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A content-adaptive unstructured grid based integral equation method with the TV regularization for SPECT reconstruction
A partial data inverse problem for the convection-diffusion equation
1. | TIFR Centre for Applicable Mathematics, Bangalore 560065, India |
2. | Beijing Computational Science Research Center, Beijing 100193, China |
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.
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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