Spectral element poisson preconditioners for heterogeneous architectures

The solution to the Poisson equation arising from the spectral element discretization of the incompressible Navier-Stokes equation requires robust preconditioning strategies. Two classes of preconditioners prove most effective: geometric p-multigrid and low-order refined methods. Low-order refined preconditioners, moreover, require the use of algebraic multigrid to approximate the inverse of the operator. The communication associated with the multigrid coarse-grid solve hinders the parallel scalability of both classes of preconditioners, especially on heterogeneous architectures. To mitigate the coarse-grid solve cost, novel smoothing strategies are considered. The fourth-kind Chebyshev polynomial smoothing proposed by James Lottes is utilized to accelerate additive Schwarz-based smoothers in a geometric p-multigrid preconditioner. Through these techniques, we develop geometric p-multigrid preconditioners capable of achieving up to an 81% speedup over the state-of-the-art p-multigrid preconditioners on the Summit supercomputer.A p-multigrid approach with an additive coarse-grid solve specifically designed for heterogeneous architectures is considered. We also propose a hybrid p-multigrid and low-order refined preconditioner that improve the time-to-solution by as much as 86% compared to the low-order preconditioner. We demonstrate the effectiveness of these novel approaches on a variety of problems arising from the spectral element discretization of the incompressible Navier-Stokes equations on GPU architectures spanning to P > 1024 NVIDIA V100 GPUs on Summit.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Computer Science

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2023

Author dc:creator

• Phillips, Malachi

Contributors dc:contributor

• Lottes, James
• Fischer, Paul
• Olson, Luke
• Kloeckner, Andreas
• Kolev, Tzanio

• Preconditioning
• Multigrid
• Poisson Equation
• Navier-stokes Equations
• Spectral Element Method
• Heterogeneous Computing
• High-performance Computing

Statement dc:rights

• Copyright 2023 Malachi Phillips

Language dc:language

en, eng

Handle dc:identifier

https://hdl.handle.net/2142/121461

OAI record oai:identifier

https://www.ideals.illinois.edu/oai-pmh?verb=GetRecord&metadataPrefix=etdms&identifier=oai%3Awww.ideals.illinois.edu%3A2142%2F121461