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April  2019, 13(2): 337-351. doi: 10.3934/ipi.2019017

Propagation of boundary-induced discontinuity in stationary radiative transfer and its application to the optical tomography

1. 

Institute of Applied Mathematical Sciences, National Taiwan University, No. 1, Sec. 4, Roosevelt Rd., Taipei 10617, Taiwan

2. 

Institute of Applied Mathematics, Inha University, 100 Inha-ro, Nam-gu, Incheon, 22212, Republic of Korea

* Corresponding author

Received  April 2018 Revised  August 2018 Published  January 2019

Fund Project: The first author was supported in part by JSPS KAKENHI grant number 15K17572.

We consider a boundary value problem of the stationary transport equation with the incoming boundary condition in two or three dimensional bounded convex domains. We discuss discontinuity of the solution to the boundary value problem arising from discontinuous incoming boundary data, which we call the boundary-induced discontinuity. In particular, we give two kinds of sufficient conditions on the incoming boundary data for the boundary-induced discontinuity. We propose a method to reconstruct the attenuation coefficient from jumps in boundary measurements.

Citation: I-Kun Chen, Daisuke Kawagoe. Propagation of boundary-induced discontinuity in stationary radiative transfer and its application to the optical tomography. Inverse Problems and Imaging, 2019, 13 (2) : 337-351. doi: 10.3934/ipi.2019017
References:
[1]

V. Agoshkov, Boundary Value Problems for Transport Equations, Birkhäuser, Boston, 1998. doi: 10.1007/978-1-4612-1994-1.

[2]

D. S. AnikonovI. V. Prokhorov and A. E. Kovtanyuk, Investigation of scattering and absorbing media by the methods of X-ray tomography, J. Inv. Ill-Posed Problems, 1 (1993), 259-281.  doi: 10.1515/jiip.1993.1.4.259.

[3]

K. AokiC. BardosC. Dogbe and F. Golse, A note on the propagation of boundary induced discontinuities in kinetic theory, Math. Models Methods Appl. Sci., 11 (2001), 1581-1595.  doi: 10.1142/S0218202501001483.

[4]

S. R. Arridge and J. C. Schotland, Optical tomography: Forward and inverse problems, Inverse Problems, 25 (2009), 123010, 59pp. doi: 10.1088/0266-5611/25/12/123010.

[5]

G. Bal and A. Jollivet, Stability estimates in stationary inverse transport, Inverse Probl. Imaging, 2 (2008), 427-454.  doi: 10.3934/ipi.2008.2.427.

[6]

M. Cessenat, Théorèmes de trace pour des espaces de fonctions de la neutronique, (French) [Trace theorems for neutronic function spaces], C. R. Acad. Sci. Paris, Sér. I, Math., 300 (1985), 89-92.

[7]

S. Chandrasekhar, Radiative Transfer, Dover Publications Inc., New York, 1960.

[8]

M. Choulli and P. Stefanov, An inverse boundary value problem for the stationary transport equation, Osaka J. Math., 36 (1999), 87-104. 

[9]

Y. Guo, Decay and continuity of the Boltzmann equation in bounded domains, Arch. Ration. Mech. Anal., 197 (2010), 713-809.  doi: 10.1007/s00205-009-0285-y.

[10]

D. Kawagoe and I.-K. Chen, Propagation of boundary-induced discontinuity in stationary radiative transfer, J. Stat. Phys., 170 (2018), 127-140.  doi: 10.1007/s10955-017-1922-8.

[11]

F. Natterer, The Mathematics of Computerized Tomography, SIAM, Germany, 2001. doi: 10.1137/1.9780898719284.

[12]

J. N. Wang, Stability estimates of an inverse problem for the stationary transport equation, Ann. Inst. Henri Poincaré, 70 (1999), 473-495. 

show all references

References:
[1]

V. Agoshkov, Boundary Value Problems for Transport Equations, Birkhäuser, Boston, 1998. doi: 10.1007/978-1-4612-1994-1.

[2]

D. S. AnikonovI. V. Prokhorov and A. E. Kovtanyuk, Investigation of scattering and absorbing media by the methods of X-ray tomography, J. Inv. Ill-Posed Problems, 1 (1993), 259-281.  doi: 10.1515/jiip.1993.1.4.259.

[3]

K. AokiC. BardosC. Dogbe and F. Golse, A note on the propagation of boundary induced discontinuities in kinetic theory, Math. Models Methods Appl. Sci., 11 (2001), 1581-1595.  doi: 10.1142/S0218202501001483.

[4]

S. R. Arridge and J. C. Schotland, Optical tomography: Forward and inverse problems, Inverse Problems, 25 (2009), 123010, 59pp. doi: 10.1088/0266-5611/25/12/123010.

[5]

G. Bal and A. Jollivet, Stability estimates in stationary inverse transport, Inverse Probl. Imaging, 2 (2008), 427-454.  doi: 10.3934/ipi.2008.2.427.

[6]

M. Cessenat, Théorèmes de trace pour des espaces de fonctions de la neutronique, (French) [Trace theorems for neutronic function spaces], C. R. Acad. Sci. Paris, Sér. I, Math., 300 (1985), 89-92.

[7]

S. Chandrasekhar, Radiative Transfer, Dover Publications Inc., New York, 1960.

[8]

M. Choulli and P. Stefanov, An inverse boundary value problem for the stationary transport equation, Osaka J. Math., 36 (1999), 87-104. 

[9]

Y. Guo, Decay and continuity of the Boltzmann equation in bounded domains, Arch. Ration. Mech. Anal., 197 (2010), 713-809.  doi: 10.1007/s00205-009-0285-y.

[10]

D. Kawagoe and I.-K. Chen, Propagation of boundary-induced discontinuity in stationary radiative transfer, J. Stat. Phys., 170 (2018), 127-140.  doi: 10.1007/s10955-017-1922-8.

[11]

F. Natterer, The Mathematics of Computerized Tomography, SIAM, Germany, 2001. doi: 10.1137/1.9780898719284.

[12]

J. N. Wang, Stability estimates of an inverse problem for the stationary transport equation, Ann. Inst. Henri Poincaré, 70 (1999), 473-495. 

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