American Institute of Mathematical Sciences

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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, New York, 2005. Google Scholar

[2]

H. T. Banks and K. Kunisch, Estimation Techniques for Distributed Parameter Systems, Birkhauser, Boston, 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), 1-26. doi: 10.2140/camcos.2009.4.1.  Google Scholar

[4]

T. Chan and T. Mathew, Domain decomposition algorithms, Acta Numerica, (1994), 61-143. 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), 881-904. 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), 5078-5094. 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), 1413-1457. doi: 10.1002/cpa.20042.  Google Scholar

[8]

H. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Kluwer Academic Publishers, The Netherlands, 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), 88-111. 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), 1001-1028. 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), 295-308. 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), 1759-1786. 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), 523-529. Google Scholar

[14]

A. Toselli and O. Widlund, Domain Decomposition Methods-Algorithms and Theory, Springer-Verlag, New York, 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., Series B, 14 (2010), 1641-1670. 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), 1504-1535. doi: 10.1137/030602551.  Google Scholar

[17]

J. Xu, Iterative methods by space decomposition and subspace correction, SIAM Review, 34 (1992), 581-613. doi: 10.1137/1034116.  Google Scholar

[18]

J. Xu and J. Zou, Some nonoverlapping domain decomposition methods, SIAM Review, 40 (1998), 857-914. 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, New York, 2005. Google Scholar

[2]

H. T. Banks and K. Kunisch, Estimation Techniques for Distributed Parameter Systems, Birkhauser, Boston, 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), 1-26. doi: 10.2140/camcos.2009.4.1.  Google Scholar

[4]

T. Chan and T. Mathew, Domain decomposition algorithms, Acta Numerica, (1994), 61-143. 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), 881-904. 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), 5078-5094. 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), 1413-1457. doi: 10.1002/cpa.20042.  Google Scholar

[8]

H. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Kluwer Academic Publishers, The Netherlands, 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), 88-111. 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), 1001-1028. 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), 295-308. 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), 1759-1786. 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), 523-529. Google Scholar

[14]

A. Toselli and O. Widlund, Domain Decomposition Methods-Algorithms and Theory, Springer-Verlag, New York, 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., Series B, 14 (2010), 1641-1670. 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), 1504-1535. doi: 10.1137/030602551.  Google Scholar

[17]

J. Xu, Iterative methods by space decomposition and subspace correction, SIAM Review, 34 (1992), 581-613. doi: 10.1137/1034116.  Google Scholar

[18]

J. Xu and J. Zou, Some nonoverlapping domain decomposition methods, SIAM Review, 40 (1998), 857-914. doi: 10.1137/S0036144596306800.  Google Scholar

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