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Detecting small low emission radiating sources
Bayesian inverse problems with Monte Carlo forward models
1.  Department of Applied Physics and Applied Mathematics, Columbia University, 200 S. W. Mudd Building, MC 4701, 500 W. 120th Street, New York, NY 10027, United States, United States 
2.  Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, United States 
Our multipleresolution forward models themselves are built around a new importance sampling scheme that allows Monte Carlo forward models to be used efficiently in inverse problems. The method is used to solve an inverse transport problem that finds applications in atmospheric remote sensing. We present a pathrecycling methodology to efficiently vary parameters in the transport equation. The forward transport equation is solved by a Monte Carlo method that is amenable to the use of $MC^3$ to solve the inverse transport problem using a Bayesian formalism.
References:
[1] 
Simon Arridge, et al., Approximation errors and model reduction with an application in optical diffusion tomography, Inverse Problems, 22 (2006). 
[2] 
Guillaume Bal, Anthony Davis and Ian Langmore, A hybrid (Monte Carlo/deterministic) approach for multidimensional radiation transport, J. Computational Physics, 230 (2011), 77237735. 
[3] 
George Casella and Robert Berger, "Statistical Inference," Duxbury, 2002. 
[4] 
Jin Chen and Xavier Intes, Timegated perturbation Monte Carlo for whole body functional imaging in small animals, Optics Express, 17 (2009). 
[5] 
J. Andrés Christen and Colin Fox, Markov chain Monte Carlo using an approximation, Journal of Computational and Graphical Statistics, 14 (2005), 795810 doi: 10.1198/106186005X76983. 
[6] 
Rick Durrett, "Probability: Theory and Examples," third edition, Brooks/Cole, 2005. 
[7] 
Yalchin Efendiev, Thomas Hou and W. Luo, Preconditioning Markov chain Monte Carlo simulations using coarsescale models, SIAM J. Sci. Comput., 28 (2006), 776803. doi: 10.1137/050628568. 
[8] 
Charles J. Geyer, Practical Markov chain Monte Carlo, Statistical Science, 7 (1992), 473511. 
[9] 
Carole K. Hayakawa and Jerome Spanier, Perturbation Monte Carlo methods for the solution of inverse problems, in "Monte Carlo and Quasi Monte Carlo Methods 2002," Springer, Berlin, (2004), 227241. 
[10] 
Carole K. Hayakawa, Jerome Spanier, and Frédéric Bevilacqua, et al., Perturbation Monte Carlo methods to solve inverse photon migration problems in heterogeneous tissues, Optics Letters, 26 (2001), 13331337. 
[11] 
Jari P. Kaipio and Erkki Somersalo, "Statistical and Computational Inverse Problems," Applied Mathematical Sciences, 160, Springer Verlag, New York, 2005. 
[12] 
Jari P. Kaipio and Erkki Somersalo, Statistical inverse problems: Discretization, model reduction, and inverse crimes, Journal of Computational and Applied Mathematics, 198 (2007), 493504. doi: 10.1016/j.cam.2005.09.027. 
[13] 
Ian Langmore, Anthony Davis and Guillaume Bal, Multipixel retrieval of structural and optical parameters in a 2D scene with a pathrecycling Monte Carlo forward model and a new Bayesian inference engine, IEEE TGRS, (2012). 
[14] 
Jun S. Liu, "Monte Carlo Strategies in Scientific Computing," Springer Series in Statistics, Springer, New York, 2008. 
[15] 
David Moulton, Colin Fox and Daniil Svyatskiy, Multilevel approximations in samplebased inversion from the DirichlettoNeumann map, Journal of Physics: Conference Series, (2008), pp. 124. 
[16] 
Hanna K. Pikkarainen, State estimation approach to nonstationary inverse problems: Discretization error and filtering problem, Inverse Problems, 22 (2006), 365379. doi: 10.1088/02665611/22/1/020. 
[17] 
Christian Robert and George Casella, "Monte Carlo Statistical Methods," Second edition, Springer Texts in Statistics, SpringerVerlag, New York, 2004. 
[18] 
Luke Tierney, Markov chains for exploring posterior distributions, The Annals of Statistics, 22 (1994), 17011762. doi: 10.1214/aos/1176325750. 
show all references
References:
[1] 
Simon Arridge, et al., Approximation errors and model reduction with an application in optical diffusion tomography, Inverse Problems, 22 (2006). 
[2] 
Guillaume Bal, Anthony Davis and Ian Langmore, A hybrid (Monte Carlo/deterministic) approach for multidimensional radiation transport, J. Computational Physics, 230 (2011), 77237735. 
[3] 
George Casella and Robert Berger, "Statistical Inference," Duxbury, 2002. 
[4] 
Jin Chen and Xavier Intes, Timegated perturbation Monte Carlo for whole body functional imaging in small animals, Optics Express, 17 (2009). 
[5] 
J. Andrés Christen and Colin Fox, Markov chain Monte Carlo using an approximation, Journal of Computational and Graphical Statistics, 14 (2005), 795810 doi: 10.1198/106186005X76983. 
[6] 
Rick Durrett, "Probability: Theory and Examples," third edition, Brooks/Cole, 2005. 
[7] 
Yalchin Efendiev, Thomas Hou and W. Luo, Preconditioning Markov chain Monte Carlo simulations using coarsescale models, SIAM J. Sci. Comput., 28 (2006), 776803. doi: 10.1137/050628568. 
[8] 
Charles J. Geyer, Practical Markov chain Monte Carlo, Statistical Science, 7 (1992), 473511. 
[9] 
Carole K. Hayakawa and Jerome Spanier, Perturbation Monte Carlo methods for the solution of inverse problems, in "Monte Carlo and Quasi Monte Carlo Methods 2002," Springer, Berlin, (2004), 227241. 
[10] 
Carole K. Hayakawa, Jerome Spanier, and Frédéric Bevilacqua, et al., Perturbation Monte Carlo methods to solve inverse photon migration problems in heterogeneous tissues, Optics Letters, 26 (2001), 13331337. 
[11] 
Jari P. Kaipio and Erkki Somersalo, "Statistical and Computational Inverse Problems," Applied Mathematical Sciences, 160, Springer Verlag, New York, 2005. 
[12] 
Jari P. Kaipio and Erkki Somersalo, Statistical inverse problems: Discretization, model reduction, and inverse crimes, Journal of Computational and Applied Mathematics, 198 (2007), 493504. doi: 10.1016/j.cam.2005.09.027. 
[13] 
Ian Langmore, Anthony Davis and Guillaume Bal, Multipixel retrieval of structural and optical parameters in a 2D scene with a pathrecycling Monte Carlo forward model and a new Bayesian inference engine, IEEE TGRS, (2012). 
[14] 
Jun S. Liu, "Monte Carlo Strategies in Scientific Computing," Springer Series in Statistics, Springer, New York, 2008. 
[15] 
David Moulton, Colin Fox and Daniil Svyatskiy, Multilevel approximations in samplebased inversion from the DirichlettoNeumann map, Journal of Physics: Conference Series, (2008), pp. 124. 
[16] 
Hanna K. Pikkarainen, State estimation approach to nonstationary inverse problems: Discretization error and filtering problem, Inverse Problems, 22 (2006), 365379. doi: 10.1088/02665611/22/1/020. 
[17] 
Christian Robert and George Casella, "Monte Carlo Statistical Methods," Second edition, Springer Texts in Statistics, SpringerVerlag, New York, 2004. 
[18] 
Luke Tierney, Markov chains for exploring posterior distributions, The Annals of Statistics, 22 (1994), 17011762. doi: 10.1214/aos/1176325750. 
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