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An adaptive total variational despeckling model based on gray level indicator frame
Stable recovery of a non-compactly supported coefficient of a Schrödinger equation on an infinite waveguide
Université de Tunis El Manar, Ecole Nationale d'Ingénieurs de Tunis, ENIT-LAMSIN, B.P. 37, 1002 Tunis, Tunisia, Aix Marseille Université, Université de Toulon, CNRS, CPT, Marseille, France |
We study the stability issue for the inverse problem of determining a coefficient appearing in a Schrödinger equation defined on an infinite cylindrical waveguide. More precisely, we prove the stable recovery of some general class of non-compactly and non periodic coefficients appearing in an unbounded cylindrical domain. We consider both results of stability from full and partial boundary measurements associated with the so called Dirichlet-to-Neumann map.
References:
[1] |
G. Alessandrini,
Stable determination of conductivity by boundary measurements, Appl. Anal., 27 (1988), 153-172.
doi: 10.1080/00036818808839730. |
[2] |
H. Ammari and G. Uhlmann,
Reconstruction from partial Cauchy data for the Schrödinger equation, Indiana University Math J., 53 (2004), 169-183.
doi: 10.1512/iumj.2004.53.2299. |
[3] |
J. Behrndt and J. Rohleder, Inverse problems with partial data for elliptic operators on unbounded Lipschitz domains, Inverse Problems, 36 (2020), 035009, 18 pp.
doi: 10.1088/1361-6420/ab603d. |
[4] |
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. |
[5] |
M. Bellassoued, Y. Kian and E. Soccorsi,
An inverse problem for the magnetic Schrödinger equation in infinite cylindrical domains, Publ. Research Institute Math. Sci., 54 (2018), 679-728.
doi: 10.4171/PRIMS/54-4-1. |
[6] |
H. Ben Joud, A stability estimate for an inverse problem for the Schrödinger equation in a magnetic field from partial boundary measurements, Inverse Problems, 25 (2009), 045012, 23 pp.
doi: 10.1088/0266-5611/25/4/045012. |
[7] |
A. L. Bukhgeim and G. Uhlmann,
Recovering a potential from partial Cauchy data, Comm. PDE, 27 (2002), 653-668.
doi: 10.1081/PDE-120002868. |
[8] |
A. P. Calderón, On an inverse boundary value problem, Seminar on Numerical Analysis and its Applications to Continuum Physics, Rio de Janeiro, Sociedade Brasileira de Matematica, (1980), 65–73.
doi: 10.1590/S0101-82052006000200002. |
[9] |
P. Caro, D. Dos Santos Ferreira and A. Ruiz,
Stability estimates for the Radon transform with restricted data and applications, Advances in Math., 267 (2014), 523-564.
doi: 10.1016/j.aim.2014.08.009. |
[10] |
P. Caro, D. Dos Santos Ferreira and A. Ruiz,
Stability estimates for the Calderón problem with partial data, J. Diff. Equa., 260 (2016), 2457-2489.
doi: 10.1016/j.jde.2015.10.007. |
[11] |
P. Caro and K. Marinov,
Stability of inverse problems in an infinite slab with partial data, Commun. Partial Diff. Equa., 41 (2016), 683-704.
doi: 10.1080/03605302.2015.1127967. |
[12] |
P. Caro and V. Pohjola, Stability estimates for an inverse problem for the magnetic Schrödinger operator, IMRN, (2015), 11083–11116.
doi: 10.1093/imrn/rnv020. |
[13] |
M. Choulli, Une Introduction Aux Problèmes Inverses Elliptiques et Paraboliques, Mathematics & Applications, Vol. 65, Springer-Verlag, Berlin, 2009.
doi: 10.1007/978-3-642-02460-3. |
[14] |
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. |
[15] |
M. Choulli, Y. Kian and E. Soccorsi,
Double logarithmic stability estimate in the identification of a scalar potential by a partial elliptic Dirichlet-to-Neumann map, Bulletin of the South Ural State University, Ser. Mathematical Modelling, Programming and Computer Software (SUSUMMCS), 8 (2015), 78-94.
