A partial data inverse problem for the convection-diffusion equation

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February 2020, 14(1): 53-75. doi: 10.3934/ipi.2019063

A partial data inverse problem for the convection-diffusion equation

Suman Kumar Sahoo ^1, and Manmohan Vashisth ^2,,

1.	TIFR Centre for Applicable Mathematics, Bangalore 560065, India
2.	Beijing Computational Science Research Center, Beijing 100193, China

^* Corresponding author: Manmohan Vashisth

Received February 2019 Revised September 2019 Published November 2019

Fund Project: First author is supported by Matrics grant MTR/2017/000837. The work of second author is supported by NSAF grant (No. U1930402).

In this article we study the inverse problem of determining the convection term and the time-dependent density coefficient appearing in the convection-diffusion equation. We prove the unique determination of these coefficients from the knowledge of solution measured on a subset of the boundary.

Keywords: Inverse problems, parabolic equation, Carleman estimates, geometric optics solutions, partial data.

Mathematics Subject Classification: 35R30, 35K20.

Citation: Suman Kumar Sahoo, Manmohan Vashisth. A partial data inverse problem for the convection-diffusion equation. Inverse Problems and Imaging, 2020, 14 (1) : 53-75. doi: 10.3934/ipi.2019063

References:

[1]	S. A. Avdonin and M. I. Belishev, Dynamical inverse problem for the Schrödinger equation (BC-method), in Proceedings of the St. Petersburg Mathematical Society, Amer. Math. Soc. Transl. Ser., 10, Amer. Math. Soc., Providence, RI, 2005, 1–14. doi: 10.1090/trans2/214/01.
[2]	S. A. Avdonin and T. I. Seidman, Identication of $q(x)$ in $u_t = \Delta u - qu$, from boundary observations, SIAM J. Control Optim., 33 (1995), 1247-1255. doi: 10.1137/S0363012993249729.
[3]	L. Baudouin and J.-P. Puel, Uniqueness and stability in an inverse problem for the Schrödinger equation, Inverse Problems, 18 (2002), 1537-1554. doi: 10.1088/0266-5611/23/3/C01.
[4]	M. I. Belishev, Recent progress in the boundary control method, Inverse Problems, 23 (2007), R1–R67. doi: 10.1088/0266-5611/23/5/R01.
[5]	M. Bellassoued and D. Dos Santos Ferreira, Stable determination of coefficients in the dynamical anisotropic Schrödinger equation from the Dirichlet-to-Neumann map, Inverse Problems, 26 (2010), 30pp. doi: 10.1088/0266-5611/26/12/125010.
[6]	M. Bellassoued, D. Jellali and M. Yamamoto, Lipschitz stability for a hyperbolic inverse problem by finite local boundary data, Appl. Anal., 85 (2006), 1219-1243. doi: 10.1080/00036810600787873.
[7]	M. Bellassoued, D. Jellali and M. Yamamoto, Stability estimate for the hyperbolic inverse boundary value problem by local Dirichlet-to-Neumann map, J. Math. Anal. Appl., 343 (2008), 1036-1046. doi: 10.1016/j.jmaa.2008.01.098.
[8]	M. Bellassoued, Y. Kian and E. Soccorsi, An inverse stability result for non-compactly supported potentials by one arbitrary lateral Neumann observation, J. Differential Equations, 260 (2016), 7535-7562. doi: 10.1016/j.jde.2016.01.033.
[9]	M. Bellassoued, Y. Kian and E. Soccorsi, An inverse problem for the magnetic Schrödinger equation in infinite cylindrical domains, Publ. Res. Inst. Math. Sci., 54 (2018), 679-728. doi: 10.4171/PRIMS/54-4-1.
[10]	I. Ben Aïcha, Stability estimate for a hyperbolic inverse problem with time-dependent coefficient, Inverse Problems, 31 (2015), 21pp. doi: 10.1088/0266-5611/31/12/125010.
[11]	I. Ben Aïcha, Stability estimate for an inverse problem for the Schrödinger equation in a magnetic field with time-dependent coefficient, J. Math. Phys., 58 (2017), 21pp. doi: 10.1063/1.4995606.
[12]	A. L. Bukhgeĭm and M. V. Klibanov, Uniqueness in the large of a class of multidimensional inverse problems, Dokl. Akad. Nauk SSSR, 260 (1981), 269-272.
