Homogenization and corrector theory for linear transport in random media

Discrete and Continuous Dynamical Systems

Volumn 28, Issue 4, 2010, Pages 1311-1343

  Bal, Guillaume ^a   Jing, Wenjia ^a  

^a Department of Earth and Environmental Sciences   (United States)

Author keywords

Central limit theorem; Corrector theory; Homogenization theory; Random media; Spatial poisson point process; Transport equation

Indexed keywords

EID: 77958051536     PISSN: 10780947     EISSN: None     Source Type: Journal    
DOI: 10.3934/dcds.2010.28.1311     Document Type: Article

References (35)

1. G. Allaire and G. Bal, Homogenization of the criticality spectral equation in neutron transport, M2AN Math. Model. Numer. Anal., 33 (1999), 721-746.
2. S. R. Arridge, Optical tomography in medical imaging, Inverse Problems, 15 (1999), R41-R93.
3. G. Bal, Kinetics of scalar wave fields in random media, Wave Motion, 43 (2005), 132-157.
4. G. Bal, Homogenization in random media and effective medium theory for high frequency waves, Disc. Cont. Dyn. Syst. B, 8 (2007), 473-492.
5. G. Bal,Central limits and homogenization in random media, Multiscale Model. Simul., 7 (2008), 677-702.
6. G. Bal, Inverse transport theory and applications, Inverse Problems, 25 (2009), 053001.
7. G. Bal, J. Garnier, S. Motsch and V. Perrier, Random integrals and correctors in homogenization, Asymptot. Anal., 59 (2008), 1-26.
8. G. Bal and W. Jing, Fluctuation theory for Radiative transfer in random media, submitted.
9. G. Bal and A. Jollivet, Stability estimates in stationary inverse transport, Inverse Probl. Imaging, 2 (2008), 427-454.
10. G. Bal and K. Ren, Transport-based imaging in random media, SIAM J. Appl. Math., 68 (2008), 1738-1762.
11. G. Bal and K. Ren, Physics-based models for measurement correlations, application to an inverse sturmliouville problem, Inverse Problems, 25 (2009), 055006.
12. A. Bensoussan, J.-L. Lions and G. C. Papanicolaou, "Asymptotic Analysis for Periodic Structures," Studies in Mathematics and its Applications, 5, North-Holland Publishing Co., Amsterdam-New York, 1978.
13. A. Bensoussan, J.-L. Lions and G. C. Papanicolaou, Boundary layers and homogenization of transport processes, Publ. Res. Inst. Math. Sci., 15 (1979), 53-157.
14. E. Bolthausen, On the central limit theorem for stationary mixing random fields, Ann. Probab., 10 (1982), 1047-1050.
15. A. Bourgeat and A. Piatnitski, Estimates in probability of the residual between the random and the homogenized solutions of one-dimensional second order operator, Asymptot. Anal., 21 (1999), 303-315.
16. S. Chandrasekhar, "Radiative Transfer," Dover Publications, New York, 1960.
17. M. Choulli and P. Stefanov, An inverse boundary value problem for the stationary transport equation, Osaka J. Math., 36 (1999), 87-104.
18. D. R. Cox and V. Isham, "Point Processes," Chapman & Hall, London, 1980, Monographs on Applied Probability and Statistics.
19. R. Dautray and J.-L. Lions, "Mathematical Analysis and Numerical Methods for Science and Technology. Vol. 6," Springer Verlag, Berlin, 1993.
20. B. Davison and J. B. Sykes, "Neutron Transport Theory," Oxford University Press, 1957.
21. P. Doukhan, "Mixing," Lecture Notes in Statistics, vol. 85, Springer-Verlag, New York, 1994.
22. L. Dumas and F. Golse, Homogenization of transport equations, SIAM J. Appl. Math., 60 (2000), 1447-1470.
23. R. Figari, E. Orlandi and G. Papanicolaou, Mean field and Gaussian approximation for partial differential equations with random coefficients, SIAM J. Appl. Math., 42 (1982), 1069-1077.
24. L. Hörmander, "The Analysis of Linear Partial Differential Operators I," Springer-Verlag, New York, 1990.
25. V. V. Jikov, S. M. Kozlov and O. A. Oleinik, "Homogenization of Differential Operators and Integral Functionals," Springer-Verlag, New York, 1994.
26. C. D. Levermore, G. C. Pomraning, D. L. Sanzo and J. Wong, Linear transport theory in a random medium, J. Math. Phys., 27 (1986), 2526-2536.
27. A. Marshak and A. B. Davis, "3d Radiative Transfer in Cloudy Atmospheres," Springer, New-York, 2005.
28. A. Mellet and A. Vasseur, Homogenization of a nonlinear transport equation, Asymptot. Anal., 51 (2007), 157-166.
29. M. Mokhtar-Kharroubi, "Mathematical Topics in Neutron Transport Theory," World Scientific, Singapore, 1997.
30. M. Reed and B. Simon, "Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-Adjointness," Academic Press, New York, 1975.
31. L. Ryzhik, G. C. Papanicolaou and J. B. Keller, Transport equations for elastic and other waves in random media, Wave Motion, 24 (1996), 327-370.
32. H. Sato and M. C. Fehler, "Seismic Wave Propagation and Scattering in the Heterogeneous Earth," AIP series in modern acoustics and signal processing, AIP Press, Springer, New York, 1998.
33. P. Stefanov and G. Uhlmann, An inverse source problem in optical modecular imaging, Anal. PDE, 1 (2008), 115-126.
34. V. V. Yurinskii, Averaging of symmetric diffusion in a random medium, Siberian Math. J., 4 (1986), 603-613,
35. English translation of: Sibirsk. Mat. Zh., 27 (1986), 167-180 (Russian).