doi: 10.14529/mmp150305. |
[16] |
M. Choulli, Y. Kian and E. Soccorsi,
On the Calderón problem in periodic cylindrical domain with partial Dirichlet and Neumann data, Mathematical Methods in the Applied Sciences, 40 (2017), 5959-5974.
doi: 10.1002/mma.4446. |
[17] |
M. Choulli, Y. Kian and E. Soccorsi,
Stability result for elliptic inverse periodic coefficient problem by partial Dirichlet-to-Neumann map, J. Spec. Theory, 8 (2018), 733-768.
doi: 10.4171/JST/212. |
[18] |
M. Choulli and E. Soccorsi,
An inverse anisotropic conductivity problem induced by twisting a homogeneous cylindrical domain, J. Spec. Theory, 5 (2015), 295-329.
doi: 10.4171/JST/99. |
[19] |
O. Yu. Èmanuvilov,
Controllability of evolution equations, Sb. Math., 186 (1995), 879-900.
doi: 10.1070/SM1995v186n06ABEH000047. |
[20] |
O. Yu. Imanuvilov, G. Uhlmann and M. Yamamoto,
The Calderón problem with partial data in two dimensions, Journal American Math. Society, 23 (2010), 655-691.
doi: 10.1090/S0894-0347-10-00656-9. |
[21] |
O. Yu. Imanuvilov and M. Yamamoto,
Lipschitz stability in inverse parabolic problems by the Carleman estimate, Inverse Problems, 14 (1998), 1229-1245.
doi: 10.1088/0266-5611/14/5/009. |
[22] |
O. Kavian, Y. Kian and E. Soccorsi,
Uniqueness and stability results for an inverse spectral problem in a periodic waveguide, J. Math. Pures Appl., 104 (2015), 1160-1189.
doi: 10.1016/j.matpur.2015.09.002. |
[23] |
C. Kenig and M. Salo,
The Calderón problem with partial data on manifolds and applications, Analysis & PDE, 6 (2013), 2003-2048.
doi: 10.2140/apde.2013.6.2003. |
[24] |
C. E. Kenig, J. Sjöstrand and G. Uhlmann,
The Calderón problem with partial data, Ann. of Math., 165 (2007), 567-591.
doi: 10.4007/annals.2007.165.567. |
[25] |
Y. Kian,
Recovery of non-compactly supported coefficients of elliptic equations on an infinite waveguide, Journal of the Institute of Mathematics of Jussieu, 19 (2020), 1573-1600.
doi: 10.1017/S1474748018000488. |
[26] |
Y. Kian,
Determination of non-compactly supported electromagnetic potentials in an unbounded closed waveguide, Revista Matemática Iberoamericana, 36 (2020), 671-710.
doi: 10.4171/rmi/1143. |
[27] |
Y. Kian,
Stability of the determination of a coefficient for wave equations in an infinite waveguide, Inverse Probl. Imaging, 8 (2014), 713-732.
doi: 10.3934/ipi.2014.8.713. |
[28] |
Y. Kian, D. Sambou and E. Soccorsi,
Logarithmic stability inequality in an inverse source problem for the heat equation on a waveguide, Applicable Analysis, 99 (2020), 2210-2228.
doi: 10.1080/00036811.2018.1557324. |
[29] |
Y. Kian, Q. S. Phan and E. Soccorsi, A carleman estimate for infinite cylindrical quantum domains and application to inverse problems, Inverse Problems, 30 (2014), 055016, 16 pp.
doi: 10.1088/0266-5611/30/5/055016. |
[30] |
Y. Kian, Q. S. Phan and E. Soccorsi,
Hölder stable determination of a quantum scalar potential in unbounded cylindrical domains, Jour. Math. Anal. Appl., 426 (2015), 194-210.