[13]	A. L. Bukhge${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} }}$m and G. Uhlmann, Recovering a potential from partial Cauchy data, Comm. Partial Differential Equations, 27 (2002), 653-668. doi: 10.1081/PDE-120002868.
[14]	P. Caro and Y. Kian, Determination of convection terms and quasi-linearities appearing in diffusion equations, preprint, arXiv: 1812.08495.
[15]	J. Cheng and M. Yamamoto, The global uniqueness for determining two convection coefficients from Dirichlet to Neumann map in two dimensions, Inverse Problems, 16 (2000), L25–L30. doi: 10.1088/0266-5611/16/3/101.
[16]	J. Cheng and M. Yamamoto, Identification of convection term in a parabolic equation with a single measurement, Nonlinear Anal., 50 (2002), 163-171. doi: 10.1016/S0362-546X(01)00742-8.
[17]	J. Cheng and M. Yamamoto, Determination of two convection coefficients from Dirichlet to Neumann map in the two-dimensional case, SIAM J. Math. Anal., 35 (2004), 1371-1393. doi: 10.1137/S0036141003422497.
[18]	M. Choulli, An abstract inverse problem, J. Appl. Math. Stochastic Anal., 4 (1991), 117-128. doi: 10.1155/S1048953391000084.
[19]	M. Choulli, Abstract inverse problem and application, J. Math. Anal. Appl., 160 (1991), 190-202. doi: 10.1016/0022-247X(91)90299-F.
[20]	M. Choulli and Y. Kian, Stability of the determination of a time-dependent coefficient in parabolic equations, Math. Control Relat. Fields, 3 (2013), 143-160. doi: 10.3934/mcrf.2013.3.143.
[21]	M. Choulli and Y. Kian, Logarithmic stability in determining the time-dependent zero order coefficient in a parabolic equation from a partial Dirichlet-to-Neumann map. Application to the determination of a nonlinear term, J. Math. Pures Appl. (9), 114 (2018), 235–261. doi: 10.1016/j.matpur.2017.12.003.
[22]	M. Choulli, Y. Kian and E. Soccorsi, Stable determination of time-dependent scalar potential from boundary measurements in a periodic quantum waveguide, SIAM J. Math. Anal., 47 (2015), 4536-4558. doi: 10.1137/140986268.
[23]	M. Choulli, Une Introduction aux Problèmes Inverses Elliptiques et Paraboliques, Mathématiques & Applications, 65, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-642-02460-3.
[24]	M. Cristofol and E. Soccorsi, Stability estimate in an inverse problem for non-autonomous magnetic Schrödinger equations, Appl. Anal., 90 (2011), 1499-1520. doi: 10.1080/00036811.2010.524161.
[25]	Z. Cha Deng, J. Ning Yu and Y. Liu, Identifying the coefficient of first-order in parabolic equation from final measurement data, Math. Comput. Simulation, 77 (2008), 421-435. doi: 10.1016/j.matcom.2008.01.002.
[26]	D. Dos Santos Ferreira, C. E. Kenig, J. Sjöstrand and G. Uhlmann, Determining a magnetic Schrödinger operator from partial Cauchy data, Comm. Math. Phys., 271 (2007), 467-488. doi: 10.1007/s00220-006-0151-9.
[27]	G. Eskin, Inverse problems for the Schrödinger equations with time-dependent electromagnetic potentials and the Aharonov-Bohm effect, J. Math. Phys., 49 (2008), 18pp. doi: 10.1063/1.2841329.
[28]	G. Eskin, Inverse hyperbolic problems with time-dependent coefficients, Comm. Partial Differential Equations, 32 (2007), 1737-1758. doi: 10.1080/03605300701382340.
[29]	G. Eskin, Inverse problems for general second order hyperbolic equations with time-dependent coefficients, Bull. Math. Sci., 7 (2017), 247-307. doi: 10.1007/s13373-017-0100-2.
[30]	P. Gaitan and Y. Kian, A stability result for a time-dependent potential in a cylindrical domain, Inverse Problems, 29 (2013), 18pp. doi: 10.1088/0266-5611/29/6/065006.
[31]	L. Hörmander, The Analysis of Linear Partial Differential Operators Ⅲ: Pseudo-Differential Operators, Classics in Mathematics, Springer, Berlin, 2007. doi: 10.1007/978-3-540-49938-1.
[32]	G. Hu and Y. Kian, Determination of singular time-dependent coefficients for wave equations from full and partial data, Inverse Probl. Imaging, 12 (2018), 745-772. doi: 10.3934/ipi.2018032.
