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 $
-
Previous Article
Sharp consistency estimates for a pressure-Poisson problem with Stokes boundary value problems
- DCDS-S Home
- This Issue
-
Next Article
3D image segmentation supported by a point cloud
March
2021, 14(3): 987-999.
doi: 10.3934/dcdss.2020381
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 |
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.
Keywords:
Geodetic boundary-value problem,
finite volume method,
MPI,
OpenMP,
domain decomposition methods,
additive Schwarz method.
Mathematics Subject Classification:
Primary: 65K05; Secondary: 49M27.
Citation:
Marek Macák, Róbert Čunderlík, Karol Mikula, Zuzana Minarechová. Computational optimization in solving the geodetic boundary value problems. Discrete and Continuous Dynamical Systems - S,
2021, 14
(3)
: 987-999.
doi: 10.3934/dcdss.2020381
![]()
References:
| [1] |
O. B. Andersen, The DTU10 Gravity field and Mean sea surface, Second International Symposium of the Gravity Field of the Earth (IGFS2), Fairbanks, Alaska, (2010). |
| [2] |
Y. Aoyama and J. Nakano, RS/6000 SP: Practical MPI programming, IBM., (1999), http://www.redbooks.ibm.com. |
| [3] |
X. Cai, Overlapping domain decomposition methods, Advanced Topics in Computational Partial Differential Equations, (2003), 57–95.
doi: 10.1007/978-3-642-18237-2_2. |
| [4] |
T. F. Chan and T. P. Mathew,
Domain decomposition algorithms, Acta Numerica, 3 (1994), 61-143.
doi: 10.1017/S0962492900002427. |
| [5] |
B. Chapman, G. Jost and R. Pas, Using OpenMP: Portable shared memory parallel programming, The MIT Press, Scientific and Engin Edition, (2007). |
| [6] |
R. Čunderlík, K. Mikula and M. Mojzeš,
Numerical solution of the linearized fixed gravimetric boundary-value problem, Journal of Geodesy, 82 (2008), 15-29.
|
| [7] |
R. Čunderlík and K. Mikula,
Direct BEM for high-resolution global gravity field modelling, Studia Geophysica et Geodaetica, 54 (2010), 219-238.
|
| [8] |
V. Dolean, P. Jolvet and F. Nataf, An Introduction to Domain Decomposition Methods. Algorithms, Theory, and Parallel Implementation, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2015.
doi: 10.1137/1.9781611974065.ch1. |
| [9] |
Z. Fašková, R. Čunderlík and K. Mikula,
Finite element method for solving geodetic boundary value problems, Journal of Geodesy, 84 (2010), 135-144.
|
| [10] |
P. Holota,
Coerciveness of the linear gravimetric boundary-value problem and a geometrical interpretation, Journal of Geodesy, 71 (1997), 640-651.
doi: 10.1007/s001900050131. |
| [11] |
P. Holota,
Neumann's boundary-value problem in studies on Earth gravity field: Weak solution, 50 years of Research Institute of Geodesy, Topography and Cartography, 50 (2005), 49-69.
|
| [12] |
W. Keller, Finite differences schemes for elliptic boundary value problems, Bulletin IAEG, 1 (1995), Section IV. |
| [13] |
R. Klees, Loesung des Fixen Geodaetischen Randwertprolems mit Hilfe der Randelementmethode, Ph.D thesis, Muenchen, 1992 |
| [14] |
R. Klees, M. van Gelderen, C. Lage and C. Schwab,
Fast numerical solution of the linearized Molodensky problem, Journal of Geodesy, 75 (2001), 349-362.
doi: 10.1007/s001900100183. |
| [15] |
K. R. Koch and A. J. Pope,
Uniqueness and existence for the geodetic boundary value problem using the known surface of the earth, Bulletin Géodésique (N.S.), 46 (1972), 467-476.
|
| [16] |
T. Mayer-Gürr and et al., The new combined satellite only model GOCO03s, International Symposium on Gravity, Geoid and Height Systems GGHS 2012, (2012). |
| [17] |
P. Meissl, The Use of Finite Elements in Physical Geodesy, Geodetic Science and Surveying, Report 313, The Ohio State University, 1981. |
| [18] |
Z. Minarechová, M. Macák, R. Čunderlík and K. Mikula,
High-resolution global gravity field modelling by the finite volume method, Studia Geophysica et Geodaetica, 59 (2015), 1-20.
