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Jacob Schroder, University of New Mexico Department of Mathematics and Statistics
Parallel-in-Time with Multigrid: Method, Theory, and Applications
The need for parallel-in-time algorithms is currently being driven by the rapidly changing nature of computer architectures. Future speedups will come through ever increasing numbers of processors (cores), but not faster clock speeds, which are stagnant. Previously, increasing clock-speeds compensated for traditional sequential time stepping algorithms when the problem size increased. However, this is no longer the case, leading to the sequential time integration bottleneck and the need to parallelize in time. In this talk, we examine an optimal-scaling parallel time integration method, multigrid reduction in time (MGRIT). MGRIT applies multigrid to the time dimension by solving the (non)linear systems that arise when solving for multiple time steps simultaneously. The result is a versatile approach that is nonintrusive and wraps existing time evolution codes. MGRIT allows for various time discretizations (e.g., Runge-Kutta and multistep) and for adaptive refinement/coarsening in time and space. Nonlinear problems are handled through full approximation storage (FAS) multigrid. Some recent theoretical results, as well as practical results for a variety of problems will be presented, e.g., nonlinear diffusion, powergrid systems, advection, and compressible Navier-Stokes. Additionally, a novel application of time-parallelism to deep learning is presented, where the layer dimension of deep neural networks is explicitly parallelized by an MGRIT method.