Multigrid Reduction in Time for Nonlinear Parabolic Problems: A Case Study

Abstract

The need for parallelism in the time dimension is being driven by changes in computer architectures, where performance increases are now provided through greater concurrency, not faster clock speeds. This creates a bottleneck for sequential time marching schemes because they lack parallelism in the time dimension. Multigrid reduction in time (MGRIT) is an iterative procedure that allows for temporal parallelism by utilizing multigrid reduction techniques and a multilevel hierarchy of coarse time grids. MGRIT has been shown to be effective for linear problems, with speedups of up to 50 times. The goal of this work is the efficient solution of nonlinear problems with MGRIT, where efficiency is defined as achieving similar performance when compared to an equivalent linear problem. The benchmark nonlinear problem is the $p$-Laplacian, where p=4 corresponds to a well-known nonlinear diffusion equation and $p=2$ corresponds to the standard linear diffusion operator, our benchmark linear problem. The key difficu...