Heat source identification based on l_1 constrained minimization

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February 2014, 8(1): 199-221. doi: 10.3934/ipi.2014.8.199

Heat source identification based on $l_1$ constrained minimization

Yingying Li ^1, , Stanley Osher ^2, and Richard Tsai ^3,

1.	University of California, Los Angeles, Los Angeles, CA 90095, United States
2.	Department of Mathematics, University of California, Los Angeles, CA 90095-1555, United States
3.	The University of Texas at Austin, Austin, TX 78712, United States

Received January 2011 Revised November 2012 Published March 2014

Abstract
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We consider the inverse problem of finding sparse initial data from the sparsely sampled solutions of the heat equation. The initial data are assumed to be a sum of an unknown but finite number of Dirac delta functions at unknown locations. Point-wise values of the heat solution at only a few locations are used in an $l_1$ constrained optimization to find the initial data. A concept of domain of effective sensing is introduced to speed up the already fast Bregman iterative algorithm for $l_1$ optimization. Furthermore, an algorithm which successively adds new measurements at specially chosen locations is introduced. By comparing the solutions of the inverse problem obtained from different number of measurements, the algorithm decides where to add new measurements in order to improve the reconstruction of the sparse initial data.

Keywords: source identification, heat equation., $L1$.

Mathematics Subject Classification: Primary: 49N45, 65M3.

Citation: Yingying Li, Stanley Osher, Richard Tsai. Heat source identification based on $l_1$ constrained minimization. Inverse Problems and Imaging, 2014, 8 (1) : 199-221. doi: 10.3934/ipi.2014.8.199

References:

[1]	L. Bregman, The relaxation method of finding the common points of convex sets and its application to the solution of problems in convex optimization, USSR Computational Mathematics and Mathematical Physics, 7 (1967), 620-631.
[2]	M. Burger, Y. Landa, N. Tanushev and R. Tsai, Discovering point sources in unknown environments, in WAFR 2008: The Eighth International Workshop on the Algorithmic Foundations of Robotics, 57 2010, 663-678. doi: 10.1007/978-3-642-00312-7_41.
[3]	J. Cai, S. Osher and Z. Shen, Convergence of the linearized Bregman iteration for $l_1$-norm minimization, Math. Comp., 78 (2009), 2127-2136. doi: 10.1090/S0025-5718-09-02242-X.
[4]	E. J. Candès and T. Tao, Decoding by linear programming, IEEE Transactions on Information Theory, 51(12) (2005).
[5]	Y. Cheng and T. Singh, Source term estimation using convex optimization, The Eleventh International Conference on Information Fusion, Cologne, Germany, (2008).
[6]	D. Donoho, Compressed sensing, IEEE Transactions on Information Theory, 52 (2006), 1289-1306. doi: 10.1109/TIT.2006.871582.
[7]	A. El Badia, T. Ha Duong and A. Hamdi, Identification of a point source in a linear advection-dispersion-reaction equation: Application to a pollution source problem, Inverse Problems, 21 (2005), 1121-1136. doi: 10.1088/0266-5611/21/3/020.
[8]	B. Farmer, C. Hall and S. Esedoglu, Source identification from line integral measurements and simple atmospheric models, Inverse Probl. Imaging, 7 (2013), 471-–490. doi: 10.3934/ipi.2013.7.471.
[9]	E. Haber, Numerical methods for optimal experimental design of large-scale ill-posed problems, Inverse Problems, 24 (2008).
[10]	Y. Landa, N. Tanushev and R. Tsai, Discovery of point sources in the Helmholtz equation posed in unknown domains with obstacles, Comm. in Math. Sci., 9 (2011), 903-928. doi: 10.4310/CMS.2011.v9.n3.a11.
[11]	Y. Li and S. Osher, Coordinate descent optimization for L1 minimization with application to compressed sensing; A greedy algorithm, Inverse Problems and Imaging, 3 (2009), 487–-503. doi: 10.3934/ipi.2009.3.487.
[12]	G. Li, Y. Tan, J. Cheng and X. Wang, Determining magnitude of groundwater pollution sources by data compatibility analysis, Inverse Problem in Science and Engineering, 14 (2006), 287-300. doi: 10.1080/17415970500485153.
[13]	L. Ling and T. Takeuchi, Point sources identification problems for heat equations, Communications in Computational Physics, 5 (2009), 897-913.
[14]	L. Ling, M. Yamamoto, Y. Hon and T. Takeuchi, Identification of source locations in two-dimensional heat equations, Inverse Problems, 22 (2006), 1289-1305. doi: 10.1088/0266-5611/22/4/011.
[15]	A. V. Mamonov and Y.-H. R. Tsai, Point source identification in non-linear advection-diffusion-reaction systems, Inverse Problems, 29 (2013). doi: 10.1088/0266-5611/29/3/035009.
[16]	S. Osher, M. Burger, D. Goldfarb, J. Xu and W. Yin, An iterative regularization method for total variation-based image restoration, MMS, 4 (2005), 460-489. doi: 10.1137/040605412.
[17]	Z. Wen, W. Yin, D. Goldfarb and Y. Zhang, A fast algorithm for sparse reconstruction based on shrinkage, subspace optimization and continuation, SIAM J. Scientific Computing, 32 (2010), 1832-1857. doi: 10.1137/090747695.
[18]	W. Yin, S. Osher, D. Goldfarb and J. Darbon, Bregman iterative algorithms for $l_1$-minimization with applications to compressed sensing, SIAM J. Imaging Sciences, (2008), 143-168. doi: 10.1137/070703983.

