A survey on direct solvers for Galerkin methods

Autores
Pardo, David; Paszynsk, Maciej; Collier, Nathan; Alvarez, Julien; Dalcin, Lisandro Daniel; Calo, Victor
Año de publicación
2012
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
In this paper we describe the history, performance, and design concepts of direct solvers for algebraic systems resulting from Galerkin discretizations of partial differential equations. Popular direct solver implementations of Gaussian elimination (also known as LU factorization) are introduced and briefly analyzed. We discuss three of the most relevant aspects influencing the performance of direct solvers on this kind of algebraic systems. First, the ordering of the degrees of freedom of the algebraic system has a significant impact on the solver performance, solution speed and memory requirements. The impact of unknowns ordering for elimination is exemplified and alternative ordering algorithms are described and compared. Second, the effect of round-off error on the simulation results is discussed. We detail this effect for uniform grids where the impact of round-off error on the solution is controlled by the condition number of the matrix in terms of the element size, but is independent of the polynomial order of approximation. Additionally, we discuss the link between unknown ordering and round-off error. Third, we describe the impact of the connectivity pattern (graph) of the basis functions on the performance of direct solvers. Variations in the connectivity structure of the resulting discrete system have severe impact on performance of the solver. That is, the resources needed to factorize the system strongly depend on its connectivity graph. Less connected graphs are cheaper to solve, that is, C0 finite element discretizations are cheaper to solve with direct solvers than Cp-1 discretizations.
Fil: Pardo, David. Universidad del Pais Vasco; España
Fil: Paszynsk, Maciej. University of Science and Technology; Polonia
Fil: Collier, Nathan. King Abdullah University of Science and Technology; Arabia Saudita
Fil: Alvarez, Julien. Universidad del Pais Vasco; España
Fil: Dalcin, Lisandro Daniel. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Santa Fe. Instituto de Desarrollo Tecnológico Para la Industria Química (i); Argentina
Fil: Calo, Victor. King Abdullah University of Science and Technology; Arabia Saudita
Materia
Lu Factorization
Gaussian Elimination
Frontal Solver
Multi-Frontal Solver
Hp-Finite Elements
Isogeometric Analysis
Cost of Regularity
Parallel Direct Solvers
Nivel de accesibilidad
acceso abierto
Condiciones de uso
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
Repositorio
CONICET Digital (CONICET)
Institución
Consejo Nacional de Investigaciones Científicas y Técnicas
OAI Identificador
oai:ri.conicet.gov.ar:11336/10965
Acceder
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oai_identifier_str oai:ri.conicet.gov.ar:11336/10965
network_acronym_str CONICETDig
repository_id_str 3498
network_name_str CONICET Digital (CONICET)
spelling A survey on direct solvers for Galerkin methodsPardo, DavidPaszynsk, MaciejCollier, NathanAlvarez, JulienDalcin, Lisandro DanielCalo, VictorLu FactorizationGaussian EliminationFrontal SolverMulti-Frontal SolverHp-Finite ElementsIsogeometric AnalysisCost of RegularityParallel Direct Solvershttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1In this paper we describe the history, performance, and design concepts of direct solvers for algebraic systems resulting from Galerkin discretizations of partial differential equations. Popular direct solver implementations of Gaussian elimination (also known as LU factorization) are introduced and briefly analyzed. We discuss three of the most relevant aspects influencing the performance of direct solvers on this kind of algebraic systems. First, the ordering of the degrees of freedom of the algebraic system has a significant impact on the solver performance, solution speed and memory requirements. The impact of unknowns ordering for elimination is exemplified and alternative ordering algorithms are described and compared. Second, the effect of round-off error on the simulation results is discussed. We detail this effect for uniform grids where the impact of round-off error on the solution is controlled by the condition number of the matrix in terms of the element size, but is independent of the polynomial order of approximation. Additionally, we discuss the link between unknown ordering and round-off error. Third, we describe the impact of the connectivity pattern (graph) of the basis functions on the performance of direct solvers. Variations in the connectivity structure of the resulting discrete system have severe impact on performance of the solver. That is, the resources needed to factorize the system strongly depend on its connectivity graph. Less connected graphs are cheaper to solve, that is, C0 finite element discretizations are cheaper to solve with direct solvers than Cp-1 discretizations.Fil: Pardo, David. Universidad del Pais Vasco; EspañaFil: Paszynsk, Maciej. University of Science and Technology; PoloniaFil: Collier, Nathan. King Abdullah University of Science and Technology; Arabia SauditaFil: Alvarez, Julien. Universidad del Pais Vasco; EspañaFil: Dalcin, Lisandro Daniel. