Hybrid Direct and Iterative Solver with Library of Multi-criteria Optimal Orderings for h Adaptive Finite Element Method Computations

Handle URI:
http://hdl.handle.net/10754/626065
Title:
Hybrid Direct and Iterative Solver with Library of Multi-criteria Optimal Orderings for h Adaptive Finite Element Method Computations
Authors:
AbouEisha, Hassan M. ( 0000-0003-4560-7175 ) ; Jopek, Konrad; Medygrał, Bartłomiej; Moshkov, Mikhail ( 0000-0003-0085-9483 ) ; Nosek, Szymon; Paszyńska, Anna; Paszyński, Maciej; Pingali, Keshav
Abstract:
In this paper we present a multi-criteria optimization of element partition trees and resulting orderings for multi-frontal solver algorithms executed for two dimensional h adaptive finite element method. In particular, the problem of optimal ordering of elimination of rows in the sparse matrices resulting from adaptive finite element method computations is reduced to the problem of finding of optimal element partition trees. Given a two dimensional h refined mesh, we find all optimal element partition trees by using the dynamic programming approach. An element partition tree defines a prescribed order of elimination of degrees of freedom over the mesh. We utilize three different metrics to estimate the quality of the element partition tree. As the first criterion we consider the number of floating point operations(FLOPs) performed by the multi-frontal solver. As the second criterion we consider the number of memory transfers (MEMOPS) performed by the multi-frontal solver algorithm. As the third criterion we consider memory usage (NONZEROS) of the multi-frontal direct solver. We show the optimization results for FLOPs vs MEMOPS as well as for the execution time estimated as FLOPs+100MEMOPS vs NONZEROS. We obtain Pareto fronts with multiple optimal trees, for each mesh, and for each refinement level. We generate a library of optimal elimination trees for small grids with local singularities. We also propose an algorithm that for a given large mesh with identified local sub-grids, each one with local singularity. We compute Schur complements over the sub-grids using the optimal trees from the library, and we submit the sequence of Schur complements into the iterative solver ILUPCG.
KAUST Department:
Computer Science, King Abdullah University of Science and Technology, Thuwal, Saudi Arabia; Applied Mathematics and Computational Science Program
Citation:
AbouEisha H, Jopek K, Medygrał B, Moshkov M, Nosek S, et al. (2016) Hybrid Direct and Iterative Solver with Library of Multi-criteria Optimal Orderings for h Adaptive Finite Element Method Computations. Procedia Computer Science 80: 865–874. Available: http://dx.doi.org/10.1016/j.procs.2016.05.312.
Publisher:
Elsevier BV
Journal:
Procedia Computer Science
Conference/Event name:
International Conference on Computational Science, ICCS 2016
Issue Date:
2-Jun-2016
DOI:
10.1016/j.procs.2016.05.312
Type:
Article
ISSN:
1877-0509
Sponsors:
The work presented in this paper has been supported by National Science Centre, Poland grant no. DEC-2015/17/B/ST6/01867 and by King Abdullah University of Science and Technology (KAUST).
Additional Links:
http://www.sciencedirect.com/science/article/pii/S1877050916306706
Appears in Collections:
Articles; Applied Mathematics and Computational Science Program

Full metadata record

DC FieldValue Language
dc.contributor.authorAbouEisha, Hassan M.en
dc.contributor.authorJopek, Konraden
dc.contributor.authorMedygrał, Bartłomiejen
dc.contributor.authorMoshkov, Mikhailen
dc.contributor.authorNosek, Szymonen
dc.contributor.authorPaszyńska, Annaen
dc.contributor.authorPaszyński, Maciejen
dc.contributor.authorPingali, Keshaven
dc.date.accessioned2017-10-31T08:20:20Z-
dc.date.available2017-10-31T08:20:20Z-
dc.date.issued2016-06-02en
dc.identifier.citationAbouEisha H, Jopek K, Medygrał B, Moshkov M, Nosek S, et al. (2016) Hybrid Direct and Iterative Solver with Library of Multi-criteria Optimal Orderings for h Adaptive Finite Element Method Computations. Procedia Computer Science 80: 865–874. Available: http://dx.doi.org/10.1016/j.procs.2016.05.312.en
dc.identifier.issn1877-0509en
dc.identifier.doi10.1016/j.procs.2016.05.312en
dc.identifier.urihttp://hdl.handle.net/10754/626065-
dc.description.abstractIn this paper we present a multi-criteria optimization of element partition trees and resulting orderings for multi-frontal solver algorithms executed for two dimensional h adaptive finite element method. In particular, the problem of optimal ordering of elimination of rows in the sparse matrices resulting from adaptive finite element method computations is reduced to the problem of finding of optimal element partition trees. Given a two dimensional h refined mesh, we find all optimal element partition trees by using the dynamic programming approach. An element partition tree defines a prescribed order of elimination of degrees of freedom over the mesh. We utilize three different metrics to estimate the quality of the element partition tree. As the first criterion we consider the number of floating point operations(FLOPs) performed by the multi-frontal solver. As the second criterion we consider the number of memory transfers (MEMOPS) performed by the multi-frontal solver algorithm. As the third criterion we consider memory usage (NONZEROS) of the multi-frontal direct solver. We show the optimization results for FLOPs vs MEMOPS as well as for the execution time estimated as FLOPs+100MEMOPS vs NONZEROS. We obtain Pareto fronts with multiple optimal trees, for each mesh, and for each refinement level. We generate a library of optimal elimination trees for small grids with local singularities. We also propose an algorithm that for a given large mesh with identified local sub-grids, each one with local singularity. We compute Schur complements over the sub-grids using the optimal trees from the library, and we submit the sequence of Schur complements into the iterative solver ILUPCG.en
dc.description.sponsorshipThe work presented in this paper has been supported by National Science Centre, Poland grant no. DEC-2015/17/B/ST6/01867 and by King Abdullah University of Science and Technology (KAUST).en
dc.publisherElsevier BVen
dc.relation.urlhttp://www.sciencedirect.com/science/article/pii/S1877050916306706en
dc.rightsUnder a Creative Commons licenseen
dc.rights.urihttp://creativecommons.org/licenses/by-nc-nd/4.0/en
dc.subjectElement partition treeen
dc.subjectFinite element methoden
dc.subjectH adaptivityen
dc.subjectILUPCGen
dc.subjectMulti-frontal direct solveren
dc.subjectOrdering algorithmsen
dc.titleHybrid Direct and Iterative Solver with Library of Multi-criteria Optimal Orderings for h Adaptive Finite Element Method Computationsen
dc.typeArticleen
dc.contributor.departmentComputer Science, King Abdullah University of Science and Technology, Thuwal, Saudi Arabiaen
dc.contributor.departmentApplied Mathematics and Computational Science Programen
dc.identifier.journalProcedia Computer Scienceen
dc.conference.date2016-06-06 to 2016-06-08en
dc.conference.nameInternational Conference on Computational Science, ICCS 2016en
dc.conference.locationSan Diego, CA, USAen
dc.eprint.versionPublisher's Version/PDFen
dc.contributor.institutionDepartment of Computer Science, AGH University of Science and Technology, Kraków, Polanden
dc.contributor.institutionFaculty of Physics, Astronomy and Applied Computer Science, Jagiellonian University, Kraków, Polanden
dc.contributor.institutionInstitute for Computational and Engineering Sciences, University of Texas, Austin, United Statesen
kaust.authorAbouEisha, Hassan M.en
kaust.authorMoshkov, Mikhailen
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