Computational optimization in solving the geodetic boundary value problems

Marek Macák; Róbert Čunderlík; Karol Mikula; Zuzana Minarechová

doi:10.3934/dcdss.2020381

Article Contents

2021, Volume 14, Issue 3: 987-999. Doi: 10.3934/dcdss.2020381

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Computational optimization in solving the geodetic boundary value problems

Slovak University of Technology in Bratislava, Faculty of Civil Engineering, Department of Mathematics and Descriptive Geometry, Radlinskeho 11, Bratislava 810 05, Slovakia

^* Corresponding author: Marek Macák
^* Corresponding author: Marek Macák

Received: December 2018

Revised: March 2020

Early access: June 2020

Published: March 2021

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Abstract

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.

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

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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 $

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

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

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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
Solver	[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
Solver	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%
8	2	70.81	5.70	308.456	+30.0%
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%

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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
$ \eta $	[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%

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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%

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