Exploiting spatial symmetries for solving Poisson's equation

This paper presents a strategy to accelerate virtually any Poisson solver by taking advantage of s spatial reflection symmetries. More precisely, we have proved the existence of an inexpensive block diagonalisation that transforms the original Poisson equation into a set of 2s fully decoupled subsys...

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Detalles Bibliográficos
Autores: Alsalti Baldellou, Àdel|||0000-0002-5331-4236, Álvarez Farré, Xavier|||0000-0002-1684-7658, Trias Miquel, Francesc Xavier|||0000-0002-5966-0703, Oliva Llena, Asensio|||0000-0002-2805-4794
Tipo de recurso: artículo
Fecha de publicación:2023
País:España
Institución:Universitat Politècnica de Catalunya (UPC)
Repositorio:UPCommons. Portal del coneixement obert de la UPC
Idioma:inglés
OAI Identifier:oai:upcommons.upc.edu:2117/386806
Acceso en línea:https://hdl.handle.net/2117/386806
https://dx.doi.org/10.1016/j.jcp.2023.112133
Access Level:acceso abierto
Palabra clave:Poisson's equation
Computational fluid dynamics
Poisson equation
Spatial symmetries
SpMM
Arithmetic intensity
Memory footprint
CFD
Poisson, Equació de
Dinàmica de fluids computacional
Àrees temàtiques de la UPC::Física::Termodinàmica
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spelling Exploiting spatial symmetries for solving Poisson's equationAlsalti Baldellou, Àdel|||0000-0002-5331-4236Álvarez Farré, Xavier|||0000-0002-1684-7658Trias Miquel, Francesc Xavier|||0000-0002-5966-0703Oliva Llena, Asensio|||0000-0002-2805-4794Poisson's equationComputational fluid dynamicsPoisson equationSpatial symmetriesSpMMArithmetic intensityMemory footprintCFDPoisson, Equació deDinàmica de fluids computacionalÀrees temàtiques de la UPC::Física::TermodinàmicaThis paper presents a strategy to accelerate virtually any Poisson solver by taking advantage of s spatial reflection symmetries. More precisely, we have proved the existence of an inexpensive block diagonalisation that transforms the original Poisson equation into a set of 2s fully decoupled subsystems then solved concurrently. This block diagonalisation is identical regardless of the mesh connectivity (structured or unstructured) and the geometric complexity of the problem, therefore applying to a wide range of academic and industrial configurations. In fact, it simplifies the task of discretising complex geometries since it only requires meshing a portion of the domain that is then mirrored implicitly by the symmetries’ hyperplanes. Thus, the resulting meshes naturally inherit the exploited symmetries, and their memory footprint becomes 2s times smaller. Thanks to the subsystems’ better spectral properties, iterative solvers converge significantly faster. Additionally, imposing an adequate grid points’ ordering allows reducing the operators’ footprint and replacing the standard sparse matrix-vector products with the sparse matrixmatrix product, a higher arithmetic intensity kernel. As a result, matrix multiplications are accelerated, and massive simulations become more affordable. Finally, we include numerical experiments based on a turbulent flow simulation and making state-of-theart solvers exploit a varying number of symmetries. On the one hand, algebraic multigrid and preconditioned Krylov subspace methods require up to 23% and 72% fewer iterations, resulting in up to 1.7x and 5.6x overall speedups, respectively. On the other, sparse direct solvers’ memory footprint, setup and solution costs are reduced by up to 48%, 58% and 46%, respectively.This work has been financially supported by two competitive R+D projects: RETOtwin (PDC2021-120970-I00), given by MCIN/AEI/10.13039/501100011033 and European Union Next GenerationEU/PRTR, and FusionCAT (001-P-001722), given by Generalitat de Catalunya RIS3CAT-FEDER. Àdel Alsalti-Baldellou has also been supported by the predoctoral grants DIN2018-010061 and 2019-DI-90, given by MCIN/AEI/10.13039/501100011033 and the Catalan Agency for Management of University and Research Grants (AGAUR), respectively.Peer ReviewedElsevier20232023-08-0120232023-04-28journal articlehttp://purl.org/coar/resource_type/c_6501VoRhttp://purl.org/coar/version/c_970fb48d4fbd8a85info:eu-repo/semantics/articleapplication/pdfhttps://hdl.handle.net/2117/386806https://dx.doi.org/10.1016/j.jcp.2023.112133reponame:UPCommons. Portal del coneixement obert de la UPCinstname:Universitat Politècnica de Catalunya (UPC)Inglésengopen accesshttp://purl.org/coar/access_right/c_abf2Attribution 4.0 Internationalhttp://creativecommons.org/licenses/by/4.0/info:eu-repo/semantics/openAccessoai:upcommons.upc.edu:2117/3868062026-05-27T15:37:01Z
dc.title.none.fl_str_mv Exploiting spatial symmetries for solving Poisson's equation
title Exploiting spatial symmetries for solving Poisson's equation
spellingShingle Exploiting spatial symmetries for solving Poisson's equation