[33]	V. Isakov, Completeness of products of solutions and some inverse problems for PDE, J. Differential Equations, 92 (1991), 305-316. doi: 10.1016/0022-0396(91)90051-A.
[34]	V. Isakov, On uniqueness in inverse problems for semilinear parabolic equations, Arch. Rational Mech. Anal., 124 (1993), 1-12. doi: 10.1007/BF00392201.
[35]	V. Isakov, Inverse Problems for Partial Differential Equations, Applied Mathematical Sciences, 127, Springer, New York, 2006. doi: 10.1007/0-387-32183-7.
[36]	A. Katchalov, Y. Kurylev and M. Lassas, Inverse Boundary Spectral Problems, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 123, Chapman & Hall/CRC, Boca Raton, FL, 2001. doi: 10.1201/9781420036220.
[37]	Y. Kian, Unique determination of a time-dependent potential for wave equations from partial data, Ann. Inst. H. Poincaré Anal. Non Linéaire, 34 (2017), 973-990. doi: 10.1016/j.anihpc.2016.07.003.
[38]	Y. Kian, Stability in the determination of a time-dependent coefficient for wave equations from partial data, J. Math. Anal. Appl., 436 (2016), 408-428. doi: 10.1016/j.jmaa.2015.12.018.
[39]	Y. Kian, Recovery of time-dependent damping coefficients and potentials appearing in wave equations from partial data, SIAM J. Math. Anal., 48 (2016), 4021-4046. doi: 10.1137/16M1076708.
[40]	Y. Kian and L. Oksanen, Recovery of time-dependent coefficient on Riemanian manifold for hyperbolic equations, Int. Math. Res. Not. IMRN, 2019 (2017), 5087-5126. doi: 10.1093/imrn/rnx263.
[41]	Y. Kian, Q. Sang Phan and E. Soccorsi, A Carleman estimate for infinite cyclindrical quantum domains and the application to inverse problems, Inverse Problems, 30 (2014), 16pp. doi: 10.1088/0266-5611/30/5/055016.
[42]	Y. Kian, Q. Sang Phan and E. Soccorsi, Hölder stable determination of a quantum scalar potential in unbounded cylindrical domains, J. Math. Anal. Appl., 426 (2015), 194-210. doi: 10.1016/j.jmaa.2015.01.028.
[43]	Y. Kian and E. Soccorsi, Hölder stably determining the time-dependent electromagnetic potential of the Schrödinger equation, SIAM J. Math. Anal., 51 (2019), 627-647. doi: 10.1137/18M1197308.
[44]	V. P. Krishnan and M. Vashisth, An inverse problem for the relativistic Schrödinger equation with partial boundary data, Applicable Analysis, (2019). doi: 10.1080/00036811.2018.1549321.
[45]	J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, Travaux et Recherches Mathématiques, 1, Dunod, Paris, 1968.
[46]	G. Nakamura and S. Sasayama, Inverse boundary value problem for the heat equation with discontinuous coefficients, J. Inverse Ill-Posed Probl., 21 (2013), 217-232. doi: 10.1515/jip-2012-0073.
[47]	V. Pohjola, A uniqueness result for an inverse problem of the steady state convection-diffusion equation, SIAM J. Math. Anal., 47 (2015), 2084-2103. doi: 10.1137/140970926.
[48]	S. RabieniaHaratbar, Support theorem for the light-ray transform of vector fields on Minkowski spaces, Inverse Probl. Imaging, 12 (2018), 293-314. doi: 10.3934/ipi.2018013.
[49]	Ra kesh and W. W. Symes, Uniqueness for an inverse problem for the wave equation, Comm. Partial Differential Equations, 13 (1988), 87-96. doi: 10.1080/03605308808820539.
[50]	R. Salazar, Determination of time-dependent coefficients for a hyperbolic inverse problem, Inverse Problems, 29 (2013), 17pp. doi: 10.1088/0266-5611/29/9/095015.
[51]	P. D. Stefanov, Uniqueness of the multi-dimensional inverse scattering problem for time dependent potentials, Math. Z., 201 (1989), 541-559. doi: 10.1007/BF01215158.
[52]	P. D. Stefanov, Support theorems for the light ray transform on analytic Lorentzian manifolds, Proc. Amer. Math. Soc., 145 (2017), 1259-1274. doi: 10.1090/proc/13117.
[53]	Z. Sun, An inverse boundary value problem for the Schrödinger operator with vector potentials, Trans. Amer. Math. Soc., 338 (1993), 953-969. doi: 10.2307/2154438.
[54]	J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math. (2), 125 (1987), 153–169. doi: 10.2307/1971291.

