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dc.contributor.authorBoukaram, Wagih Halim
dc.contributor.authorLucchesi, Marco
dc.contributor.authorTurkiyyah, George
dc.contributor.authorLe Maître, Olivier
dc.contributor.authorKnio, Omar
dc.contributor.authorKeyes, David E.
dc.date.accessioned2020-06-14T10:50:59Z
dc.date.available2020-06-14T10:50:59Z
dc.date.issued2020-06-11
dc.date.submitted2020-01-18
dc.identifier.citationBoukaram, W., Lucchesi, M., Turkiyyah, G., Le Maître, O., Knio, O., & Keyes, D. (2020). Hierarchical matrix approximations for space-fractional diffusion equations. Computer Methods in Applied Mechanics and Engineering, 369, 113191. doi:10.1016/j.cma.2020.113191
dc.identifier.issn0045-7825
dc.identifier.doi10.1016/j.cma.2020.113191
dc.identifier.urihttp://hdl.handle.net/10754/663531
dc.description.abstractSpace fractional diffusion models generally lead to dense discrete matrix operators, which lead to substantial computational challenges when the system size becomes large. For a state of size N, full representation of a fractional diffusion matrix would require O(N2) memory storage requirement, with a similar estimate for matrix–vector products. In this work, we present H2 matrix representation and algorithms that are amenable to efficient implementation on GPUs, and that can reduce the cost of storing these operators to O(N) asymptotically. Matrix–vector multiplications can be performed in asymptotically linear time as well. Performance of the algorithms is assessed in light of 2D simulations of space fractional diffusion equation with constant diffusivity. Attention is focused on smooth particle approximation of the governing equations, which lead to discrete operators involving explicit radial kernels. The algorithms are first tested using the fundamental solution of the unforced space fractional diffusion equation in an unbounded domain, and then for the steady, forced, fractional diffusion equation in a bounded domain. Both matrix-inverse and pseudo-transient solution approaches are considered in the latter case. Our experiments show that the construction of the fractional diffusion matrix, the matrix–vector multiplication, and the generation of an approximate inverse pre-conditioner all perform very well on a single GPU on 2D problems with N in the range 105 – 106. In addition, the tests also showed that, for the entire range of parameters and fractional orders considered, results obtained using the H2 approximations were in close agreement with results obtained using dense operators, and exhibited the same spatial order of convergence. Overall, the present experiences showed that the H2 matrix framework promises to provide practical means to handle large-scale space fractional diffusion models in several space dimensions, at a computational cost that is asymptotically similar to the cost of handling classical diffusion equations.
dc.description.sponsorshipResearch reported in this publication was supported by research funding from King Abdullah University of Science and Technology (KAUST).
dc.publisherElsevier BV
dc.relation.urlhttps://linkinghub.elsevier.com/retrieve/pii/S0045782520303765
dc.rightsNOTICE: this is the author’s version of a work that was accepted for publication in Computer Methods in Applied Mechanics and Engineering. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Computer Methods in Applied Mechanics and Engineering, [369, , (2020-06-11)] DOI: 10.1016/j.cma.2020.113191 . © 2020. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/
dc.titleHierarchical matrix approximations for space-fractional diffusion equations
dc.typeArticle
dc.contributor.departmentComputer Science Program
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
dc.contributor.departmentApplied Mathematics and Computational Science Program
dc.contributor.departmentExtreme Computing Research Center
dc.contributor.departmentOffice of the President
dc.identifier.journalComputer Methods in Applied Mechanics and Engineering
dc.rights.embargodate2022-06-11
dc.eprint.versionPost-print
dc.contributor.institutionDepartment of Computer Science, American University of Beirut, Beirut, Lebanon.
dc.contributor.institutionCentre de Mathématiques Appliquées, CNRS, Inria, Ecole Polytechnique, Palaiseau, France.
dc.identifier.volume369
dc.identifier.pages113191
kaust.personBoukaram, Wagih Halim
kaust.personLucchesi, Marco
kaust.personKnio, Omar
kaust.personKeyes, David E.
dc.date.accepted2020-05-29
dc.date.published-online2020-06-11
dc.date.published-print2020-09


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