Alsalti Baldellou, Àdel|||0000-0002-5331-4236
Poisson's equation
Computational fluid dynamics
Poisson equation
Spatial symmetries
SpMM
Arithmetic intensity
Memory footprint
CFD
Poisson, Equació de
Dinàmica de fluids computacional
Àrees temàtiques de la UPC::Física::Termodinàmica
title_short Exploiting spatial symmetries for solving Poisson's equation
title_full Exploiting spatial symmetries for solving Poisson's equation
title_fullStr Exploiting spatial symmetries for solving Poisson's equation
title_full_unstemmed Exploiting spatial symmetries for solving Poisson's equation
title_sort Exploiting spatial symmetries for solving Poisson's equation
dc.creator.none.fl_str_mv Alsalti Baldellou, Àdel|||0000-0002-5331-4236
Álvarez Farré, Xavier|||0000-0002-1684-7658
Trias Miquel, Francesc Xavier|||0000-0002-5966-0703
Oliva Llena, Asensio|||0000-0002-2805-4794
author Alsalti Baldellou, Àdel|||0000-0002-5331-4236
author_facet Alsalti Baldellou, Àdel|||0000-0002-5331-4236
Álvarez Farré, Xavier|||0000-0002-1684-7658
Trias Miquel, Francesc Xavier|||0000-0002-5966-0703
Oliva Llena, Asensio|||0000-0002-2805-4794
author_role author
author2 Álvarez Farré, Xavier|||0000-0002-1684-7658
Trias Miquel, Francesc Xavier|||0000-0002-5966-0703
Oliva Llena, Asensio|||0000-0002-2805-4794
author2_role author
author
author
dc.subject.none.fl_str_mv Poisson's equation
Computational fluid dynamics
Poisson equation
Spatial symmetries
SpMM
Arithmetic intensity
Memory footprint
CFD
Poisson, Equació de
Dinàmica de fluids computacional
Àrees temàtiques de la UPC::Física::Termodinàmica
topic Poisson's equation
Computational fluid dynamics
Poisson equation
Spatial symmetries
SpMM
Arithmetic intensity
Memory footprint
CFD
Poisson, Equació de
Dinàmica de fluids computacional
Àrees temàtiques de la UPC::Física::Termodinàmica
description This paper presents a strategy to accelerate virtually any Poisson solver by taking advantage of s spatial reflection symmetries. More precisely, we have proved the existence of an inexpensive block diagonalisation that transforms the original Poisson equation into a set of 2s fully decoupled subsystems then solved concurrently. This block diagonalisation is identical regardless of the mesh connectivity (structured or unstructured) and the geometric complexity of the problem, therefore applying to a wide range of academic and industrial configurations. In fact, it simplifies the task of discretising complex geometries since it only requires meshing a portion of the domain that is then mirrored implicitly by the symmetries’ hyperplanes. Thus, the resulting meshes naturally inherit the exploited symmetries, and their memory footprint becomes 2s times smaller. Thanks to the subsystems’ better spectral properties, iterative solvers converge significantly faster. Additionally, imposing an adequate grid points’ ordering allows reducing the operators’ footprint and replacing the standard sparse matrix-vector products with the sparse matrixmatrix product, a higher arithmetic intensity kernel. As a result, matrix multiplications are accelerated, and massive simulations become more affordable. Finally, we include numerical experiments based on a turbulent flow simulation and making state-of-theart solvers exploit a varying number of symmetries. On the one hand, algebraic multigrid and preconditioned Krylov subspace methods require up to 23% and 72% fewer iterations, resulting in up to 1.7x and 5.6x overall speedups, respectively. On the other, sparse direct solvers’ memory footprint, setup and solution costs are reduced by up to 48%, 58% and 46%, respectively.
publishDate 2023
dc.date.none.fl_str_mv 2023
2023-08-01
2023
2023-04-28
dc.type.none.fl_str_mv journal article
http://purl.org/coar/resource_type/c_6501
VoR
http://purl.org/coar/version/c_970fb48d4fbd8a85
dc.type.openaire.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv https://hdl.handle.net/2117/386806
https://dx.doi.org/10.1016/j.jcp.2023.112133
url https://hdl.handle.net/2117/386806
https://dx.doi.org/10.1016/j.jcp.2023.112133
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2
Attribution 4.0 International
http://creativecommons.org/licenses/by/4.0/
dc.rights.openaire.fl_str_mv info:eu-repo/semantics/openAccess
rights_invalid_str_mv open access
http://purl.org/coar/access_right/c_abf2
Attribution 4.0 International
http://creativecommons.org/licenses/by/4.0/
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.publisher.none.fl_str_mv Elsevier
publisher.none.fl_str_mv Elsevier
dc.source.none.fl_str_mv reponame:UPCommons. Portal del coneixement obert de la UPC
instname:Universitat Politècnica de Catalunya (UPC)
instname_str Universitat Politècnica de Catalunya (UPC)
reponame_str UPCommons. Portal del coneixement obert de la UPC
collection UPCommons. Portal del coneixement obert de la UPC
repository.name.fl_str_mv
repository.mail.fl_str_mv
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