show all references

References:

[1]	S. A. Avdonin and M. I. Belishev, Dynamical inverse problem for the Schrödinger equation (BC-method), in Proceedings of the St. Petersburg Mathematical Society, Amer. Math. Soc. Transl. Ser., 10, Amer. Math. Soc., Providence, RI, 2005, 1–14. doi: 10.1090/trans2/214/01.
[2]	S. A. Avdonin and T. I. Seidman, Identication of $q(x)$ in $u_t = \Delta u - qu$, from boundary observations, SIAM J. Control Optim., 33 (1995), 1247-1255. doi: 10.1137/S0363012993249729.
[3]	L. Baudouin and J.-P. Puel, Uniqueness and stability in an inverse problem for the Schrödinger equation, Inverse Problems, 18 (2002), 1537-1554. doi: 10.1088/0266-5611/23/3/C01.
[4]	M. I. Belishev, Recent progress in the boundary control method, Inverse Problems, 23 (2007), R1–R67. doi: 10.1088/0266-5611/23/5/R01.
[5]	M. Bellassoued and D. Dos Santos Ferreira, Stable determination of coefficients in the dynamical anisotropic Schrödinger equation from the Dirichlet-to-Neumann map, Inverse Problems, 26 (2010), 30pp. doi: 10.1088/0266-5611/26/12/125010.
[6]	M. Bellassoued, D. Jellali and M. Yamamoto, Lipschitz stability for a hyperbolic inverse problem by finite local boundary data, Appl. Anal., 85 (2006), 1219-1243. doi: 10.1080/00036810600787873.
[7]	M. Bellassoued, D. Jellali and M. Yamamoto, Stability estimate for the hyperbolic inverse boundary value problem by local Dirichlet-to-Neumann map, J. Math. Anal. Appl., 343 (2008), 1036-1046. doi: 10.1016/j.jmaa.2008.01.098.
[8]	M. Bellassoued, Y. Kian and E. Soccorsi, An inverse stability result for non-compactly supported potentials by one arbitrary lateral Neumann observation, J. Differential Equations, 260 (2016), 7535-7562. doi: 10.1016/j.jde.2016.01.033.
[9]	M. Bellassoued, Y. Kian and E. Soccorsi, An inverse problem for the magnetic Schrödinger equation in infinite cylindrical domains, Publ. Res. Inst. Math. Sci., 54 (2018), 679-728. doi: 10.4171/PRIMS/54-4-1.
[10]	I. Ben Aïcha, Stability estimate for a hyperbolic inverse problem with time-dependent coefficient, Inverse Problems, 31 (2015), 21pp. doi: 10.1088/0266-5611/31/12/125010.
[11]	I. Ben Aïcha, Stability estimate for an inverse problem for the Schrödinger equation in a magnetic field with time-dependent coefficient, J. Math. Phys., 58 (2017), 21pp. doi: 10.1063/1.4995606.
[12]	A. L. Bukhgeĭm and M. V. Klibanov, Uniqueness in the large of a class of multidimensional inverse problems, Dokl. Akad. Nauk SSSR, 260 (1981), 269-272.
[13]	A. L. Bukhge${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} }}$m and G. Uhlmann, Recovering a potential from partial Cauchy data, Comm. Partial Differential Equations, 27 (2002), 653-668. doi: 10.1081/PDE-120002868.
[14]	P. Caro and Y. Kian, Determination of convection terms and quasi-linearities appearing in diffusion equations, preprint, arXiv: 1812.08495.
[15]	J. Cheng and M. Yamamoto, The global uniqueness for determining two convection coefficients from Dirichlet to Neumann map in two dimensions, Inverse Problems, 16 (2000), L25–L30. doi: 10.1088/0266-5611/16/3/101.
[16]	J. Cheng and M. Yamamoto, Identification of convection term in a parabolic equation with a single measurement, Nonlinear Anal., 50 (2002), 163-171. doi: 10.1016/S0362-546X(01)00742-8.
[17]	J. Cheng and M. Yamamoto, Determination of two convection coefficients from Dirichlet to Neumann map in the two-dimensional case, SIAM J. Math. Anal., 35 (2004), 1371-1393. doi: 10.1137/S0036141003422497.
