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Computational optimization in solving the geodetic boundary value problems

  • * Corresponding author: Marek Macák

    * Corresponding author: Marek Macák 
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  • The finite volume method (FVM) as a numerical method can be straightforwardly applied for global as well as local gravity field modelling. However, to obtain precise numerical solutions it requires very refined discretization which leads to large-scale parallel computations. To optimize such computations, we present a special class of numerical techniques that are based on a physical decomposition of the computational domain. The domain decomposition (DD) methods like the Additive Schwarz Method are very efficient methods for solving partial differential equations. We briefly present their mathematical formulations, and we test their efficiency in numerical experiments dealing with gravity field modelling. Since there is no need to solve special interface problems between neighbouring subdomains, in our applications we use the overlapping DD methods. Finally, we present the numerical experiment using the FVM approach with 93 312 000 000 unknowns that would not be possible to perform using available computing facilities without aforementioned methods that can efficiently reduce a numerical complexity of the problem.

    Mathematics Subject Classification: Primary: 65K05; Secondary: 49M27.

    Citation:

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  • Figure 1.  Illustration of solution after: a) $ 1^{st} $ iteration of domain decomposition with $ \eta = 15 $, b) $ 10^{th} $ iteration of domain decomposition with $ \eta = 15 $

    Figure 2.  Illustration of data management in parallel DD implementations where blue color illustrate subdomains and yellow color illustrate parallelization

    Figure 3.  Global gravity field model with the resolution $ 1\times 1\ arc\; min $ on the Earth's surface, $ [m^2 s^{-2}] $

    Table 1.  Efficiency comparison of the stationary and nonstationary methods in the experiment with 259 200 unknowns, tested on one CPU core

    Solver CPU time Number of
    [s] iterations
    GS 703 10000
    SOR 92 1136
    BiCG 68 568
    Bi-CGSTAB 41 348
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    Table 2.  Efficiency comparison for various Bi-CGSTAB linear solvers in the experiment with 4 374 000 unknowns, tested on one CPU core

    Solver Number of CPU time Additional memory
    iterations [s] for solver [MB]
    Bi-CGSTAB 1053 403.82 184.26
    BiCGstab(2) 554 494.14 258.02
    BiCGstab(4) 272 629.01 405.46
    BiCGstab(8) 130 860.86 700.34
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    Table 3.  Comparison of Processes/Threads parallelization in the experiment with 4 374 000 unknowns, tested on 4 quad-core CPUs

    MPI OpenMP CPU time Speedup RAM Memory
    Processes Threads [s] ratio [MB] increase
    1 1 403.82 - 237.108 -
    2 232.40 1.73
    4 191.36 2.11
    8 87.31 4.63
    16 57.51 7.02
    2 1 216.84 1.86 245.868 +3.7%
    2 126.17 3.20
    4 98.46 4.10
    8 85.88 4.70
    4 1 114.01 3.54 266.040 +12.2%
    2 79.72 5.06
    4 55.56 7.26
    8 1 79.34 5.09 308.456 +30.0%
    2 70.81 5.70
    16 1 59.51 6.78 390.068 +64.5%
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    Table 4.  Efficiency comparison for the different number of subdomains in the DD experiment with 4 374 000 unknowns, tested on one CPU core

    Number of CPU time Speedup RAM Memory
    subdomains [s] ratio [MB] saving
    1 403.82 - 237.108 -
    5 1651.68 0.24 89.868 -62.1%
    10 907.99 0.44 71.308 -69.9%
    15 856.04 0.46 65.248 -72.5%
    30 854.24 0.47 57.816 -75.6%
     | Show Table
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    Table 5.  Efficiency comparison for the different number $ \eta $ in the experiment with 4 374 000 unknowns for case of 30 subdomains, tested on one CPU core

    $ \eta $ CPU time Speedup
    [s] ratio
    1 854.24 -
    5 308.02 2.77
    10 252.33 3.38
    15 224.65 3.80
    20 236.56 3.61
    25 265.73 3.21
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    Table 6.  Comparison for the different number of subdomains using parallel DD method in the experiment with 4 374 000 unknowns with $ \eta = 15 $, tested on 4 quad-core CPUs

    Number of CPU time Speedup RAM Memory
    subdomains [s] ratio [MB] saving
    1 55.56 - 266.040 -
    5 55.52 1.00 115.508 -56.6%
    10 28.47 1.95 97.568 -63.3%
    15 17.44 3.18 91.156 -65.7%
    30 18.67 2.97 84.128 -68.3%
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    Table 7.  Efficiency comparison for different computation strategies in the experiment with 4 374 000 unknowns where we use 30 subdomains and $ \eta = 15 $

    Computation CPU time Speedup RAM Memory
    strategies [s] ratio [MB] saving
    Serial without DD 403.82 - 237.108 -
    Serial with DD 224.65 1.79 57.816 -75.6%
    Parallel without DD 55.56 7.26 266.040 +10.8%
    Parallel with DD 18.67 21.6 84.128 -64.5%
     | Show Table
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    Table 8.  Comparison for the different number of subdomains using Parallel-Domain decomposition method in the experiment with 34 992 000 000 unknowns, tested on 28 octo-core CPUs

    No. sub. CPU time CPU time RAM Memory
    domains [s] saving [GB] saving
    1 706.8 - 1 652 -
    2 683.6 1.03 968 -41.4%
    5 703.5 1.00 557 -66.3%
    10 700.9 1.01 420 -74.5%
    15 710.0 0.99 375 -77.3%
    30 718.5 0.98 329 -80.0%
     | Show Table
    DownLoad: CSV
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