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Santa Fe. Instituto de Desarrollo Tecnológico Para la Industria Química (i); ArgentinaFil: Calo, Victor. King Abdullah University of Science and Technology; Arabia SauditaSociedad Española de Matemática Aplicada2012-01info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/10965Pardo, David; Paszynsk, Maciej; Collier, Nathan; Alvarez, Julien; Dalcin, Lisandro Daniel; et al.; A survey on direct solvers for Galerkin methods; Sociedad Española de Matemática Aplicada; SeMA Journal; 57; 1-2012; 107-1341575-9822enginfo:eu-repo/semantics/altIdentifier/doi/10.1007/BF03322602info:eu-repo/semantics/altIdentifier/url/http://www.sema.org.es/ojs/index.php?journal=journal&page=article&op=view&path%5B%5D=616info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-sa/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-09-29T09:37:54Zoai:ri.conicet.gov.ar:11336/10965instacron:CONICETInstitucionalhttp://ri.conicet.gov.ar/Organismo científico-tecnológicoNo correspondehttp://ri.conicet.gov.ar/oai/requestdasensio@conicet.gov.ar; lcarlino@conicet.gov.arArgentinaNo correspondeNo correspondeNo correspondeopendoar:34982025-09-29 09:37:55.181CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv A survey on direct solvers for Galerkin methods
title A survey on direct solvers for Galerkin methods
spellingShingle A survey on direct solvers for Galerkin methods
Pardo, David
Lu Factorization
Gaussian Elimination
Frontal Solver
Multi-Frontal Solver
Hp-Finite Elements
Isogeometric Analysis
Cost of Regularity
Parallel Direct Solvers
title_short A survey on direct solvers for Galerkin methods
title_full A survey on direct solvers for Galerkin methods
title_fullStr A survey on direct solvers for Galerkin methods
title_full_unstemmed A survey on direct solvers for Galerkin methods
title_sort A survey on direct solvers for Galerkin methods
dc.creator.none.fl_str_mv Pardo, David
Paszynsk, Maciej
Collier, Nathan
Alvarez, Julien
Dalcin, Lisandro Daniel
Calo, Victor
author Pardo, David
author_facet Pardo, David
Paszynsk, Maciej
Collier, Nathan
Alvarez, Julien
Dalcin, Lisandro Daniel
Calo, Victor
author_role author
author2 Paszynsk, Maciej
Collier, Nathan
Alvarez, Julien
Dalcin, Lisandro Daniel
Calo, Victor
author2_role author
author
author
author
author
dc.subject.none.fl_str_mv Lu Factorization
Gaussian Elimination
Frontal Solver
Multi-Frontal Solver
Hp-Finite Elements
Isogeometric Analysis
Cost of Regularity
Parallel Direct Solvers
topic Lu Factorization
Gaussian Elimination
Frontal Solver
Multi-Frontal Solver
Hp-Finite Elements
Isogeometric Analysis
Cost of Regularity
Parallel Direct Solvers
purl_subject.fl_str_mv https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
dc.description.none.fl_txt_mv In this paper we describe the history, performance, and design concepts of direct solvers for algebraic systems resulting from Galerkin discretizations of partial differential equations. Popular direct solver implementations of Gaussian elimination (also known as LU factorization) are introduced and briefly analyzed. We discuss three of the most relevant aspects influencing the performance of direct solvers on this kind of algebraic systems. First, the ordering of the degrees of freedom of the algebraic system has a significant impact on the solver performance, solution speed and memory requirements. The impact of unknowns ordering for elimination is exemplified and alternative ordering algorithms are described and compared. Second, the effect of round-off error on the simulation results is discussed. We detail this effect for uniform grids where the impact of round-off error on the solution is controlled by the condition number of the matrix in terms of the element size, but is independent of the polynomial order of approximation. Additionally, we discuss the link between unknown ordering and round-off error. Third, we describe the impact of the connectivity pattern (graph) of the basis functions on the performance of direct solvers. Variations in the connectivity structure of the resulting discrete system have severe impact on performance of the solver. That is, the resources needed to factorize the system strongly depend on its connectivity graph. Less connected graphs are cheaper to solve, that is, C0 finite element discretizations are cheaper to solve with direct solvers than Cp-1 discretizations.
