November  2015, 9(4): 1069-1091. doi: 10.3934/ipi.2015.9.1069

A parallel space-time domain decomposition method for unsteady source inversion problems

1. 

Laboratory for Engineering and Scientific Computing, Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences, Shenzhen, Guangdong 518055, China

2. 

Department of Computer Science, University of Colorado, Boulder, CO 80309, United States

3. 

Department of Mathematics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong

Received  June 2014 Revised  November 2014 Published  October 2015

In this paper, we propose a parallel space-time domain decomposition method for solving an unsteady source identification problem governed by the linear convection-diffusion equation. Traditional approaches require to solve repeatedly a forward parabolic system, an adjoint system and a system with respect to the unknown sources. The three systems have to be solved one after another. These sequential steps are not desirable for large scale parallel computing. A space-time restrictive additive Schwarz method is proposed for a fully implicit space-time coupled discretization scheme to recover the time-dependent pollutant source intensity functions. We show with numerical experiments that the scheme works well with noise in the observation data. More importantly it is demonstrated that the parallel space-time Schwarz preconditioner is scalable on a supercomputer with over $10^3$ processors, thus promising for large scale applications.
Citation: Xiaomao Deng, Xiao-Chuan Cai, Jun Zou. A parallel space-time domain decomposition method for unsteady source inversion problems. Inverse Problems & Imaging, 2015, 9 (4) : 1069-1091. doi: 10.3934/ipi.2015.9.1069
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show all references

References:
[1]

Proceedings of Supercomputing 2005, Seattle, WA, 2005, p43. doi: 10.1109/SC.2005.25.  Google Scholar

[2]

Finite Elements in Analysis and Design, 39 (2003), 683-705. doi: 10.1016/S0168-874X(03)00054-4.  Google Scholar

[3]

Environmental Forensics, 2 (2001), 205-214. doi: 10.1006/enfo.2001.0055.  Google Scholar

[4]

Physical Review E, 66 (2002), 2-5. Google Scholar

[5]

Resonance, 8 (2003), 48-58. doi: 10.1007/BF02866759.  Google Scholar

[6]

Technical report, Argonne National Laboratory, 2014. Google Scholar

[7]

Master Thesis, Virginia Polytechnic Institute and State University, Blacksburg, Virginia, 1996. Google Scholar

[8]

in Parallel Computational Fluid Dynamics 1999, Towards Teraflops, Optimization and Novel Formulations, 2000, 131-138. doi: 10.1016/B978-044482851-4.50017-7.  Google Scholar

[9]

Communications in Applied Mathematics and Computational Science, 4 (2009), 1-26. doi: 10.2140/camcos.2009.4.1.  Google Scholar

[10]

SIAM Journal on Scientific Computing, 21 (1999), 792-797. doi: 10.1137/S106482759732678X.  Google Scholar

[11]

SIAM Journal on Control and Optimization, 37 (1999), 892-910. doi: 10.1137/S0363012997318602.  Google Scholar

[12]

$1^{st}$ edition, North-Holland Pub. Co., Amsterdam/New York, 1978.  Google Scholar

[13]

International Journal of Numerical Analysis and Modeling, 10 (2013), 588-602.  Google Scholar

[14]

Mathematics and its Applications, 375, Kluwer Academic Publishers Group, Dordrecht, 1996. doi: 10.1007/978-94-009-1740-8.  Google Scholar

[15]

International Journal for Numerical Methods in Engineering, 58 (2003), 1397-1434. doi: 10.1002/nme.860.  Google Scholar

[16]

Domain Decomposition Methods in Science and Engineering XVII, Lect. Notes Comput. Sci. Eng., 60, Springer, Berlin, 2008, 45-56. doi: 10.1007/978-3-540-75199-1_4.  Google Scholar

[17]

Water Resources Research, 19 (1983), 779-790. Google Scholar

[18]

Applied Numerical Mathematics, 58 (2008), 422-434. doi: 10.1016/j.apnum.2007.01.017.  Google Scholar

[19]

Inverse Problems, 25 (2009), 075006, 18pp. doi: 10.1088/0266-5611/25/7/075006.  Google Scholar

[20]

Inverse Problems, 25 (2009), 115009, 21pp. doi: 10.1088/0266-5611/25/11/115009.  Google Scholar

[21]

IMA Journal of Numerical Analysis, 30 (2010), 677-701. doi: 10.1093/imanum/drn066.  Google Scholar

[22]

Inverse Problems, 14 (1998), 83-100. doi: 10.1088/0266-5611/14/1/009.  Google Scholar

[23]

SIAM Journal on Scientific Computing, 22 (2000), 1511-1526. doi: 10.1137/S1064827598346740.  Google Scholar

[24]

in Proceedings of 2nd Berkeley Symposium, University of California Press, Berkeley, 1951, 481-492.  Google Scholar

[25]

ComptesRendus de l'Academie des Sciences Series I Mathematics, 332 (2001), 661-668. doi: 10.1016/S0764-4442(00)01793-6.  Google Scholar

[26]

Ph.D. Thesis, University of Colorado, 2008. Google Scholar

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Indoor Air, 17 (2007), 419-438. doi: 10.1111/j.1600-0668.2007.00497.x.  Google Scholar

[28]

International Journal of Quantum Chemistry, 93 (2003), 223-228. doi: 10.1002/qua.10554.  Google Scholar

[29]

in Domain Decomposition Methods in Science and Engineering, Lect. Notes Comput. Sci. Eng., 40, Springer, Berlin, 2005, 441-448. doi: 10.1007/3-540-26825-1_45.  Google Scholar

[30]

Ecological Modelling, 111 (1998), 187-205. doi: 10.1016/S0304-3800(98)00118-5.  Google Scholar

[31]

SIAM Journal on Scientific Computing, 27 (2006), 1305-1328. doi: 10.1137/040602997.  Google Scholar

[32]

Mathematical and Computer Modelling, 42 (2005), 601-612. doi: 10.1016/j.mcm.2004.06.023.  Google Scholar

[33]

$2^{nd}$ edition, Society for Industrial and Applied Mathematics, 2003. doi: 10.1137/1.9780898718003.  Google Scholar

[34]

Walter de Gruyter, Berlin, 2007. doi: 10.1515/9783110205794.  Google Scholar

[35]

Water Resources Research, 30 (1994), 71-79. doi: 10.1029/93WR02656.  Google Scholar

[36]

Water Resources Research, 31 (1995), 2669-2673. doi: 10.1029/95WR02383.  Google Scholar

[37]

Water Resources Research, 33 (1997), 537-546. doi: 10.1029/96WR03753.  Google Scholar

[38]

in Domain Decomposition Methods in Science and Engineering, Lect. Notes Comput. Sci. Eng., {40}, Springer, Berlin, 2005, 449-456. doi: 10.1007/3-540-26825-1_46.  Google Scholar

[39]

Springer, 2005.  Google Scholar

[40]

International Journal for Numerical Methods in Fluids, 68 (2012), 48-82. doi: 10.1002/fld.2494.  Google Scholar

[41]

CRC Press, 2003.  Google Scholar

[42]

SIAM Journal on Numerical Analysis, 43 (2005), 1504-1535. doi: 10.1137/030602551.  Google Scholar

[43]

SIAM Journal on Scientific Computing, 32 (2010), 418-438. doi: 10.1137/080727348.  Google Scholar

[44]

International Journal for Numerical Methods in Engineering, 91 (2012), 644-665. doi: 10.1002/nme.4286.  Google Scholar

[45]

Journal of Agro-Environment Science, 30 (2011), 1880-1887. Google Scholar

[46]

Applied Geochemistry, 16 (2001), 409-417. Google Scholar

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