Benjamin Ong is the principal investigator (PI) on a project that has received a $45,000 research and development contract from the Lawrence Livermore National Laboratory, entitled “Systematic Approaches to Construct Coarse-Grid Operators for Multigrid Reduction in Time.”
Multigrid Reduction in Time (MGRIT) [2] uses multigrid reduction techniques to enable temporal parallelism for solving initial value problems. It is known that the convergence rate of MGRIT [3] depends in part on the choice of time-stepping operators on the fine- and coarse-grid, which we call the fine-grid operator and coarse-grid operator respectively. An “ideal” coarse-grid operator is the fine-grid operator applied to approximate the solution on the coarse time interval.
In practice, the ideal coarse-grid operator is never used as the computational cost destroys any parallel speed-up that could be obtained using MGRIT. Instead, a common choice for a coarse-grid operator is a simple re-discretization of the fine-grid operator, i.e., if a single-step method is used on the fine-grid with time-step size h, then the same single-step method is used on the coarse-grid with time-step size m h, where m is a specified coarsening factor.
Numerical simulations are increasingly important in the study of complex systems in engineering, life sciences, medicine, chemistry, physics, and even non-traditional fields such as social sciences. Dr. Ong is working to solve these large-scale evolution problems on modern supercomputing architectures by using a hierarchy of space-time grids to accelerate the solution on the finest time grid.
References
Time permitting, Dr. Ong will explore the connection between the proposed sequences of generated coarse-grid operators to those recently proposed by Vargas et al. [4].
[1] Daniel Crane. The Singular Value Expansion for Compact and Non-Compact Operators. PhD thesis, Michigan Technological University, 2020.
[2] R. D. Falgout, S. Friedhoff, Tz. V. Kolev, S. P. MacLachlan, and J. B. Schroder. Parallel time integration with multigrid. SIAM Journal on Scientific Computing, 36(6):C635–C661, 2014.
[3] Andreas Hessenthaler, Ben S. Southworth, David Nordsletten, Oliver RÅNohrle, Robert D. Falgout, and Jacob B. Schroder. Multilevel convergence analysis of multigrid-reduction-in-time. SIAM Journal on Scientific Computing, 42(2):A771–A796, 2020.
[4] David. A. Vargas. A general framework for deriving coarse grid operators for multigrid reduction in time. Technical report, Lawrence Livermore National Laboratory, 2023.
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