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References:

[1]	L. Bregman, The relaxation method of finding the common points of convex sets and its application to the solution of problems in convex optimization, USSR Computational Mathematics and Mathematical Physics, 7 (1967), 620-631.
[2]	M. Burger, Y. Landa, N. Tanushev and R. Tsai, Discovering point sources in unknown environments, in WAFR 2008: The Eighth International Workshop on the Algorithmic Foundations of Robotics, 57 2010, 663-678. doi: 10.1007/978-3-642-00312-7_41.
[3]	J. Cai, S. Osher and Z. Shen, Convergence of the linearized Bregman iteration for $l_1$-norm minimization, Math. Comp., 78 (2009), 2127-2136. doi: 10.1090/S0025-5718-09-02242-X.
[4]	E. J. Candès and T. Tao, Decoding by linear programming, IEEE Transactions on Information Theory, 51(12) (2005).
[5]	Y. Cheng and T. Singh, Source term estimation using convex optimization, The Eleventh International Conference on Information Fusion, Cologne, Germany, (2008).
[6]	D. Donoho, Compressed sensing, IEEE Transactions on Information Theory, 52 (2006), 1289-1306. doi: 10.1109/TIT.2006.871582.
[7]	A. El Badia, T. Ha Duong and A. Hamdi, Identification of a point source in a linear advection-dispersion-reaction equation: Application to a pollution source problem, Inverse Problems, 21 (2005), 1121-1136. doi: 10.1088/0266-5611/21/3/020.
[8]	B. Farmer, C. Hall and S. Esedoglu, Source identification from line integral measurements and simple atmospheric models, Inverse Probl. Imaging, 7 (2013), 471-–490. doi: 10.3934/ipi.2013.7.471.
[9]	E. Haber, Numerical methods for optimal experimental design of large-scale ill-posed problems, Inverse Problems, 24 (2008).
[10]	Y. Landa, N. Tanushev and R. Tsai, Discovery of point sources in the Helmholtz equation posed in unknown domains with obstacles, Comm. in Math. Sci., 9 (2011), 903-928. doi: 10.4310/CMS.2011.v9.n3.a11.
[11]	Y. Li and S. Osher, Coordinate descent optimization for L1 minimization with application to compressed sensing; A greedy algorithm, Inverse Problems and Imaging, 3 (2009), 487–-503. doi: 10.3934/ipi.2009.3.487.
[12]	G. Li, Y. Tan, J. Cheng and X. Wang, Determining magnitude of groundwater pollution sources by data compatibility analysis, Inverse Problem in Science and Engineering, 14 (2006), 287-300. doi: 10.1080/17415970500485153.
[13]	L. Ling and T. Takeuchi, Point sources identification problems for heat equations, Communications in Computational Physics, 5 (2009), 897-913.
[14]	L. Ling, M. Yamamoto, Y. Hon and T. Takeuchi, Identification of source locations in two-dimensional heat equations, Inverse Problems, 22 (2006), 1289-1305. doi: 10.1088/0266-5611/22/4/011.
[15]	A. V. Mamonov and Y.-H. R. Tsai, Point source identification in non-linear advection-diffusion-reaction systems, Inverse Problems, 29 (2013). doi: 10.1088/0266-5611/29/3/035009.
[16]	S. Osher, M. Burger, D. Goldfarb, J. Xu and W. Yin, An iterative regularization method for total variation-based image restoration, MMS, 4 (2005), 460-489. doi: 10.1137/040605412.
[17]	Z. Wen, W. Yin, D. Goldfarb and Y. Zhang, A fast algorithm for sparse reconstruction based on shrinkage, subspace optimization and continuation, SIAM J. Scientific Computing, 32 (2010), 1832-1857. doi: 10.1137/090747695.
[18]	W. Yin, S. Osher, D. Goldfarb and J. Darbon, Bregman iterative algorithms for $l_1$-minimization with applications to compressed sensing, SIAM J. Imaging Sciences, (2008), 143-168. doi: 10.1137/070703983.

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