TY - JOUR

T1 - Hierarchical matrix approximations for space-fractional diffusion equations

AU - Boukaram, Wagih Halim

AU - Lucchesi, Marco

AU - Turkiyyah, George

AU - Le Maître, Olivier

AU - Knio, Omar

AU - Keyes, David E.

N1 - KAUST Repository Item: Exported on 2020-10-01
Acknowledgements: Research reported in this publication was supported by research funding from King Abdullah University of Science and Technology (KAUST).

PY - 2020/6/11

Y1 - 2020/6/11

N2 - 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.

AB - 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.

UR - http://hdl.handle.net/10754/663531

UR - https://linkinghub.elsevier.com/retrieve/pii/S0045782520303765

UR - http://www.scopus.com/inward/record.url?scp=85086142076&partnerID=8YFLogxK

U2 - 10.1016/j.cma.2020.113191

DO - 10.1016/j.cma.2020.113191

M3 - Article

SN - 0045-7825

VL - 369

SP - 113191

JO - Computer Methods in Applied Mechanics and Engineering

JF - Computer Methods in Applied Mechanics and Engineering

ER -