American Institute of Mathematical Sciences

Advanced
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, (2005), 43.  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.  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.  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.  doi: 10.1088/0266-5611/21/3/020.  Google Scholar

[5]

A. Blake and A. Zisserman, "Visual Reconstruction,", MIT Press, (1987).   Google Scholar

[6]

M. Burger, Y. Landa, N. M. Tanushev and R. Tsai, Discovering a point source in unknown environments,, in, (2009), 663.  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.  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.  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.  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, (2004), 112.  doi: 10.1117/12.570442.  Google Scholar

[12]

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

[13]

C. M. Gittins and W. J. Marinelli, AIRIS multispectral imaging chemical sensor,, in, (1998), 65.  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.  doi: 10.1137/070698920.  Google Scholar

[15]

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

[17]

Y. Li, S. Osher and R. Tsai, Heat source identification based on $L_1$ constrained minimization,, UCLA CAM Report, (2011), 11.   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.  doi: 10.1088/0266-5611/22/4/011.  Google Scholar

[19]

J. L. Lions, Pointwise control for distributed systems,, in, (1992), 1.  doi: 10.1137/1.9781611970982.ch1.  Google Scholar

[20]

J. Rauch, "Partial Differential Equations,", Springer-Verlag, (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, (2005), 43.  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.  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.  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.  doi: 10.1088/0266-5611/21/3/020.  Google Scholar

[5]

A. Blake and A. Zisserman, "Visual Reconstruction,", MIT Press, (1987).   Google Scholar

[6]

M. Burger, Y. Landa, N. M. Tanushev and R. Tsai, Discovering a point source in unknown environments,, in, (2009), 663.  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.  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.  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.  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, (2004), 112.  doi: 10.1117/12.570442.  Google Scholar

[12]

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

[13]

C. M. Gittins and W. J. Marinelli, AIRIS multispectral imaging chemical sensor,, in, (1998), 65.  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.  doi: 10.1137/070698920.  Google Scholar

[15]

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

[17]

Y. Li, S. Osher and R. Tsai, Heat source identification based on $L_1$ constrained minimization,, UCLA CAM Report, (2011), 11.   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.  doi: 10.1088/0266-5611/22/4/011.  Google Scholar

[19]

J. L. Lions, Pointwise control for distributed systems,, in, (1992), 1.  doi: 10.1137/1.9781611970982.ch1.  Google Scholar

[20]

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

[1]

Qing Tang. On an optimal control problem of time-fractional advection-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020, 25 (2) : 761-779. doi: 10.3934/dcdsb.2019266
[2]

Assyr Abdulle. Multiscale methods for advection-diffusion problems. Conference Publications, 2005, 2005 (Special) : 11-21. doi: 10.3934/proc.2005.2005.11
[3]

Lena-Susanne Hartmann, Ilya Pavlyukevich. Advection-diffusion equation on a half-line with boundary Lévy noise. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 637-655. doi: 10.3934/dcdsb.2018200
[4]

Michael Taylor. Random walks, random flows, and enhanced diffusivity in advection-diffusion equations. Discrete & Continuous Dynamical Systems - B, 2012, 17 (4) : 1261-1287. doi: 10.3934/dcdsb.2012.17.1261
[5]

Steven L. Brunton, Joshua L. Proctor, Jonathan H. Tu, J. Nathan Kutz. Compressed sensing and dynamic mode decomposition. Journal of Computational Dynamics, 2015, 2 (2) : 165-191. doi: 10.3934/jcd.2015002
[6]

Alexandre Caboussat, Roland Glowinski. A Numerical Method for a Non-Smooth Advection-Diffusion Problem Arising in Sand Mechanics. Communications on Pure & Applied Analysis, 2009, 8 (1) : 161-178. doi: 10.3934/cpaa.2009.8.161
[7]

Patrick Henning, Mario Ohlberger. A-posteriori error estimate for a heterogeneous multiscale approximation of advection-diffusion problems with large expected drift. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : 1393-1420. doi: 10.3934/dcdss.2016056
[8]

Patrick Henning, Mario Ohlberger. The heterogeneous multiscale finite element method for advection-diffusion problems with rapidly oscillating coefficients and large expected drift. Networks & Heterogeneous Media, 2010, 5 (4) : 711-744. doi: 10.3934/nhm.2010.5.711
[9]

Ying Zhang, Ling Ma, Zheng-Hai Huang. On phaseless compressed sensing with partially known support. Journal of Industrial & Management Optimization, 2020, 16 (3) : 1519-1526. doi: 10.3934/jimo.2019014
[10]

Shousheng Luo, Tie Zhou. Superiorization of EM algorithm and its application in Single-Photon Emission Computed Tomography(SPECT). Inverse Problems & Imaging, 2014, 8 (1) : 223-246. doi: 10.3934/ipi.2014.8.223
[11]

Vikram Krishnamurthy, William Hoiles. Information diffusion in social sensing. Numerical Algebra, Control & Optimization, 2016, 6 (3) : 365-411. doi: 10.3934/naco.2016017
[12]

Yingying Li, Stanley Osher. Coordinate descent optimization for l1 minimization with application to compressed sensing; a greedy algorithm. Inverse Problems & Imaging, 2009, 3 (3) : 487-503. doi: 10.3934/ipi.2009.3.487
[13]

Song Li, Junhong Lin. Compressed sensing with coherent tight frames via $l_q$-minimization for $0 < q \leq 1$. Inverse Problems & Imaging, 2014, 8 (3) : 761-777. doi: 10.3934/ipi.2014.8.761
[14]

Henrik Garde, Kim Knudsen. 3D reconstruction for partial data electrical impedance tomography using a sparsity prior. Conference Publications, 2015, 2015 (special) : 495-504. doi: 10.3934/proc.2015.0495
[15]

Linfei Wang, Dapeng Tao, Ruonan Wang, Ruxin Wang, Hao Li. Big Map R-CNN for object detection in large-scale remote sensing images. Mathematical Foundations of Computing, 2019, 2 (4) : 299-314. doi: 10.3934/mfc.2019019
[16]

Patrick De Kepper, István Szalai. An effective design method to produce stationary chemical reaction-diffusion patterns. Communications on Pure & Applied Analysis, 2012, 11 (1) : 189-207. doi: 10.3934/cpaa.2012.11.189
[17]

Ivan Gentil, Bogusław Zegarlinski. Asymptotic behaviour of reversible chemical reaction-diffusion equations. Kinetic & Related Models, 2010, 3 (3) : 427-444. doi: 10.3934/krm.2010.3.427
[18]

Jifa Jiang, Junping Shi. Dynamics of a reaction-diffusion system of autocatalytic chemical reaction. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 245-258. doi: 10.3934/dcds.2008.21.245
[19]

Jan Haškovec, Dietmar Oelz. A free boundary problem for aggregation by short range sensing and differentiated diffusion. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1461-1480. doi: 10.3934/dcdsb.2015.20.1461
[20]

M. B. A. Mansour. Computation of traveling wave fronts for a nonlinear diffusion-advection model. Mathematical Biosciences & Engineering, 2009, 6 (1) : 83-91. doi: 10.3934/mbe.2009.6.83

2018 Impact Factor: 1.469

Tools

Metrics

  • PDF downloads (17)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences