functionals of the high-dimensional solutions and therefore retain important aspects of the high-fidelity information present in the original, computed solutions. Our approach is rooted in regarding each state as a vertex on a graph and identifying edges via processes that are induced either by numerical solution strategies or by the physics. Correspondences are drawn between the sampling of stationary states or the time evolution of dynamic phenomena and the machinery of graph theory.
A collection of computations is examined in this framework and new insights to them are presented through the lens of the graph theoretic representation.
Krishnakumar Garikipati obtained his bachelor’s degree from the Indian Institute of Technology, Bombay, in 1991, and his master’s and Ph.D. from Stanford University in 1992 and 1996, respectively. After a few years of post-doctoral work, he joined the faculty at University of Michigan in 2000, where he has been a professor in the Department of Mechanical Engineering and Department of Mathematics since 2012. His research draws on applied mathematics and numerical methods to explain phenomena in materials physics and biophysics. A recent interest is in using data-driven methods to enhance our ability to solve computational physics problems. In 2016 he was appointed director of the Michigan Institute for Computational Discovery and Engineering (MICDE), a research institute focused on developing new paradigms of computational science that cut across application areas. He has been awarded the DOE Early Career Award, the Presidential Early Career Award for Scientists and Engineers, and a Humboldt Research Fellowship.