doi: 10.1016/j.jmaa.2015.01.028. |
[31] |
K. Krupchyk, M. Lassas and G. Uhlmann,
Inverse problems with Partial data for a magnetic Schrödinger operator in an infinite slab or bounded domain, Comm. Math. Phys., 312 (2012), 87-126.
doi: 10.1007/s00220-012-1431-1. |
[32] |
K. Krupchyk and G. Uhlmann,
Stability estimates for partial data inverse problems for Schrödinger operators in the high frequency limit, J. Math. Pures Appl., 126 (2019), 273-291.
doi: 10.1016/j.matpur.2019.02.017. |
[33] |
X. Li, Inverse boundary value problems with partial data in unbounded domains, Inverse Problems, 28 (2012), 085003, 23 pp.
doi: 10.1088/0266-5611/28/8/085003. |
[34] |
X. Li,
Inverse problem for Schrödinger equations with Yang-Mills potentials in a slab, J. Diff. Equat., 253 (2012), 694-726.
doi: 10.1016/j.jde.2012.04.001. |
[35] |
X. Li and G. Uhlmann,
Inverse Problems with partial data in a Slab, Inverse Problems and Imaging, 4 (2010), 449-462.
doi: 10.3934/ipi.2010.4.449. |
[36] |
J-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Die Grundlehren der mathematischen Wissenschaften, Band 181. Springer-Verlag, New York-Heidelberg, 1972.
doi: 10.1007/978-3-642-65161-8. |
[37] |
J. Sylvester and G. Uhlmann,
A global uniqueness theorem for an inverse boundary value problem, Ann. of Math., 125 (1987), 153-169.
doi: 10.2307/1971291. |
[38] |
G. Uhlmann, Electrical impedance tomography and Calderón's problem, Inverse problems, 25 (2009), 123011, 39 pp.
doi: 10.1088/0266-5611/25/12/123011. |
[39] |
M. S. Zhdanov and G. V. Keller, The Geoelectrical Methods in Geophysical Exploration, Methods in Geochemistry and Geophysics, vol 31 (Amsterdam: Elsevier), (1994). Google Scholar |
[40] |
Y. Zou and Z. Guo,
A review of electrical impedance techniques for breast cancer detection, Med. Eng. Phys., 25 (2003), 79-90.
doi: 10.1016/S1350-4533(02)00194-7. |
show all references
References:
[1] |
G. Alessandrini,
Stable determination of conductivity by boundary measurements, Appl. Anal., 27 (1988), 153-172.
doi: 10.1080/00036818808839730. |
[2] |
H. Ammari and G. Uhlmann,
Reconstruction from partial Cauchy data for the Schrödinger equation, Indiana University Math J., 53 (2004), 169-183.
doi: 10.1512/iumj.2004.53.2299. |
[3] |
J. Behrndt and J. Rohleder, Inverse problems with partial data for elliptic operators on unbounded Lipschitz domains, Inverse Problems, 36 (2020), 035009, 18 pp.
doi: 10.1088/1361-6420/ab603d. |
[4] |
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. |
[5] |
M. Bellassoued, Y. Kian and E. Soccorsi,
An inverse problem for the magnetic Schrödinger equation in infinite cylindrical domains, Publ. Research Institute Math. Sci., 54 (2018), 679-728.
doi: 10.4171/PRIMS/54-4-1. |
[6] |
H. Ben Joud, A stability estimate for an inverse problem for the Schrödinger equation in a magnetic field from partial boundary measurements, Inverse Problems, 25 (2009), 045012, 23 pp.
doi: 10.1088/0266-5611/25/4/045012. |
[7] |
A. L. Bukhgeim and G. Uhlmann,
Recovering a potential from partial Cauchy data, Comm. PDE, 27 (2002), 653-668.
doi: 10.1081/PDE-120002868. |
[8] |
A. P. Calderón, On an inverse boundary value problem, Seminar on Numerical Analysis and its Applications to Continuum Physics, Rio de Janeiro, Sociedade Brasileira de Matematica, (1980), 65–73.
