# American Institue of Mathematical Sciences

2018, 12(1): 59-90. doi: 10.3934/ipi.2018003

## Generalized stability estimates in inverse transport theory

 1 University of Chicago, 5747 S. Ellis Avenue, Jones 120B, Chicago, IL 60637, USA 2 Laboratoire de Mathématiques Paul Painlevé, CNRS UMR 8524/Université Lille 1 Sciences et Technologies, 59655 Villeneuve d'Ascq Cedex, France

* Corresponding author: Alexandre Jollivet

Received  March 2017 Published  December 2017

Inverse transport theory concerns the reconstruction of the absorption and scattering coefficients in a transport equation from knowledge of the albedo operator, which models all possible boundary measurements. Uniqueness and stability results are well known and are typically obtained for errors of the albedo operator measured in the $L^1$ sense. We claim that such error estimates are not always very informative. For instance, arbitrarily small blurring and misalignment of detectors result in $O(1)$ errors of the albedo operator and hence in $O(1)$ error predictions on the reconstruction of the coefficients, which are not useful.

This paper revisit such stability estimates by introducing a more forgiving metric on the measurements errors, namely the $1-$Wasserstein distances, which penalize blurring or misalignment by an amount proportional to the width of the blurring kernel or to the amount of misalignment. We obtain new stability estimates in this setting.

We also consider the effect of errors, still measured in the $1-$ Wasserstein distance, on the generation of the probing source. This models blurring and misalignment in the design of (laser) probes and allows us to consider discretized sources. Under appropriate assumptions on the coefficients, we quantify the effect of such errors on the reconstructions.