Fil: Pardo, David. Universidad del Pais Vasco; España
Fil: Paszynsk, Maciej. University of Science and Technology; Polonia
Fil: Collier, Nathan. King Abdullah University of Science and Technology; Arabia Saudita
Fil: Alvarez, Julien. Universidad del Pais Vasco; España
Fil: Dalcin, Lisandro Daniel. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Santa Fe. Instituto de Desarrollo Tecnológico Para la Industria Química (i); Argentina
Fil: Calo, Victor. King Abdullah University of Science and Technology; Arabia Saudita
description In this paper we describe the history, performance, and design concepts of direct solvers for algebraic systems resulting from Galerkin discretizations of partial differential equations. Popular direct solver implementations of Gaussian elimination (also known as LU factorization) are introduced and briefly analyzed. We discuss three of the most relevant aspects influencing the performance of direct solvers on this kind of algebraic systems. First, the ordering of the degrees of freedom of the algebraic system has a significant impact on the solver performance, solution speed and memory requirements. The impact of unknowns ordering for elimination is exemplified and alternative ordering algorithms are described and compared. Second, the effect of round-off error on the simulation results is discussed. We detail this effect for uniform grids where the impact of round-off error on the solution is controlled by the condition number of the matrix in terms of the element size, but is independent of the polynomial order of approximation. Additionally, we discuss the link between unknown ordering and round-off error. Third, we describe the impact of the connectivity pattern (graph) of the basis functions on the performance of direct solvers. Variations in the connectivity structure of the resulting discrete system have severe impact on performance of the solver. That is, the resources needed to factorize the system strongly depend on its connectivity graph. Less connected graphs are cheaper to solve, that is, C0 finite element discretizations are cheaper to solve with direct solvers than Cp-1 discretizations.
publishDate 2012
dc.date.none.fl_str_mv 2012-01
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
http://purl.org/coar/resource_type/c_6501
info:ar-repo/semantics/articulo
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/11336/10965
Pardo, David; Paszynsk, Maciej; Collier, Nathan; Alvarez, Julien; Dalcin, Lisandro Daniel; et al.; A survey on direct solvers for Galerkin methods; Sociedad Española de Matemática Aplicada; SeMA Journal; 57; 1-2012; 107-134
1575-9822
url http://hdl.handle.net/11336/10965
identifier_str_mv Pardo, David; Paszynsk, Maciej; Collier, Nathan; Alvarez, Julien; Dalcin, Lisandro Daniel; et al.; A survey on direct solvers for Galerkin methods; Sociedad Española de Matemática Aplicada; SeMA Journal; 57; 1-2012; 107-134
1575-9822
dc.language.none.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv info:eu-repo/semantics/altIdentifier/doi/10.1007/BF03322602
info:eu-repo/semantics/altIdentifier/url/http://www.sema.org.es/ojs/index.php?journal=journal&page=article&op=view&path%5B%5D=616
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
eu_rights_str_mv openAccess
rights_invalid_str_mv https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
dc.format.none.fl_str_mv application/pdf
application/pdf
dc.publisher.none.fl_str_mv Sociedad Española de Matemática Aplicada
publisher.none.fl_str_mv Sociedad Española de Matemática Aplicada
dc.source.none.fl_str_mv reponame:CONICET Digital (CONICET)
instname:Consejo Nacional de Investigaciones Científicas y Técnicas
reponame_str CONICET Digital (CONICET)
collection CONICET Digital (CONICET)
instname_str Consejo Nacional de Investigaciones Científicas y Técnicas
repository.name.fl_str_mv CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicas
repository.mail.fl_str_mv dasensio@conicet.gov.ar; lcarlino@conicet.gov.ar
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score 13.070432

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