American Institute of Mathematical Sciences

Advanced
February  2015, 9(1): 163-188. doi: 10.3934/ipi.2015.9.163

Overlapping domain decomposition methods for linear inverse problems

Daijun Jiang 1, , Hui Feng 2, and  Jun Zou 3,

1. 

School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China
2. 

School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China
3. 

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

Received  October 2013 Revised  May 2014 Published  January 2015

We shall derive and propose several efficient overlapping domain decomposition methods for solving some typical linear inverse problems, including the identification of the flux, the source strength and the initial temperature in second order elliptic and parabolic systems. The methods are iterative, and computationally very efficient: only local forward and adjoint problems need to be solved in each subdomain, and the local minimizations have explicit solutions. Numerical experiments are provided to demonstrate the robustness and efficiency of the methods: the algorithms converge globally, even with rather poor initial guesses; and their convergences do not deteriorate or deteriorate only slightly when the meshes are refined.
Keywords: domain decomposition, explicit subdomain solver., Inverse problems, parameter identification.
Mathematics Subject Classification: Primary: 31A25, 65M55; Secondary: 90C2.
Citation: Daijun Jiang, Hui Feng, Jun Zou. Overlapping domain decomposition methods for linear inverse problems. Inverse Problems & Imaging, 2015, 9 (1) : 163-188. doi: 10.3934/ipi.2015.9.163
References:
[1]

R. C. Aster, B. Borchers and C. H. Thurber, Parameter Estimation and Inverse Problems,, Elsevier Academic Press, (2005). Google Scholar

[2]

H. T. Banks and K. Kunisch, Estimation Techniques for Distributed Parameter Systems,, Birkhauser, (1989). Google Scholar

[3]

X. Cai, S. Liu and J. Zou, Parallel overlapping domain decomposition methods for coupled inverse elliptic problems,, Comm. Appl. Math. Comput. Sci., 4 (2009), 1. doi: 10.2140/camcos.2009.4.1. Google Scholar

[4]

T. Chan and T. Mathew, Domain decomposition algorithms,, Acta Numerica, (1994), 61. Google Scholar

[5]

T. Chan and X. Tai, Identification of discontinuous coefficients from elliptic problems using total variation regularization,, SIAM J. Sci. Comput., 25 (2003), 881. doi: 10.1137/S1064827599326020. Google Scholar

[6]

H. Chang and D. Yang, A Schwarz domain decomposition method with gradient projection for optimal control governed by elliptic partial differential equations,, J. Comput. Appl. Math., 235 (2011), 5078. doi: 10.1016/j.cam.2011.04.037. Google Scholar

[7]

I. Daubechies, M. Defrise and C. Demol, An iterative thresholding algorithm for linear inverse problems,, Comm. Pure Appl. Math., 57 (2004), 1413. doi: 10.1002/cpa.20042. Google Scholar

[8]

H. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems,, Kluwer Academic Publishers, (2000). Google Scholar

[9]

M. Heinkenschloss and M. Herty, A spatial domain decomposition method for parabolic optimal control problems,, J. Comput. Appl. Math., 201 (2007), 88. doi: 10.1016/j.cam.2006.02.002. Google Scholar

[10]

M. Heinkenschloss and H. Nguyen, Neumann-Neumann domain decomposition preconditioners for linear-quadratic elliptic optimal control problems,, SIAM Journal on Scientific Computing, 28 (2006), 1001. doi: 10.1137/040612774. Google Scholar

[11]

K. Ito and J. Zou, Identification of some source densities of the distribution type,, J. Comput. Appl. Math., 132 (2001), 295. doi: 10.1016/S0377-0427(00)00332-0. Google Scholar

[12]

J. Li and J. Zou, A multilevel model correction method for parameter identification,, Inverse Problems, 23 (2007), 1759. doi: 10.1088/0266-5611/23/5/001. Google Scholar

[13]

X. Tai, J. Froyen, M. Espedal and T. Chan, Overlapping domain decomposition and multigrid methods for inverse problems,, Contemporary Mathematics, 218 (1998), 523. Google Scholar

[14]

A. Toselli and O. Widlund, Domain Decomposition Methods-Algorithms and Theory,, Springer-Verlag, (2004). Google Scholar

[15]

L. Wang and J. Zou, Error estimates of finite element methods for parameter identifications in elliptic and parabolic systems,, Disc. Cont. Dynam. Sys., 14 (2010), 1641. doi: 10.3934/dcdsb.2010.14.1641. Google Scholar

