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    Hierarchical matrix approximations for space-fractional diffusion equations

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    Type
    Article
    Authors
    Boukaram, Wagih Halim
    Lucchesi, Marco
    Turkiyyah, George
    Le Maître, Olivier cc
    Knio, Omar cc
    Keyes, David E. cc
    KAUST Department
    Computer Science Program
    Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
    Applied Mathematics and Computational Science Program
    Extreme Computing Research Center
    Office of the President
    Date
    2020-06-11
    Online Publication Date
    2020-06-11
    Print Publication Date
    2020-09
    Embargo End Date
    2022-06-11
    Submitted Date
    2020-01-18
    Permanent link to this record
    http://hdl.handle.net/10754/663531
    
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    Abstract
    Space 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.
    Citation
    Boukaram, 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
    Sponsors
    Research reported in this publication was supported by research funding from King Abdullah University of Science and Technology (KAUST).
    Publisher
    Elsevier BV
    Journal
    Computer Methods in Applied Mechanics and Engineering
    DOI
    10.1016/j.cma.2020.113191
    Additional Links
    https://linkinghub.elsevier.com/retrieve/pii/S0045782520303765
    ae974a485f413a2113503eed53cd6c53
    10.1016/j.cma.2020.113191
    Scopus Count
    Collections
    Articles; Applied Mathematics and Computational Science Program; Extreme Computing Research Center; Computer Science Program; Computer, Electrical and Mathematical Science and Engineering (CEMSE) Division

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