Scheduling massively parallel multigrid for multilevel Monte Carlo methods

TL;DR

This paper develops and analyzes advanced scheduling strategies for parallel multigrid solvers within multilevel Monte Carlo methods, significantly improving efficiency and scalability for 3D uncertainty quantification tasks.

Contribution

It introduces optimized concurrent execution across MLMC levels, samples, and spatial grids using load balancing and scalability analysis of multigrid solvers.

Findings

Enhanced parallel efficiency through optimized scheduling.

Demonstrated large-scale 3D scaling with multigrid methods.

Identified the importance of the multigrid solver's scalability window.

Abstract

The computational complexity of naive, sampling-based uncertainty quantification for 3D partial differential equations is extremely high. Multilevel approaches, such as multilevel Monte Carlo (MLMC), can reduce the complexity significantly, but to exploit them fully in a parallel environment, sophisticated scheduling strategies are needed. Often fast algorithms that are executed in parallel are essential to compute fine level samples in 3D, whereas to compute individual coarse level samples only moderate numbers of processors can be employed efficiently. We make use of multiple instances of a parallel multigrid solver combined with advanced load balancing techniques. In particular, we optimize the concurrent execution across the three layers of the MLMC method: parallelization across levels, across samples, and across the spatial grid. The overall efficiency and performance of these methods will be analyzed. Here the scalability window of the multigrid solver is revealed as being essential, i.e., the property that the solution can be computed with a range of process numbers while maintaining good parallel efficiency. We evaluate the new scheduling strategies in a series of numerical tests, and conclude the paper demonstrating large 3D scaling experiments.