[16]

J. Xie and J. Zou, Numerical reconstruction of heat fluxes,, SIAM J. Numer. Anal., 43 (2005), 1504. doi: 10.1137/030602551. Google Scholar

[17]

J. Xu, Iterative methods by space decomposition and subspace correction,, SIAM Review, 34 (1992), 581. doi: 10.1137/1034116. Google Scholar

[18]

J. Xu and J. Zou, Some nonoverlapping domain decomposition methods,, SIAM Review, 40 (1998), 857. doi: 10.1137/S0036144596306800. Google Scholar

show all references

References:
[1]

R. C. Aster, B. Borchers and C. H. Thurber, Parameter Estimation and Inverse Problems,, Elsevier Academic Press, (2005). Google Scholar

[2]

H. T. Banks and K. Kunisch, Estimation Techniques for Distributed Parameter Systems,, Birkhauser, (1989). Google Scholar

[3]

X. Cai, S. Liu and J. Zou, Parallel overlapping domain decomposition methods for coupled inverse elliptic problems,, Comm. Appl. Math. Comput. Sci., 4 (2009), 1. doi: 10.2140/camcos.2009.4.1. Google Scholar

[4]

T. Chan and T. Mathew, Domain decomposition algorithms,, Acta Numerica, (1994), 61. Google Scholar

[5]

T. Chan and X. Tai, Identification of discontinuous coefficients from elliptic problems using total variation regularization,, SIAM J. Sci. Comput., 25 (2003), 881. doi: 10.1137/S1064827599326020. Google Scholar

[6]

H. Chang and D. Yang, A Schwarz domain decomposition method with gradient projection for optimal control governed by elliptic partial differential equations,, J. Comput. Appl. Math., 235 (2011), 5078. doi: 10.1016/j.cam.2011.04.037. Google Scholar

[7]

I. Daubechies, M. Defrise and C. Demol, An iterative thresholding algorithm for linear inverse problems,, Comm. Pure Appl. Math., 57 (2004), 1413. doi: 10.1002/cpa.20042. Google Scholar

[8]

H. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems,, Kluwer Academic Publishers, (2000). Google Scholar

[9]

M. Heinkenschloss and M. Herty, A spatial domain decomposition method for parabolic optimal control problems,, J. Comput. Appl. Math., 201 (2007), 88. doi: 10.1016/j.cam.2006.02.002. Google Scholar

[10]

M. Heinkenschloss and H. Nguyen, Neumann-Neumann domain decomposition preconditioners for linear-quadratic elliptic optimal control problems,, SIAM Journal on Scientific Computing, 28 (2006), 1001. doi: 10.1137/040612774. Google Scholar

[11]

K. Ito and J. Zou, Identification of some source densities of the distribution type,, J. Comput. Appl. Math., 132 (2001), 295. doi: 10.1016/S0377-0427(00)00332-0. Google Scholar

[12]

J. Li and J. Zou, A multilevel model correction method for parameter identification,, Inverse Problems, 23 (2007), 1759. doi: 10.1088/0266-5611/23/5/001. Google Scholar

[13]

X. Tai, J. Froyen, M. Espedal and T. Chan, Overlapping domain decomposition and multigrid methods for inverse problems,, Contemporary Mathematics, 218 (1998), 523. Google Scholar

[14]

A. Toselli and O. Widlund, Domain Decomposition Methods-Algorithms and Theory,, Springer-Verlag, (2004). Google Scholar

[15]

L. Wang and J. Zou, Error estimates of finite element methods for parameter identifications in elliptic and parabolic systems,, Disc. Cont. Dynam. Sys., 14 (2010), 1641. doi: 10.3934/dcdsb.2010.14.1641. Google Scholar

[16]

J. Xie and J. Zou, Numerical reconstruction of heat fluxes,, SIAM J. Numer. Anal., 43 (2005), 1504. doi: 10.1137/030602551. Google Scholar

[17]

J. Xu, Iterative methods by space decomposition and subspace correction,, SIAM Review, 34 (1992), 581. doi: 10.1137/1034116. Google Scholar

[18]

J. Xu and J. Zou, Some nonoverlapping domain decomposition methods,, SIAM Review, 40 (1998), 857. doi: 10.1137/S0036144596306800. Google Scholar

[1]

Mikko Orispää, Markku Lehtinen. Fortran linear inverse problem solver. Inverse Problems & Imaging, 2010, 4 (3) : 485-503. doi: 10.3934/ipi.2010.4.485
[2]

