Extreme-scale Multigrid Components within PETSc

TL;DR

This paper introduces a new agglomeration component in PETSc to improve the scalability and performance of multilevel preconditioners for solving elliptic PDEs on massively parallel systems.

Contribution

A novel software module for agglomeration in PETSc that enhances multigrid solver scalability and efficiency for high-resolution PDE simulations.

Findings

Agglomeration improves multigrid solver scalability on large parallel systems.

Numerical experiments show significant performance gains with the new implementation.

The approach maintains efficiency with structured meshes in geometric multigrid.

Abstract

Elliptic partial differential equations (PDEs) frequently arise in continuum descriptions of physical processes relevant to science and engineering. Multilevel preconditioners represent a family of scalable techniques for solving discrete PDEs of this type and thus are the method of choice for high-resolution simulations. The scalability and time-to-solution of massively parallel multilevel preconditioners can be adversely effected by using a coarse-level solver with sub-optimal algorithmic complexity. To maintain scalability, agglomeration techniques applied to the coarse level have been shown to be necessary. In this work, we present a new software component introduced within the Portable Extensible Toolkit for Scientific computation (PETSc) which permits agglomeration. We provide an overview of the design and implementation of this functionality, together with several use cases highlighting the benefits of agglomeration. Lastly, we demonstrate via numerical experiments employing geometric multigrid with structured meshes, the flexibility and performance gains possible using our MPI-rank agglomeration implementation.