Computational cost of isogeometric multi-frontal solvers on parallel distributed memory machines

Handle URI:
http://hdl.handle.net/10754/564032
Title:
Computational cost of isogeometric multi-frontal solvers on parallel distributed memory machines
Authors:
Woźniak, Maciej; Paszyński, Maciej R.; Pardo, D.; Dalcin, Lisandro ( 0000-0001-8086-0155 ) ; Calo, Victor M. ( 0000-0002-1805-4045 )
Abstract:
This paper derives theoretical estimates of the computational cost for isogeometric multi-frontal direct solver executed on parallel distributed memory machines. We show theoretically that for the Cp-1 global continuity of the isogeometric solution, both the computational cost and the communication cost of a direct solver are of order O(log(N)p2) for the one dimensional (1D) case, O(Np2) for the two dimensional (2D) case, and O(N4/3p2) for the three dimensional (3D) case, where N is the number of degrees of freedom and p is the polynomial order of the B-spline basis functions. The theoretical estimates are verified by numerical experiments performed with three parallel multi-frontal direct solvers: MUMPS, PaStiX and SuperLU, available through PETIGA toolkit built on top of PETSc. Numerical results confirm these theoretical estimates both in terms of p and N. For a given problem size, the strong efficiency rapidly decreases as the number of processors increases, becoming about 20% for 256 processors for a 3D example with 1283 unknowns and linear B-splines with C0 global continuity, and 15% for a 3D example with 643 unknowns and quartic B-splines with C3 global continuity. At the same time, one cannot arbitrarily increase the problem size, since the memory required by higher order continuity spaces is large, quickly consuming all the available memory resources even in the parallel distributed memory version. Numerical results also suggest that the use of distributed parallel machines is highly beneficial when solving higher order continuity spaces, although the number of processors that one can efficiently employ is somehow limited.
KAUST Department:
Numerical Porous Media SRI Center (NumPor); Applied Mathematics and Computational Science Program; Earth Science and Engineering Program; Physical Sciences and Engineering (PSE) Division; Environmental Science and Engineering Program
Publisher:
Elsevier BV
Journal:
Computer Methods in Applied Mechanics and Engineering
Issue Date:
Feb-2015
DOI:
10.1016/j.cma.2014.11.020
Type:
Article
ISSN:
00457825
Appears in Collections:
Articles; Environmental Science and Engineering Program; Applied Mathematics and Computational Science Program; Physical Sciences and Engineering (PSE) Division; Earth Science and Engineering Program

Full metadata record

DC FieldValue Language
dc.contributor.authorWoźniak, Maciejen
dc.contributor.authorPaszyński, Maciej R.en
dc.contributor.authorPardo, D.en
dc.contributor.authorDalcin, Lisandroen
dc.contributor.authorCalo, Victor M.en
dc.date.accessioned2015-08-03T12:29:10Zen
dc.date.available2015-08-03T12:29:10Zen
dc.date.issued2015-02en
dc.identifier.issn00457825en
dc.identifier.doi10.1016/j.cma.2014.11.020en
dc.identifier.urihttp://hdl.handle.net/10754/564032en
dc.description.abstractThis paper derives theoretical estimates of the computational cost for isogeometric multi-frontal direct solver executed on parallel distributed memory machines. We show theoretically that for the Cp-1 global continuity of the isogeometric solution, both the computational cost and the communication cost of a direct solver are of order O(log(N)p2) for the one dimensional (1D) case, O(Np2) for the two dimensional (2D) case, and O(N4/3p2) for the three dimensional (3D) case, where N is the number of degrees of freedom and p is the polynomial order of the B-spline basis functions. The theoretical estimates are verified by numerical experiments performed with three parallel multi-frontal direct solvers: MUMPS, PaStiX and SuperLU, available through PETIGA toolkit built on top of PETSc. Numerical results confirm these theoretical estimates both in terms of p and N. For a given problem size, the strong efficiency rapidly decreases as the number of processors increases, becoming about 20% for 256 processors for a 3D example with 1283 unknowns and linear B-splines with C0 global continuity, and 15% for a 3D example with 643 unknowns and quartic B-splines with C3 global continuity. At the same time, one cannot arbitrarily increase the problem size, since the memory required by higher order continuity spaces is large, quickly consuming all the available memory resources even in the parallel distributed memory version. Numerical results also suggest that the use of distributed parallel machines is highly beneficial when solving higher order continuity spaces, although the number of processors that one can efficiently employ is somehow limited.en
dc.publisherElsevier BVen
dc.subjectCommunication costen
dc.subjectComputational costen
dc.subjectIsogeometric analysisen
dc.subjectMulti-frontal direct solveren
dc.subjectParallel distributed memory machineen
dc.titleComputational cost of isogeometric multi-frontal solvers on parallel distributed memory machinesen
dc.typeArticleen
dc.contributor.departmentNumerical Porous Media SRI Center (NumPor)en
dc.contributor.departmentApplied Mathematics and Computational Science Programen
dc.contributor.departmentEarth Science and Engineering Programen
dc.contributor.departmentPhysical Sciences and Engineering (PSE) Divisionen
dc.contributor.departmentEnvironmental Science and Engineering Programen
dc.identifier.journalComputer Methods in Applied Mechanics and Engineeringen
dc.contributor.institutionAGH University of Sciences and Technology, Faculty of Computer Science, Electronics and Telecommunication, Department of Computer Science, al. A Mickiewicza 30Krakow, Polanden
dc.contributor.institutionDepartment of Applied Mathematics, Statistics, and Operational Research, University of the Basque Country (UPV/EHU)Bilbao, Spainen
dc.contributor.institutionBasque Center for Applied Mathematics (BCAM)Bilbao, Spainen
dc.contributor.institutionIKERBASQUE, Basque Foundation for ScienceBilbao, Spainen
kaust.authorDalcin, Lisandroen
kaust.authorCalo, Victor M.en
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