Qingping Deng. A nonoverlapping domain decomposition method for nonconforming finite element problems. Communications on Pure & Applied Analysis, 2003, 2 (3) : 297-310. doi: 10.3934/cpaa.2003.2.297
[3]

Davide Guidetti. Some inverse problems of identification for integrodifferential parabolic systems with a boundary memory term. Discrete & Continuous Dynamical Systems - S, 2015, 8 (4) : 749-756. doi: 10.3934/dcdss.2015.8.749
[4]

Laurent Bourgeois, Houssem Haddar. Identification of generalized impedance boundary conditions in inverse scattering problems. Inverse Problems & Imaging, 2010, 4 (1) : 19-38. doi: 10.3934/ipi.2010.4.19
[5]

Chuang Zheng. Inverse problems for the fourth order Schrödinger equation on a finite domain. Mathematical Control & Related Fields, 2015, 5 (1) : 177-189. doi: 10.3934/mcrf.2015.5.177
[6]

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
[7]

H. T. Banks, D. Rubio, N. Saintier, M. I. Troparevsky. Optimal design for parameter estimation in EEG problems in a 3D multilayered domain. Mathematical Biosciences & Engineering, 2015, 12 (4) : 739-760. doi: 10.3934/mbe.2015.12.739
[8]

Yuepeng Wang, Yue Cheng, I. Michael Navon, Yuanhong Guan. Parameter identification techniques applied to an environmental pollution model. Journal of Industrial & Management Optimization, 2018, 14 (2) : 817-831. doi: 10.3934/jimo.2017077
[9]

Pingping Niu, Shuai Lu, Jin Cheng. On periodic parameter identification in stochastic differential equations. Inverse Problems & Imaging, 2019, 13 (3) : 513-543. doi: 10.3934/ipi.2019025
[10]

Simone Cacace, Maurizio Falcone. A dynamic domain decomposition for the eikonal-diffusion equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 109-123. doi: 10.3934/dcdss.2016.9.109
[11]

Fang Zeng, Pablo Suarez, Jiguang Sun. A decomposition method for an interior inverse scattering problem. Inverse Problems & Imaging, 2013, 7 (1) : 291-303. doi: 10.3934/ipi.2013.7.291
[12]

Qinqin Chai, Ryan Loxton, Kok Lay Teo, Chunhua Yang. A unified parameter identification method for nonlinear time-delay systems. Journal of Industrial & Management Optimization, 2013, 9 (2) : 471-486. doi: 10.3934/jimo.2013.9.471
[13]

Qing Hong, Guorong Hu. Molecular decomposition and a class of Fourier multipliers for bi-parameter modulation spaces. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3103-3120. doi: 10.3934/cpaa.2019139
[14]

Wenxiong Chen, Congming Li. Indefinite elliptic problems in a domain. Discrete & Continuous Dynamical Systems - A, 1997, 3 (3) : 333-340. doi: 10.3934/dcds.1997.3.333
[15]

Jing Xu, Xue-Cheng Tai, Li-Lian Wang. A two-level domain decomposition method for image restoration. Inverse Problems & Imaging, 2010, 4 (3) : 523-545. doi: 10.3934/ipi.2010.4.523
[16]

Rongliang Chen, Jizu Huang, Xiao-Chuan Cai. A parallel domain decomposition algorithm for large scale image denoising. Inverse Problems & Imaging, 2019, 13 (6) : 1259-1282. doi: 10.3934/ipi.2019055
[17]

Jin-Mun Jeong, Seong-Ho Cho. Identification problems of retarded differential systems in Hilbert spaces. Evolution Equations & Control Theory, 2017, 6 (1) : 77-91. doi: 10.3934/eect.2017005
[18]

Alfredo Lorenzi. Identification problems related to cylindrical dielectrics **in presence of polarization**. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 2247-2265. doi: 10.3934/dcdsb.2014.19.2247
[19]

Leonid Berlyand, Petru Mironescu. Two-parameter homogenization for a Ginzburg-Landau problem in a perforated domain. Networks & Heterogeneous Media, 2008, 3 (3) : 461-487. doi: 10.3934/nhm.2008.3.461
[20]

Colin Guillarmou, Antônio Sá Barreto. Inverse problems for Einstein manifolds. Inverse Problems & Imaging, 2009, 3 (1) : 1-15. doi: 10.3934/ipi.2009.3.1

2018 Impact Factor: 1.469

Tools

Metrics

  • PDF downloads (7)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences