# American Institute of Mathematical Sciences

May  2013, 7(2): 471-490. doi: 10.3934/ipi.2013.7.471

## Source identification from line integral measurements and simple atmospheric models

 1 Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, MI 48109-1043, United States 2 Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208-2370, United States 3 Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, MI 48109

Received  May 2011 Revised  August 2012 Published  May 2013

We consider the problem of estimating the sparse initial condition of a solution to the advection-diffusion equation based on line integrals of the solution at a later time. We propose models for locating a single and multiple point sources. We also propose algorithms for the efficient implementation of these models. In practice, the models are relevant also for reconstructing the solution of the PDE at the observation time from a very sparse Radon transform; in this case, our models improve on more standard Radon inversion techniques by utilizing the specialized information about how the observed function was generated.
Citation: Brittan Farmer, Cassandra Hall, Selim Esedoḡlu. Source identification from line integral measurements and simple atmospheric models. Inverse Problems and Imaging, 2013, 7 (2) : 471-490. doi: 10.3934/ipi.2013.7.471
##### References:
 [1] V. Akçelik, G. Biros, A. Draganeascu, J. Hill, O. Ghattas and B. van Bloemen Waanders, Dynamic data-driven inversion for terascale simulations: Real-time identification of airborne contaminants, in "Proceedings of the 2005 ACM/IEEE conference on Supercomputing," IEEE Computer Society, (2005), 43-57. doi: 10.1109/SC.2005.25. [2] V. Akçelik, G. Biros, O. Ghattas, K. R. Long and B. van Bloemen Waanders, A variational finite element method for source inversion for convective-diffusive transport, Finite Elem. Anal. Des., 39 (2003), 683-705. doi: 10.1016/S0168-874X(03)00054-4. [3] A. El Badia and T. Ha Duong, Some remarks on the problem of source identification from boundary measurements, Inverse Problems, 14 (1998), 883-891. doi: 10.1088/0266-5611/14/4/008. [4] 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. [5] A. Blake and A. Zisserman, "Visual Reconstruction," MIT Press, Cambridge, MA, 1987. [6] M. Burger, Y. Landa, N. M. Tanushev and R. Tsai, Discovering a point source in unknown environments, in "Algorithmic Foundation of Robotics VII" (eds. G. S. Chirikjian, H. Choset, M. Morales and T. Murphy), Springer, (2009), 663-678. doi: 10.1007/978-3-642-00312-7_41. [7] E. J. Candes and T. Tao, Decoding by linear programming, IEEE Trans. Inform. Theory, 51 (2005), 4203-4215. doi: 10.1109/TIT.2005.858979. [8] T. F. Chan and H.-M. Zhou, Total variation wavelet thresholding, J. Sci. Comput., 32 (2007), 315-341. doi: 10.1007/s10915-007-9133-0. [9] R. Chartrand, Exact reconstruction of sparse signals via nonconvex minimization, Signal Processing Letters, 14 (2007), 707-710. doi: 10.1109/LSP.2007.898300. [10] R. Chartrand and V. Staneva, Restricted isometry properties and nonconvex compressive sensing, Inverse Problems, 24 (2008). doi: 10.1088/0266-5611/24/3/035020. [11] B. R. Cosofret, C. M. Gittins and W. J. Marinelli, Visualization and tomographic analysis of chemical vapor plumes via LWIR imaging Fabry-Perot spectrometry, in "Proc. SPIE: Chemical and Biological Standoff Detection II" (eds. J. O. Jensen and J.-M. Theriault), SPIE, (2004), 112-121. doi: 10.1117/12.570442. [12] D. L. Donoho, Sparse components of images and optimal atomic decompositions, Constructive Approximation, 17 (2001), 353-382. doi: 10.1007/s003650010032. [13] C. M. Gittins and W. J. Marinelli, AIRIS multispectral imaging chemical sensor, in "Proc. SPIE: Electro-Optical Technology for Remote Chemical Detection and Identification III" (eds. M. Fallahi and E. A. Howden), SPIE, (1998), 65-74. doi: 10.1117/12.317637. [14] E. T. Hale, W. Yin and Y. Zhang, Fixed-point continuation for $l_1$-minimization: Methodology and convergence, SIAM J. Optim., 19 (2008), 1107-1130. doi: 10.1137/070698920. [15] J. P. Kernévez, "The Sentinel Method and Its Application to Environmental Pollution Problems," CRC Press, New York, 1997. [16] Y. Landa, H. Shen, R. Takei and Y.-H. R. Tsai, Autonomous source discovery and navigation in complicated environments, UCLA CAM Report, (2009), 09-73. [17] Y. Li, S. Osher and R. Tsai, Heat source identification based on $L_1$ constrained minimization, UCLA CAM Report, (2011), 11-04. [18] L. Ling, M. Yamamoto, Y. C. 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. [19] J. L. Lions, Pointwise control for distributed systems, in "Control and Estimation in Distributed Parameters Systems" (ed. H. T. Banks), SIAM, (1992), 1-39. doi: 10.1137/1.9781611970982.ch1. [20] J. Rauch, "Partial Differential Equations," Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-0953-9.