|
| [19] |
O. Nesvadba, P. Holota and R. Klees,
A direct method and its numerical interpretation in the determination of the gravity field of the Earth from terrestrial data, Proceedings Dynamic Planet 2005, Monitoring and Understanding a Dynamic Planet with Geodetic and Oceanographic Tools, 130 (2007), 370-376.
|
| [20] |
N. K. Pavlis, S. A. Holmes, S. C. Kenyon and J. K. Factor,
The development and evaluation of the Earth Gravitational Model 2008 (EGM2008), Journal of Geophysical Research: Solid Earth, 117 (2012), 1-38.
doi: 10.1029/2011JB008916. |
| [21] |
B. Shaofeng and B. Dingbo,
The finite element method for the geodetic boundary value problem, Manuscripta Geodetica, 16 (1991), 353-359.
|
| [22] |
G. L. G. Sleijpen and D. R. Fokkema,
BiCGstab$(l)$ for linear equations involving unsymmetric matrices with complex spectrum, Electron. Trans. Numer. Anal., 1 (1993), 11-32.
|
| [23] |
M. Šprlák, Z. Fašková and K. Mikula,
On the application of the coupled finite-infinite element method to geodetic boundary-value problem, Studia Geophysica et Geodaetica, 55 (2011), 479-487.
|
show all references
References:
| [1] |
O. B. Andersen, The DTU10 Gravity field and Mean sea surface, Second International Symposium of the Gravity Field of the Earth (IGFS2), Fairbanks, Alaska, (2010). |
| [2] |
Y. Aoyama and J. Nakano, RS/6000 SP: Practical MPI programming, IBM., (1999), http://www.redbooks.ibm.com. |
| [3] |
X. Cai, Overlapping domain decomposition methods, Advanced Topics in Computational Partial Differential Equations, (2003), 57–95.
doi: 10.1007/978-3-642-18237-2_2. |
| [4] |
T. F. Chan and T. P. Mathew,
Domain decomposition algorithms, Acta Numerica, 3 (1994), 61-143.
doi: 10.1017/S0962492900002427. |
| [5] |
B. Chapman, G. Jost and R. Pas, Using OpenMP: Portable shared memory parallel programming, The MIT Press, Scientific and Engin Edition, (2007). |
| [6] |
R. Čunderlík, K. Mikula and M. Mojzeš,
Numerical solution of the linearized fixed gravimetric boundary-value problem, Journal of Geodesy, 82 (2008), 15-29.
|
| [7] |
R. Čunderlík and K. Mikula,
Direct BEM for high-resolution global gravity field modelling, Studia Geophysica et Geodaetica, 54 (2010), 219-238.
|
| [8] |
V. Dolean, P. Jolvet and F. Nataf, An Introduction to Domain Decomposition Methods. Algorithms, Theory, and Parallel Implementation, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2015.
doi: 10.1137/1.9781611974065.ch1. |
| [9] |
Z. Fašková, R. Čunderlík and K. Mikula,
Finite element method for solving geodetic boundary value problems, Journal of Geodesy, 84 (2010), 135-144.
|
| [10] |
P. Holota,
Coerciveness of the linear gravimetric boundary-value problem and a geometrical interpretation, Journal of Geodesy, 71 (1997), 640-651.
doi: 10.1007/s001900050131. |
| [11] |
P. Holota,
Neumann's boundary-value problem in studies on Earth gravity field: Weak solution, 50 years of Research Institute of Geodesy, Topography and Cartography, 50 (2005), 49-69.
|
| [12] |
W. Keller, Finite differences schemes for elliptic boundary value problems, Bulletin IAEG, 1 (1995), Section IV. |
| [13] |
R. Klees, Loesung des Fixen Geodaetischen Randwertprolems mit Hilfe der Randelementmethode, Ph.D thesis, Muenchen, 1992 |
| [14] |
R. Klees, M. van Gelderen, C. Lage and C. Schwab,
Fast numerical solution of the linearized Molodensky problem, Journal of Geodesy, 75 (2001), 349-362.
