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

Source identification from line integral measurements and simple atmospheric models

Brittan Farmer 1, , Cassandra Hall 2, and  Selim Esedoḡlu 3,

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.
Keywords: compressed sensing, Advection-diffusion equation, computed tomography, standoff chemical detection., sparsity.
Mathematics Subject Classification: Primary: 49N45, 49M20; Secondary: 35K0.
Citation: Brittan Farmer, Cassandra Hall, Selim Esedoḡlu. Source identification from line integral measurements and simple atmospheric models. Inverse Problems & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

A. Blake and A. Zisserman, "Visual Reconstruction," MIT Press, Cambridge, MA, 1987.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

R. Chartrand, Exact reconstruction of sparse signals via nonconvex minimization, Signal Processing Letters, 14 (2007), 707-710. doi: 10.1109/LSP.2007.898300.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

D. L. Donoho, Sparse components of images and optimal atomic decompositions, Constructive Approximation, 17 (2001), 353-382. doi: 10.1007/s003650010032.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

J. P. Kernévez, "The Sentinel Method and Its Application to Environmental Pollution Problems," CRC Press, New York, 1997.  Google Scholar

[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. Google Scholar

[17]

Y. Li, S. Osher and R. Tsai, Heat source identification based on $L_1$ constrained minimization, UCLA CAM Report, (2011), 11-04. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

J. Rauch, "Partial Differential Equations," Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-0953-9.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

A. Blake and A. Zisserman, "Visual Reconstruction," MIT Press, Cambridge, MA, 1987.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

R. Chartrand, Exact reconstruction of sparse signals via nonconvex minimization, Signal Processing Letters, 14 (2007), 707-710. doi: 10.1109/LSP.2007.898300.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

D. L. Donoho, Sparse components of images and optimal atomic decompositions, Constructive Approximation, 17 (2001), 353-382. doi: 10.1007/s003650010032.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

J. P. Kernévez, "The Sentinel Method and Its Application to Environmental Pollution Problems," CRC Press, New York, 1997.  Google Scholar

[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. Google Scholar

[17]

Y. Li, S. Osher and R. Tsai, Heat source identification based on $L_1$ constrained minimization, UCLA CAM Report, (2011), 11-04. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

J. Rauch, "Partial Differential Equations," Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-0953-9.  Google Scholar

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