Citation: Guillaume Bal, Alexandre Jollivet. Generalized stability estimates in inverse transport theory. Inverse Problems & Imaging, 2018, 12 (1) : 59-90. doi: 10.3934/ipi.2018003
##### References:
 [1] D. S. Anikonov, A. E. Kovtanyuk and I. V. Prokhorov, Transport Equation and Tomography, Inverse and Ill-posed Problems Series, VSP, Utrecht, 2002. [2] G. Bal, Inverse transport theory and applications, Inverse Problems, 25 (2009), 053001, 48 pp. [3] G. Bal, A. Jollivet, Stability estimates in stationary inverse transport, Inverse Probl. Imaging, 2 (2008), 427-454. doi: 10.3934/ipi.2008.2.427. [4] -, Approximate stability in inverse transport, in Biomedical Mathematics: Promising Directions in Imaging, Therapy Planning and Inverse Problems (eds. Y. Censor, M. Jiang and G. Wang), Medical Physics Publishing, Madison, WI, USA, 2009. [5] -, G. Bal, A. Jollivet, Time-dependent angularly averaged inverse transport, Inverse Problems, 25 (2009), 075010, 32 pp. [6] -, Stability for time-dependent inverse transport, SIAM J. Math. Anal., 42 (2010), 679-700. doi: 10.1137/080734480. [7] G. Bal, A. Jollivet and V. Jugnon, Inverse transport theory of Photoacoustics, Inverse Problems, 26 (2010), 025011, 35 pp. [8] G. Bal, A. Jollivet, I. Langmore, F. Monard, Angular average of time-harmonic transport solutions, Comm. Partial Differential Equations, 36 (2011), 1044-1070. doi: 10.1080/03605302.2010.540608. [9] G. Bal, I. Langmore, F. Monard, Inverse transport with isotropic sources and angularly averaged measurements, Inverse Probl. Imaging, 2 (2008), 23-42. doi: 10.3934/ipi.2008.2.23. [10] G. Bal, A. Tamasan, Inverse source problems in transport equations, SIAM J. Math. Anal., 39 (2007), 57-76. doi: 10.1137/050647177. [11] A. N. Bondarenko, Singular structure of the fundamental solution of the transport equation, and inverse problems in particle scattering theory, (Russian), Dokl. Akad. Nauk SSSR, 322 (1992), 274-276; translation in Soviet Phys. Dokl. 37(1) (1992), 21-22. [12] -, The structure of the fundamental solution of the time-independent transport equation, J. Math. Anal. Appl. , 221 (1998), 430-451. doi: 10.1006/jmaa.1997.5842. [13] M. Choulli, P. Stefanov, Inverse scattering and inverse boundary value problems for the linear Boltzmann equation, Comm. Partial Diff. Equ., 21 (1996), 763-785. doi: 10.1080/03605309608821207. [14] -, An inverse boundary value problem for the stationary transport equation, Osaka J. Math., 36 (1999), 87-104. [15] N. S. Dairbekov, G. P. Paternain, P. Stefanov, G. Uhlmann, The boundary rigidity problem in the presence of a magnetic field, Adv. Math., 216 (2007), 535-609. doi: 10.1016/j.aim.2007.05.014. [16] H. Egger, M. Schlottbom, An Lp theory for stationary radiative transfer, Appl. Anal., 93 (2014), 1283-1296. doi: 10.1080/00036811.2013.826798. [17] G. B. Folland, Lectures on Partial Differential Equations at the Tata Inst. of Bombay, Springer Verlag, Berlin, 1983. [18] I. Langmore, The stationary transport equation with angularly averaged measurements, Inverse Problems, 24 (2008), 015024, 23 pp. [19] I. Langmore, S. McDowall, Optical tomography for variable refractive index with angularly averaged measurements, Comm. Partial Diff. Equ., 33 (2008), 2180-2207. doi: 10.1080/03605300802523453. [20] E. W. Larsen, Solution to multidimensional inverse transport problems, J. Math. Phys., 25 (1984), 131-135. doi: 10.1063/1.526007. [21] -, Solution of three-dimensional inverse transport problems, Transport Theory Statist. Phys. , 17 (1988), 147-167. doi: 10.1080/00411458808230860. [22] N. J. McCormick, Methods for solving inverse problems for radiation transport -an update, Transport Theory Statist. Phys., 15 (1986), 758-772. doi: 10.1080/00411458608212714. [23] -, Inverse radiative transfer problems: A review, Nucl. Sci. Eng., 112 (1992), 185-198. [24] S. R. McDowall, A. Tamasan and P. Stefanov, Stability of the gauge equivalent classes in stationary inverse transport,, Inverse Problems, 26 (2010), 025006, 19pp. [25] S. G. Mikhlin and S. Prössdorf, Singular Integral Operators, Translated from the German by Albrecht Böttcher and Reinhard Lehmann. Mathematische Lehrbücher und Monographien, Ⅱ. Abteilung: Mathematische Monographien [Mathematical Textbooks and Monographs, Part II: Mathematical Monographs], 68. Akademie-Verlag, Berlin, 1986. 528 pp. [26] F. Natterer, The Mathematics of Computerized Tomography, Wiley, New York, 1986. [27] P. Stefanov, Inverse problems in transport theory, in Inside Out: Inverse Problems and Applications (ed. G. Uhlmann), vol. 47 of MSRI publications, Cambridge University Press, Cambridge, UK, (2003), 111-131. [28] P. Stefanov, A. Tamasan, Uniqueness and non-uniqueness in inverse radiative transfer, Proc. Amer. Math. Soc., 137 (2009), 2335-2344. doi: 10.1090/S0002-9939-09-09839-6. [29] P. Stefanov, G. Uhlmann, Optical tomography in two dimensions, Methods Appl. Anal., 10 (2003), 1-9. doi: 10.4310/MAA.2003.v10.n1.a1. [30] -, An inverse source problem in optical molecular imaging, Analysis and PDE, 1 (2008), 115-126. doi: 10.2140/apde.2008.1.115. [31] A. Tamasan, An inverse boundary value problem in two-dimensional transport, Inverse Problems, 18 (2002), 209-219. doi: 10.1088/0266-5611/18/1/314. [32] C. Villani, Optimal Transport. Old and New, Vol. 338 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 