doi: 10.1590/S0101-82052006000200002. |
[9] |
P. Caro, D. Dos Santos Ferreira and A. Ruiz,
Stability estimates for the Radon transform with restricted data and applications, Advances in Math., 267 (2014), 523-564.
doi: 10.1016/j.aim.2014.08.009. |
[10] |
P. Caro, D. Dos Santos Ferreira and A. Ruiz,
Stability estimates for the Calderón problem with partial data, J. Diff. Equa., 260 (2016), 2457-2489.
doi: 10.1016/j.jde.2015.10.007. |
[11] |
P. Caro and K. Marinov,
Stability of inverse problems in an infinite slab with partial data, Commun. Partial Diff. Equa., 41 (2016), 683-704.
doi: 10.1080/03605302.2015.1127967. |
[12] |
P. Caro and V. Pohjola, Stability estimates for an inverse problem for the magnetic Schrödinger operator, IMRN, (2015), 11083–11116.
doi: 10.1093/imrn/rnv020. |
[13] |
M. Choulli, Une Introduction Aux Problèmes Inverses Elliptiques et Paraboliques, Mathematics & Applications, Vol. 65, Springer-Verlag, Berlin, 2009.
doi: 10.1007/978-3-642-02460-3. |
[14] |
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. |
[15] |
M. Choulli, Y. Kian and E. Soccorsi,
Double logarithmic stability estimate in the identification of a scalar potential by a partial elliptic Dirichlet-to-Neumann map, Bulletin of the South Ural State University, Ser. Mathematical Modelling, Programming and Computer Software (SUSUMMCS), 8 (2015), 78-94.
doi: 10.14529/mmp150305. |
[16] |
M. Choulli, Y. Kian and E. Soccorsi,
On the Calderón problem in periodic cylindrical domain with partial Dirichlet and Neumann data, Mathematical Methods in the Applied Sciences, 40 (2017), 5959-5974.
doi: 10.1002/mma.4446. |
[17] |
M. Choulli, Y. Kian and E. Soccorsi,
Stability result for elliptic inverse periodic coefficient problem by partial Dirichlet-to-Neumann map, J. Spec. Theory, 8 (2018), 733-768.
doi: 10.4171/JST/212. |
[18] |
M. Choulli and E. Soccorsi,
An inverse anisotropic conductivity problem induced by twisting a homogeneous cylindrical domain, J. Spec. Theory, 5 (2015), 295-329.
doi: 10.4171/JST/99. |
[19] |
O. Yu. Èmanuvilov,
Controllability of evolution equations, Sb. Math., 186 (1995), 879-900.
doi: 10.1070/SM1995v186n06ABEH000047. |
[20] |
O. Yu. Imanuvilov, G. Uhlmann and M. Yamamoto,
The Calderón problem with partial data in two dimensions, Journal American Math. Society, 23 (2010), 655-691.
doi: 10.1090/S0894-0347-10-00656-9. |
[21] |
O. Yu. Imanuvilov and M. Yamamoto,
Lipschitz stability in inverse parabolic problems by the Carleman estimate, Inverse Problems, 14 (1998), 1229-1245.
doi: 10.1088/0266-5611/14/5/009. |
[22] |
O. Kavian, Y. Kian and E. Soccorsi,
Uniqueness and stability results for an inverse spectral problem in a periodic waveguide, J. Math. Pures Appl., 104 (2015), 1160-1189.
doi: 10.1016/j.matpur.2015.09.002. |
[23] |
C. Kenig and M. Salo,
The Calderón problem with partial data on manifolds and applications, Analysis & PDE, 6 (2013), 2003-2048.
doi: 10.2140/apde.2013.6.2003. |
[24] |
C. E. Kenig, J. Sjöstrand and G. Uhlmann,
The Calderón problem with partial data, Ann. of Math., 165 (2007), 567-591.