doi: 10.1007/s001900100183. |
| [15] |
K. R. Koch and A. J. Pope,
Uniqueness and existence for the geodetic boundary value problem using the known surface of the earth, Bulletin Géodésique (N.S.), 46 (1972), 467-476.
|
| [16] |
T. Mayer-Gürr and et al., The new combined satellite only model GOCO03s, International Symposium on Gravity, Geoid and Height Systems GGHS 2012, (2012). |
| [17] |
P. Meissl, The Use of Finite Elements in Physical Geodesy, Geodetic Science and Surveying, Report 313, The Ohio State University, 1981. |
| [18] |
Z. Minarechová, M. Macák, R. Čunderlík and K. Mikula,
High-resolution global gravity field modelling by the finite volume method, Studia Geophysica et Geodaetica, 59 (2015), 1-20.
|
| [19] |
O. Nesvadba, P. Holota and R. Klees,
A direct method and its numerical interpretation in the determination of the gravity field of the Earth from terrestrial data, Proceedings Dynamic Planet 2005, Monitoring and Understanding a Dynamic Planet with Geodetic and Oceanographic Tools, 130 (2007), 370-376.
|
| [20] |
N. K. Pavlis, S. A. Holmes, S. C. Kenyon and J. K. Factor,
The development and evaluation of the Earth Gravitational Model 2008 (EGM2008), Journal of Geophysical Research: Solid Earth, 117 (2012), 1-38.
doi: 10.1029/2011JB008916. |
| [21] |
B. Shaofeng and B. Dingbo,
The finite element method for the geodetic boundary value problem, Manuscripta Geodetica, 16 (1991), 353-359.
|
| [22] |
G. L. G. Sleijpen and D. R. Fokkema,
BiCGstab$(l)$ for linear equations involving unsymmetric matrices with complex spectrum, Electron. Trans. Numer. Anal., 1 (1993), 11-32.
|
| [23] |
M. Šprlák, Z. Fašková and K. Mikula,
On the application of the coupled finite-infinite element method to geodetic boundary-value problem, Studia Geophysica et Geodaetica, 55 (2011), 479-487.
|
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 |
| Solver | CPU time | Number of |
| [s] | iterations | |
| GS | 703 | 10000 |
| SOR | 92 | 1136 |
| BiCG | 68 | 568 |
| Bi-CGSTAB | 41 | 348 |
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 |
| 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 |
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% |
| 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% |
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% |
| 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% |
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
| 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 |
| 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 |
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% |
| 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% |
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% |
| 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% |
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% |
| 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% |
| [1] |
Qingping Deng. A nonoverlapping domain decomposition method for nonconforming finite element problems. Communications on Pure and Applied Analysis, 2003, 2 (3) : 297-310. doi: 10.3934/cpaa.2003.2.297 |
| [2] |
John R. Graef, Lingju Kong, Bo Yang. Positive solutions of a nonlinear higher order boundary-value problem. Conference Publications, 2009, 2009 (Special) : 276-285. doi: 10.3934/proc.2009.2009.276 |
| [3] |
Kateryna Marynets. Study of a nonlinear boundary-value problem of geophysical relevance. Discrete and Continuous Dynamical Systems, 2019, 39 (8) : 4771-4781. doi: 10.3934/dcds.2019194 |
| [4] |
Christos V. Nikolopoulos, Georgios E. Zouraris. Numerical solution of a non-local elliptic problem modeling a thermistor with a finite element and a finite volume method. Conference Publications, 2007, 2007 (Special) : 768-778. doi: 10.3934/proc.2007.2007.768 |
