Space-time Parallelization of Numerical Methods for Partial Differential Equations

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The field of high-performance scientific computing is in the midst of a disruptive paradigm shift brought about by changes in computer architecture. Increased computational through-put is now achieved mainly by increasing concurrency with the number of cores per processor increasing exponentially and plans for the first exascale system suggesting massive cores. In addition, limitations to system power, increased memory hierarchy, and fundamental physical barriers will make memory access and communication relatively more expensive than floating point operations. These changes are necessitating the reconsideration of numerical algorithms of nearly every type in terms of the new efficiency metrics emerging architectures require. Motivated by the challenge of increasing concurrency, this research centers around the analysis, implementation, and application of new algorithms to enable parallelization of numerical methods for partial differential equations in both space and time. His approach builds on the parallel full approximation scheme in space and time (PFASST) algorithm recently developed by the PIs. PFASST combines iterative temporal integration schemes and a hierarchy of spatial and temporal discretizations to allow work on multiple time steps of a PDE to be done concurrently. Preliminary studies of the PFASST algorithm by the PIs have demonstrated temporal parallel efficiencies in excess of fifty percent on a range of representative model PDEs of differing type, however, space-time parallelism to this point is largely unexplored and certainly not widely adopted in the broader community. This project outlines a program of research necessary to move the use of space-time parallelism from the proof of concept stage to an effective way of increasing computational speed in large scale applications across computational science. Specific major issues addressed in the research include mathematical analysis of the convergence of PFASST in various discretization regimes, load-balancing and optimization of the trade-off between space and time parallelization, adapting PFASST to unstructured grids and particle based simulations, and the use of multiple physical models within the discretization hierarchies. The research has the potential to increase the available computational concurrency in virtually all time-dependent numerical methods. Hence this research project could impact a broad spectrum of fields such as computational chemistry, biology, physics, engineering, and computer science. In the area of education and outreach, the PI includes activities at the graduate and undergraduate level, with a component designed to reach high-school teachers and students.