[18]	M. Choulli, An abstract inverse problem, J. Appl. Math. Stochastic Anal., 4 (1991), 117-128. doi: 10.1155/S1048953391000084.
[19]	M. Choulli, Abstract inverse problem and application, J. Math. Anal. Appl., 160 (1991), 190-202. doi: 10.1016/0022-247X(91)90299-F.
[20]	M. Choulli and Y. Kian, Stability of the determination of a time-dependent coefficient in parabolic equations, Math. Control Relat. Fields, 3 (2013), 143-160. doi: 10.3934/mcrf.2013.3.143.
[21]	M. Choulli and Y. Kian, Logarithmic stability in determining the time-dependent zero order coefficient in a parabolic equation from a partial Dirichlet-to-Neumann map. Application to the determination of a nonlinear term, J. Math. Pures Appl. (9), 114 (2018), 235–261. doi: 10.1016/j.matpur.2017.12.003.
[22]	M. Choulli, Y. Kian and E. Soccorsi, Stable determination of time-dependent scalar potential from boundary measurements in a periodic quantum waveguide, SIAM J. Math. Anal., 47 (2015), 4536-4558. doi: 10.1137/140986268.
[23]	M. Choulli, Une Introduction aux Problèmes Inverses Elliptiques et Paraboliques, Mathématiques & Applications, 65, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-642-02460-3.
[24]	M. Cristofol and E. Soccorsi, Stability estimate in an inverse problem for non-autonomous magnetic Schrödinger equations, Appl. Anal., 90 (2011), 1499-1520. doi: 10.1080/00036811.2010.524161.
[25]	Z. Cha Deng, J. Ning Yu and Y. Liu, Identifying the coefficient of first-order in parabolic equation from final measurement data, Math. Comput. Simulation, 77 (2008), 421-435. doi: 10.1016/j.matcom.2008.01.002.
[26]	D. Dos Santos Ferreira, C. E. Kenig, J. Sjöstrand and G. Uhlmann, Determining a magnetic Schrödinger operator from partial Cauchy data, Comm. Math. Phys., 271 (2007), 467-488. doi: 10.1007/s00220-006-0151-9.
[27]	G. Eskin, Inverse problems for the Schrödinger equations with time-dependent electromagnetic potentials and the Aharonov-Bohm effect, J. Math. Phys., 49 (2008), 18pp. doi: 10.1063/1.2841329.
[28]	G. Eskin, Inverse hyperbolic problems with time-dependent coefficients, Comm. Partial Differential Equations, 32 (2007), 1737-1758. doi: 10.1080/03605300701382340.
[29]	G. Eskin, Inverse problems for general second order hyperbolic equations with time-dependent coefficients, Bull. Math. Sci., 7 (2017), 247-307. doi: 10.1007/s13373-017-0100-2.
[30]	P. Gaitan and Y. Kian, A stability result for a time-dependent potential in a cylindrical domain, Inverse Problems, 29 (2013), 18pp. doi: 10.1088/0266-5611/29/6/065006.
[31]	L. Hörmander, The Analysis of Linear Partial Differential Operators Ⅲ: Pseudo-Differential Operators, Classics in Mathematics, Springer, Berlin, 2007. doi: 10.1007/978-3-540-49938-1.
[32]	G. Hu and Y. Kian, Determination of singular time-dependent coefficients for wave equations from full and partial data, Inverse Probl. Imaging, 12 (2018), 745-772. doi: 10.3934/ipi.2018032.
[33]	V. Isakov, Completeness of products of solutions and some inverse problems for PDE, J. Differential Equations, 92 (1991), 305-316. doi: 10.1016/0022-0396(91)90051-A.
[34]	V. Isakov, On uniqueness in inverse problems for semilinear parabolic equations, Arch. Rational Mech. Anal., 124 (1993), 1-12. doi: 10.1007/BF00392201.
[35]	V. Isakov, Inverse Problems for Partial Differential Equations, Applied Mathematical Sciences, 127, Springer, New York, 2006. doi: 10.1007/0-387-32183-7.