| [5] |
Thi-Thao-Phuong Hoang. Optimized Ventcel-Schwarz waveform relaxation and mixed hybrid finite element method for transport problems. Discrete and Continuous Dynamical Systems - S, 2022 doi: 10.3934/dcdss.2022060 |
| [6] |
Bastian Gebauer, Nuutti Hyvönen. Factorization method and inclusions of mixed type in an inverse elliptic boundary value problem. Inverse Problems and Imaging, 2008, 2 (3) : 355-372. doi: 10.3934/ipi.2008.2.355 |
| [7] |
Yu-Feng Sun, Zheng Zeng, Jie Song. Quasilinear iterative method for the boundary value problem of nonlinear fractional differential equation. Numerical Algebra, Control and Optimization, 2020, 10 (2) : 157-164. doi: 10.3934/naco.2019045 |
| [8] |
Jing Xu, Xue-Cheng Tai, Li-Lian Wang. A two-level domain decomposition method for image restoration. Inverse Problems and Imaging, 2010, 4 (3) : 523-545. doi: 10.3934/ipi.2010.4.523 |
| [9] |
Fang Zeng, Pablo Suarez, Jiguang Sun. A decomposition method for an interior inverse scattering problem. Inverse Problems and Imaging, 2013, 7 (1) : 291-303. doi: 10.3934/ipi.2013.7.291 |
| [10] |
Hideo Ikeda, Koji Kondo, Hisashi Okamoto, Shoji Yotsutani. On the global branches of the solutions to a nonlocal boundary-value problem arising in Oseen's spiral flows. Communications on Pure and Applied Analysis, 2003, 2 (3) : 381-390. doi: 10.3934/cpaa.2003.2.381 |
| [11] |
Ivonne Rivas, Muhammad Usman, Bing-Yu Zhang. Global well-posedness and asymptotic behavior of a class of initial-boundary-value problem of the Korteweg-De Vries equation on a finite domain. Mathematical Control and Related Fields, 2011, 1 (1) : 61-81. doi: 10.3934/mcrf.2011.1.61 |
| [12] |
Zhousheng Ruan, Sen Zhang, Sican Xiong. Solving an inverse source problem for a time fractional diffusion equation by a modified quasi-boundary value method. Evolution Equations and Control Theory, 2018, 7 (4) : 669-682. doi: 10.3934/eect.2018032 |
| [13] |
Yacheng Liu, Runzhang Xu. Potential well method for initial boundary value problem of the generalized double dispersion equations. Communications on Pure and Applied Analysis, 2008, 7 (1) : 63-81. doi: 10.3934/cpaa.2008.7.63 |
| [14] |
Daijun Jiang, Hui Feng, Jun Zou. Overlapping domain decomposition methods for linear inverse problems. Inverse Problems and Imaging, 2015, 9 (1) : 163-188. doi: 10.3934/ipi.2015.9.163 |
| [15] |
Oleksandr Boichuk, Victor Feruk. Boundary-value problems for weakly singular integral equations. Discrete and Continuous Dynamical Systems - B, 2022, 27 (3) : 1379-1395. doi: 10.3934/dcdsb.2021094 |
| [16] |
Caterina Calgaro, Meriem Ezzoug, Ezzeddine Zahrouni. Stability and convergence of an hybrid finite volume-finite element method for a multiphasic incompressible fluid model. Communications on Pure and Applied Analysis, 2018, 17 (2) : 429-448. doi: 10.3934/cpaa.2018024 |
| [17] |
Xiaomao Deng, Xiao-Chuan Cai, Jun Zou. A parallel space-time domain decomposition method for unsteady source inversion problems. Inverse Problems and Imaging, 2015, 9 (4) : 1069-1091. doi: 10.3934/ipi.2015.9.1069 |
| [18] |
Xu Yang, François Golse, Zhongyi Huang, Shi Jin. Numerical study of a domain decomposition method for a two-scale linear transport equation. Networks and Heterogeneous Media, 2006, 1 (1) : 143-166. doi: 10.3934/nhm.2006.1.143 |
| [19] |
Shi Jin, Xu Yang, Guangwei Yuan. A domain decomposition method for a two-scale transport equation with energy flux conserved at the interface. Kinetic and Related Models, 2008, 1 (1) : 65-84. doi: 10.3934/krm.2008.1.65 |
| [20] |
Zhongyi Huang. Tailored finite point method for the interface problem. Networks and Heterogeneous Media, 2009, 4 (1) : 91-106. doi: 10.3934/nhm.2009.4.91 |
2020 Impact Factor: 2.425
Tools
Metrics
Other articles
by authors
[Back to Top]