show all references

##### References:
 [1] V. Akçelik, G. Biros, A. Draganeascu, J. Hill, O. Ghattas and B. van Bloemen Waanders, Dynamic data-driven inversion for terascale simulations: Real-time identification of airborne contaminants, in "Proceedings of the 2005 ACM/IEEE conference on Supercomputing," IEEE Computer Society, (2005), 43-57. doi: 10.1109/SC.2005.25. [2] V. Akçelik, G. Biros, O. Ghattas, K. R. Long and B. van Bloemen Waanders, A variational finite element method for source inversion for convective-diffusive transport, Finite Elem. Anal. Des., 39 (2003), 683-705. doi: 10.1016/S0168-874X(03)00054-4. [3] A. El Badia and T. Ha Duong, Some remarks on the problem of source identification from boundary measurements, Inverse Problems, 14 (1998), 883-891. doi: 10.1088/0266-5611/14/4/008. [4] 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. [5] A. Blake and A. Zisserman, "Visual Reconstruction," MIT Press, Cambridge, MA, 1987. [6] M. Burger, Y. Landa, N. M. Tanushev and R. Tsai, Discovering a point source in unknown environments, in "Algorithmic Foundation of Robotics VII" (eds. G. S. Chirikjian, H. Choset, M. Morales and T. Murphy), Springer, (2009), 663-678. doi: 10.1007/978-3-642-00312-7_41. [7] E. J. Candes and T. Tao, Decoding by linear programming, IEEE Trans. Inform. Theory, 51 (2005), 4203-4215. doi: 10.1109/TIT.2005.858979. [8] T. F. Chan and H.-M. Zhou, Total variation wavelet thresholding, J. Sci. Comput., 32 (2007), 315-341. doi: 10.1007/s10915-007-9133-0. [9] R. Chartrand, Exact reconstruction of sparse signals via nonconvex minimization, Signal Processing Letters, 14 (2007), 707-710. doi: 10.1109/LSP.2007.898300. [10] R. Chartrand and V. Staneva, Restricted isometry properties and nonconvex compressive sensing, Inverse Problems, 24 (2008). doi: 10.1088/0266-5611/24/3/035020. [11] B. R. Cosofret, C. M. Gittins and W. J. Marinelli, Visualization and tomographic analysis of chemical vapor plumes via LWIR imaging Fabry-Perot spectrometry, in "Proc. SPIE: Chemical and Biological Standoff Detection II" (eds. J. O. Jensen and J.-M. Theriault), SPIE, (2004), 112-121. doi: 10.1117/12.570442. [12] D. L. Donoho, Sparse components of images and optimal atomic decompositions, Constructive Approximation, 17 (2001), 353-382. doi: 10.1007/s003650010032. [13] C. M. Gittins and W. J. Marinelli, AIRIS multispectral imaging chemical sensor, in "Proc. SPIE: Electro-Optical Technology for Remote Chemical Detection and Identification III" (eds. M. Fallahi and E. A. Howden), SPIE, (1998), 65-74. doi: 10.1117/12.317637. [14] E. T. Hale, W. Yin and Y. Zhang, Fixed-point continuation for $l_1$-minimization: Methodology and convergence, SIAM J. Optim., 19 (2008), 1107-1130. doi: 10.1137/070698920. [15] J. P. Kernévez, "The Sentinel Method and Its Application to Environmental Pollution Problems," CRC Press, New York, 1997. [16] Y. Landa, H. Shen, R. Takei and Y.-H. R. Tsai, Autonomous source discovery and navigation in complicated environments, UCLA CAM Report, (2009), 09-73. [17] Y. Li, S. Osher and R. Tsai, Heat source identification based on $L_1$ constrained minimization, UCLA CAM Report, (2011), 11-04. [18] L. Ling, M. Yamamoto, Y. C. 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. [19] J. L. Lions, Pointwise control for distributed systems, in "Control and Estimation in Distributed Parameters Systems" (ed. H. T. Banks), SIAM, (1992), 1-39. doi: 10.1137/1.9781611970982.ch1. [20] J. Rauch, "Partial Differential Equations," Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-0953-9.
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