doi: 10.4007/annals.2007.165.567. |
[25] |
Y. Kian,
Recovery of non-compactly supported coefficients of elliptic equations on an infinite waveguide, Journal of the Institute of Mathematics of Jussieu, 19 (2020), 1573-1600.
doi: 10.1017/S1474748018000488. |
[26] |
Y. Kian,
Determination of non-compactly supported electromagnetic potentials in an unbounded closed waveguide, Revista Matemática Iberoamericana, 36 (2020), 671-710.
doi: 10.4171/rmi/1143. |
[27] |
Y. Kian,
Stability of the determination of a coefficient for wave equations in an infinite waveguide, Inverse Probl. Imaging, 8 (2014), 713-732.
doi: 10.3934/ipi.2014.8.713. |
[28] |
Y. Kian, D. Sambou and E. Soccorsi,
Logarithmic stability inequality in an inverse source problem for the heat equation on a waveguide, Applicable Analysis, 99 (2020), 2210-2228.
doi: 10.1080/00036811.2018.1557324. |
[29] |
Y. Kian, Q. S. Phan and E. Soccorsi, A carleman estimate for infinite cylindrical quantum domains and application to inverse problems, Inverse Problems, 30 (2014), 055016, 16 pp.
doi: 10.1088/0266-5611/30/5/055016. |
[30] |
Y. Kian, Q. S. Phan and E. Soccorsi,
Hölder stable determination of a quantum scalar potential in unbounded cylindrical domains, Jour. Math. Anal. Appl., 426 (2015), 194-210.
doi: 10.1016/j.jmaa.2015.01.028. |
[31] |
K. Krupchyk, M. Lassas and G. Uhlmann,
Inverse problems with Partial data for a magnetic Schrödinger operator in an infinite slab or bounded domain, Comm. Math. Phys., 312 (2012), 87-126.
doi: 10.1007/s00220-012-1431-1. |
[32] |
K. Krupchyk and G. Uhlmann,
Stability estimates for partial data inverse problems for Schrödinger operators in the high frequency limit, J. Math. Pures Appl., 126 (2019), 273-291.
doi: 10.1016/j.matpur.2019.02.017. |
[33] |
X. Li, Inverse boundary value problems with partial data in unbounded domains, Inverse Problems, 28 (2012), 085003, 23 pp.
doi: 10.1088/0266-5611/28/8/085003. |
[34] |
X. Li,
Inverse problem for Schrödinger equations with Yang-Mills potentials in a slab, J. Diff. Equat., 253 (2012), 694-726.
doi: 10.1016/j.jde.2012.04.001. |
[35] |
X. Li and G. Uhlmann,
Inverse Problems with partial data in a Slab, Inverse Problems and Imaging, 4 (2010), 449-462.
doi: 10.3934/ipi.2010.4.449. |
[36] |
J-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Die Grundlehren der mathematischen Wissenschaften, Band 181. Springer-Verlag, New York-Heidelberg, 1972.
doi: 10.1007/978-3-642-65161-8. |
[37] |
J. Sylvester and G. Uhlmann,
A global uniqueness theorem for an inverse boundary value problem, Ann. of Math., 125 (1987), 153-169.
doi: 10.2307/1971291. |
[38] |
G. Uhlmann, Electrical impedance tomography and Calderón's problem, Inverse problems, 25 (2009), 123011, 39 pp.
doi: 10.1088/0266-5611/25/12/123011. |
[39] |
M. S. Zhdanov and G. V. Keller, The Geoelectrical Methods in Geophysical Exploration, Methods in Geochemistry and Geophysics, vol 31 (Amsterdam: Elsevier), (1994). Google Scholar |
[40] |
Y. Zou and Z. Guo,
A review of electrical impedance techniques for breast cancer detection, Med. Eng. Phys., 25 (2003), 79-90.
doi: 10.1016/S1350-4533(02)00194-7. |
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