2009. [33] 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] D. S. Anikonov, A. E. Kovtanyuk and I. V. Prokhorov, Transport Equation and Tomography, Inverse and Ill-posed Problems Series, VSP, Utrecht, 2002. [2] G. Bal, Inverse transport theory and applications, Inverse Problems, 25 (2009), 053001, 48 pp. [3] G. Bal, A. Jollivet, Stability estimates in stationary inverse transport, Inverse Probl. Imaging, 2 (2008), 427-454. doi: 10.3934/ipi.2008.2.427. [4] -, Approximate stability in inverse transport, in Biomedical Mathematics: Promising Directions in Imaging, Therapy Planning and Inverse Problems (eds. Y. Censor, M. Jiang and G. Wang), Medical Physics Publishing, Madison, WI, USA, 2009. [5] -, G. Bal, A. Jollivet, Time-dependent angularly averaged inverse transport, Inverse Problems, 25 (2009), 075010, 32 pp. [6] -, Stability for time-dependent inverse transport, SIAM J. Math. Anal., 42 (2010), 679-700. doi: 10.1137/080734480. [7] G. Bal, A. Jollivet and V. Jugnon, Inverse transport theory of Photoacoustics, Inverse Problems, 26 (2010), 025011, 35 pp. [8] G. Bal, A. Jollivet, I. Langmore, F. Monard, Angular average of time-harmonic transport solutions, Comm. Partial Differential Equations, 36 (2011), 1044-1070. doi: 10.1080/03605302.2010.540608. [9] G. Bal, I. Langmore, F. Monard, Inverse transport with isotropic sources and angularly averaged measurements, Inverse Probl. Imaging, 2 (2008), 23-42. doi: 10.3934/ipi.2008.2.23. [10] G. Bal, A. Tamasan, Inverse source problems in transport equations, SIAM J. Math. Anal., 39 (2007), 57-76. doi: 10.1137/050647177. [11] A. N. Bondarenko, Singular structure of the fundamental solution of the transport equation, and inverse problems in particle scattering theory, (Russian), Dokl. Akad. Nauk SSSR, 322 (1992), 274-276; translation in Soviet Phys. Dokl. 37(1) (1992), 21-22. [12] -, The structure of the fundamental solution of the time-independent transport equation, J. Math. Anal. Appl. , 221 (1998), 430-451. doi: 10.1006/jmaa.1997.5842. [13] M. Choulli, P. Stefanov, Inverse scattering and inverse boundary value problems for the linear Boltzmann equation, Comm. Partial Diff. Equ., 21 (1996), 763-785. doi: 10.1080/03605309608821207. [14] -, An inverse boundary value problem for the stationary transport equation, Osaka J. Math., 36 (1999), 87-104. [15] N. S. Dairbekov, G. P. Paternain, P. Stefanov, G. Uhlmann, The boundary rigidity problem in the presence of a magnetic field, Adv. Math., 216 (2007), 535-609. doi: 10.1016/j.aim.2007.05.014. [16] H. Egger, M. Schlottbom, An Lp theory for stationary radiative transfer, Appl. Anal., 93 (2014), 1283-1296. doi: 10.1080/00036811.2013.826798. [17] G. B. Folland, Lectures on Partial Differential Equations at the Tata Inst. of Bombay, Springer Verlag, Berlin, 1983. [18] I. Langmore, The stationary transport equation with angularly averaged measurements, Inverse Problems, 24 (2008), 015024, 23 pp. [19] I. Langmore, S. McDowall, Optical tomography for variable refractive index with angularly averaged measurements, Comm. Partial Diff. Equ., 33 (2008), 2180-2207. doi: 10.1080/03605300802523453. [20] E. W. Larsen, Solution to multidimensional inverse transport problems, J. Math. Phys., 25 (1984), 131-135. doi: 10.1063/1.526007. [21] -, Solution of three-dimensional inverse transport problems, Transport Theory Statist. Phys. , 17 (1988), 147-167. doi: 10.1080/00411458808230860. [22] N. J. McCormick, Methods for solving inverse problems for radiation transport -an update, Transport Theory Statist. Phys., 15 (1986), 758-772. doi: 10.1080/00411458608212714. [23] -, Inverse radiative transfer problems: A review, Nucl. Sci. Eng., 112 (1992), 185-198. [24] S. R. McDowall, A. Tamasan and P. Stefanov, Stability of the gauge equivalent classes in stationary inverse transport,, Inverse Problems, 26 (2010), 025006, 19pp. [25] S. G. Mikhlin and S. Prössdorf, Singular Integral Operators, Translated from the German by Albrecht Böttcher and Reinhard Lehmann. Mathematische Lehrbücher und Monographien, Ⅱ. Abteilung: Mathematische Monographien [Mathematical Textbooks and Monographs, Part II: Mathematical Monographs], 68. Akademie-Verlag, Berlin, 1986. 528 pp. [26] F. Natterer, The Mathematics of Computerized Tomography, Wiley, New York, 1986. [27] P. Stefanov, Inverse problems in transport theory, in Inside Out: Inverse Problems and Applications (ed. G. Uhlmann), vol. 47 of MSRI publications, Cambridge University Press, Cambridge, UK, (2003), 111-131. [28] P. Stefanov, A. Tamasan, Uniqueness and non-uniqueness in inverse radiative transfer, Proc. Amer. Math. Soc., 137 (2009), 2335-2344. doi: 10.1090/S0002-9939-09-09839-6. [29] P. Stefanov, G. Uhlmann, Optical tomography in two dimensions, Methods Appl. Anal., 10 (2003), 1-9. doi: 10.4310/MAA.2003.v10.n1.a1. [30] -, An inverse source problem in optical molecular imaging, Analysis and PDE, 1 (2008), 115-126. doi: 10.2140/apde.2008.1.115. [31] A. Tamasan, An inverse boundary value problem in two-dimensional transport, Inverse Problems, 18 (2002), 209-219. doi: 10.1088/0266-5611/18/1/314. [32] C. Villani, Optimal Transport. Old and New, Vol. 338 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 2009. [33] 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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