[36]	A. Katchalov, Y. Kurylev and M. Lassas, Inverse Boundary Spectral Problems, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 123, Chapman & Hall/CRC, Boca Raton, FL, 2001. doi: 10.1201/9781420036220.
[37]	Y. Kian, Unique determination of a time-dependent potential for wave equations from partial data, Ann. Inst. H. Poincaré Anal. Non Linéaire, 34 (2017), 973-990. doi: 10.1016/j.anihpc.2016.07.003.
[38]	Y. Kian, Stability in the determination of a time-dependent coefficient for wave equations from partial data, J. Math. Anal. Appl., 436 (2016), 408-428. doi: 10.1016/j.jmaa.2015.12.018.
[39]	Y. Kian, Recovery of time-dependent damping coefficients and potentials appearing in wave equations from partial data, SIAM J. Math. Anal., 48 (2016), 4021-4046. doi: 10.1137/16M1076708.
[40]	Y. Kian and L. Oksanen, Recovery of time-dependent coefficient on Riemanian manifold for hyperbolic equations, Int. Math. Res. Not. IMRN, 2019 (2017), 5087-5126. doi: 10.1093/imrn/rnx263.
[41]	Y. Kian, Q. Sang Phan and E. Soccorsi, A Carleman estimate for infinite cyclindrical quantum domains and the application to inverse problems, Inverse Problems, 30 (2014), 16pp. doi: 10.1088/0266-5611/30/5/055016.
[42]	Y. Kian, Q. Sang Phan and E. Soccorsi, Hölder stable determination of a quantum scalar potential in unbounded cylindrical domains, J. Math. Anal. Appl., 426 (2015), 194-210. doi: 10.1016/j.jmaa.2015.01.028.
[43]	Y. Kian and E. Soccorsi, Hölder stably determining the time-dependent electromagnetic potential of the Schrödinger equation, SIAM J. Math. Anal., 51 (2019), 627-647. doi: 10.1137/18M1197308.
[44]	V. P. Krishnan and M. Vashisth, An inverse problem for the relativistic Schrödinger equation with partial boundary data, Applicable Analysis, (2019). doi: 10.1080/00036811.2018.1549321.
[45]	J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, Travaux et Recherches Mathématiques, 1, Dunod, Paris, 1968.
[46]	G. Nakamura and S. Sasayama, Inverse boundary value problem for the heat equation with discontinuous coefficients, J. Inverse Ill-Posed Probl., 21 (2013), 217-232. doi: 10.1515/jip-2012-0073.
[47]	V. Pohjola, A uniqueness result for an inverse problem of the steady state convection-diffusion equation, SIAM J. Math. Anal., 47 (2015), 2084-2103. doi: 10.1137/140970926.
[48]	S. RabieniaHaratbar, Support theorem for the light-ray transform of vector fields on Minkowski spaces, Inverse Probl. Imaging, 12 (2018), 293-314. doi: 10.3934/ipi.2018013.
[49]	Ra kesh and W. W. Symes, Uniqueness for an inverse problem for the wave equation, Comm. Partial Differential Equations, 13 (1988), 87-96. doi: 10.1080/03605308808820539.
[50]	R. Salazar, Determination of time-dependent coefficients for a hyperbolic inverse problem, Inverse Problems, 29 (2013), 17pp. doi: 10.1088/0266-5611/29/9/095015.
[51]	P. D. Stefanov, Uniqueness of the multi-dimensional inverse scattering problem for time dependent potentials, Math. Z., 201 (1989), 541-559. doi: 10.1007/BF01215158.
[52]	P. D. Stefanov, Support theorems for the light ray transform on analytic Lorentzian manifolds, Proc. Amer. Math. Soc., 145 (2017), 1259-1274. doi: 10.1090/proc/13117.
[53]	Z. Sun, An inverse boundary value problem for the Schrödinger operator with vector potentials, Trans. Amer. Math. Soc., 338 (1993), 953-969. doi: 10.2307/2154438.
[54]	J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math. (2), 125 (1987), 153–169. doi: 10.2307/1971291.

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