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Chaotic multigrid methods for the solution of elliptic equations

Chaotic multigrid methods for the solution of elliptic equations
Chaotic multigrid methods for the solution of elliptic equations
Supercomputer power has been doubling approximately every 14 months for several decades, increasing the capabilities of scientific modelling at a similar rate. However, to utilize these machines effectively for applications such as computational fluid dynamics, improvements to strong scalability are required. Here,the particular focus is on semi-implicit, viscous-flow CFD, where the largest bottleneck to strong scalability is the parallel solution of the linear pressure-correction equation – an elliptic Poisson equation. State-of-the-art linear solvers, such as Krylov subspace or multigrid methods, provide excellent numerical performance for elliptic equations, but do not scale efficiently due to frequent synchronization between processes. Complete desynchronization is possible for basic, Jacobi-like solvers using the theory of 'chaotic relaxations'. These non-deterministic, chaotic solvers scale superbly, as demonstrated herein, but lack the numerical performance to converge elliptic equations – even with the relatively lax convergence requirements of the example CFD application. However, these chaotic principles can also be applied to multigrid solvers. In this paper, a 'chaotic-cycle' algebraic multigrid method is described and implemented as an open-source library. It is tested on a model Poisson equation, and also within the context of CFD. Two CFD test cases are used: the canonical lid-driven cavity flow and the flow simulation of a ship (KVLCC2). The chaotic-cycle multigrid shows good scalability and numerical performance compared to classical V-, W- and F-cycles. On 2048 cores the chaotic-cycle multigrid solver performs up to 7.7x faster than Flexible-GMRES and 13.3x faster than classical V-cycle multigrid. Further improvements to chaotic-cycle multigrid can be made, relating to coarse-grid communications and desynchronized residual computations. It is expected that the chaotic-cycle multigrid could be applied to other scientific fields, wherever a scalable elliptic-equation solver is required.
Chaotic Cycle, Multigrid, Chaotic Relaxation, Exascale, Strong Scalability, Elliptic Equations
0010-4655
26-36
Hawkes, J.
3882428e-838b-4e54-9fbf-ba85ba080dc9
Vaz, G.
7fe43ac2-1513-440b-8332-aa64d61fce9d
Phillips, A.B.
f565b1da-6881-4e2a-8729-c082b869028f
Klaij, C.M.
54797642-32ab-49ab-9085-5caed19137e6
Cox, S.J.
0e62aaed-24ad-4a74-b996-f606e40e5c55
Turnock, S.R.
d6442f5c-d9af-4fdb-8406-7c79a92b26ce
Hawkes, J.
3882428e-838b-4e54-9fbf-ba85ba080dc9
Vaz, G.
7fe43ac2-1513-440b-8332-aa64d61fce9d
Phillips, A.B.
f565b1da-6881-4e2a-8729-c082b869028f
Klaij, C.M.
54797642-32ab-49ab-9085-5caed19137e6
Cox, S.J.
0e62aaed-24ad-4a74-b996-f606e40e5c55
Turnock, S.R.
d6442f5c-d9af-4fdb-8406-7c79a92b26ce

Hawkes, J., Vaz, G., Phillips, A.B., Klaij, C.M., Cox, S.J. and Turnock, S.R. (2019) Chaotic multigrid methods for the solution of elliptic equations. Computer Physics Communications, 237, 26-36. (doi:10.1016/j.cpc.2018.10.031).

Record type: Article

Abstract

Supercomputer power has been doubling approximately every 14 months for several decades, increasing the capabilities of scientific modelling at a similar rate. However, to utilize these machines effectively for applications such as computational fluid dynamics, improvements to strong scalability are required. Here,the particular focus is on semi-implicit, viscous-flow CFD, where the largest bottleneck to strong scalability is the parallel solution of the linear pressure-correction equation – an elliptic Poisson equation. State-of-the-art linear solvers, such as Krylov subspace or multigrid methods, provide excellent numerical performance for elliptic equations, but do not scale efficiently due to frequent synchronization between processes. Complete desynchronization is possible for basic, Jacobi-like solvers using the theory of 'chaotic relaxations'. These non-deterministic, chaotic solvers scale superbly, as demonstrated herein, but lack the numerical performance to converge elliptic equations – even with the relatively lax convergence requirements of the example CFD application. However, these chaotic principles can also be applied to multigrid solvers. In this paper, a 'chaotic-cycle' algebraic multigrid method is described and implemented as an open-source library. It is tested on a model Poisson equation, and also within the context of CFD. Two CFD test cases are used: the canonical lid-driven cavity flow and the flow simulation of a ship (KVLCC2). The chaotic-cycle multigrid shows good scalability and numerical performance compared to classical V-, W- and F-cycles. On 2048 cores the chaotic-cycle multigrid solver performs up to 7.7x faster than Flexible-GMRES and 13.3x faster than classical V-cycle multigrid. Further improvements to chaotic-cycle multigrid can be made, relating to coarse-grid communications and desynchronized residual computations. It is expected that the chaotic-cycle multigrid could be applied to other scientific fields, wherever a scalable elliptic-equation solver is required.

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ChaoticMultigridHawkes - Accepted Manuscript
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More information

Accepted/In Press date: 29 October 2018
e-pub ahead of print date: 10 November 2018
Published date: 1 April 2019
Keywords: Chaotic Cycle, Multigrid, Chaotic Relaxation, Exascale, Strong Scalability, Elliptic Equations

Identifiers

Local EPrints ID: 426758
URI: http://eprints.soton.ac.uk/id/eprint/426758
ISSN: 0010-4655
PURE UUID: 4eaa4120-0a0c-4041-b87a-759dbf546839
ORCID for A.B. Phillips: ORCID iD orcid.org/0000-0003-3234-8506
ORCID for S.R. Turnock: ORCID iD orcid.org/0000-0001-6288-0400

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Date deposited: 11 Dec 2018 17:31
Last modified: 07 Oct 2020 06:08

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