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February  2019, 13(1): 81-91. doi: 10.3934/ipi.2019005

Recovering two coefficients in an elliptic equation via phaseless information

Vladimir G. Romanov 1, and  Masahiro Yamamoto 2,3,

1. 

Sobolev Institute of Mathematics, Siberian Division of Russian Academy of Sciences, Acad. Koptyug prospekt 4, 630090 Novosibirsk, Russia
2. 

Department of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo, 153, Japan
3. 

Peoples' Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation

Received  January 2018 Revised  April 2018 Published  December 2018

For fixed $y \in \mathbb{R}^3$, we consider the equation $L u+k^2u = - δ(x-y), \>x \in \mathbb{R}^3$, where $L=\text{div}(n(x)^{-2}\nabla)+q(x)$, $k >0$ is a frequency, $n(x)$ is a refraction index and $q(x)$ is a potential. Assuming that the refraction index $n(x)$ is different from $1$ only inside a bounded compact domain $Ω$ with a smooth boundary $S$ and the potential $q(x)$ vanishes outside of the same domain, we study an inverse problem of finding both coefficients inside $Ω$ from some given information on solutions of the elliptic equation. Namely, it is supposed that the point source located at point $y \in S$ is a variable parameter of the problem. Then for the solution $u(x,y,k)$ of the above equation satisfying the radiation condition, we assume to be given the following phaseless information $f(x,y,k)=|u(x,y,k)|^2$ for all $x,y \in S$ and for all $k≥ k_0>0$, where $k_0$ is some constant. We prove that this phaseless information uniquely determines both coefficients $n(x)$ and $q(x)$ inside $Ω$.

Keywords: Inverse problem, elliptic equation, uniqueness, phaseless information.
Mathematics Subject Classification: 35R30.
Citation: Vladimir G. Romanov, Masahiro Yamamoto. Recovering two coefficients in an elliptic equation via phaseless information. Inverse Problems and Imaging, 2019, 13 (1) : 81-91. doi: 10.3934/ipi.2019005
References:
[1]

J. N. Bernstein and M. L. Gerver, On the problem of integral geometry for a family of geodesics and the inverse kinematics seismic problem, Dokl. Akad. Nauk SSSR, 243 (1978), 302-305 (in Russian).

[2]

K. Chadan and P. S. Sabatier, Inverse Problems in Quantum Scattering Theory, Texts and Monographs in Physics, Springer-Verlag, New York - Berlin, 1977.

[3]

M. V. Klibanov, Phaseless inverse scattering problems in three dimensions, SIAM J. Appl. Math., 74 (2014), 392-410.  doi: 10.1137/130926250.

[4]

M. V. Klibanov, On the first solution of a long standing problem: Uniqueness of the phaseless quantum inverse scattering problem in 3-d, Applied Mathematics Letters, 37 (2014), 82-85.  doi: 10.1016/j.aml.2014.06.005.

[5]

M. V. Klibanov, Uniqueness of two phaseless non-overdetermined inverse acoustics problems in 3-d, Applicable Analysis, 93 (2014), 1135-1149.  doi: 10.1080/00036811.2013.818136.

[6]

M. V. Klibanov, A phaseless inverse scattering problem for the 3-D Helmholtz equation, Inverse Problems and Imaging, 11 (2017), 263-276.  doi: 10.3934/ipi.2017013.

[7]

M. V. Klibanov and V. G. Romanov, The first solution of a long standing problem: Reconstruction formula for a 3-d phaseless inverse scattering problem for the Schrödinger equation, J. Inverse and Ill-Posed Problems, 23 (2015), 415-428.  doi: 10.1515/jiip-2015-0025.

[8]

M. V. Klibanov and V. G. Romanov, Explicit solution of 3-D phaseless inverse scattering problems for the Schrödinger equation: The plane wave case, Eurasian J. of Math. and Comp. Appl., 3 (2015), 48-63. 

[9]

M. V. Klibanov and V. G. Romanov, Explicit formula for the solution of the phaseless inverse scattering problem of imaging of nano structures, J. of Inverse and Ill-Posed Problems, 23 (2015), 187-193.  doi: 10.1515/jiip-2015-0004.

[10]

M. V. Klibanov and V. G. Romanov, Two reconstruction procedures for a 3-d phaseless inverse scattering problem for the generalized Helmholtz equation, Inverse Problems, 32 (2016), 015005 (16pp). doi: 10.1088/0266-5611/32/1/015005.

[11]

M. V. Klibanov and V. G. Romanov, Reconstruction procedures for two inverse scattering problem without the phase information, SIAM J. Appl. Math., 76 (2016), 178-196.  doi: 10.1137/15M1022367.

[12]

M. V. Klibanov and V. G. Romanov, Uniqueness of a 3-D coefficient inverse scattering problem without the phase information, Inverse Problems, 33 (2017), 095007 (10 pp), https://doi.org/10.1088/1361-6420/aa7a18 doi: 10.1088/1361-6420/aa7a18.

[13]

M. M. Lavrent'ev and V. G. Romanov, On three linearized inverse problems for hyperbolic equations, Soviet Math. Dokl., 7 (1966), 1650-1652. 

[14]

M. M. Lavrent'ev, V. G. Romanov and S. P. Shishat·skiǐ, Ill-Posed Problems of Mathematical Physics and Analysis, Transl. of Math. Monographs, Vol. 64, AMS, Providence, Rhode Island, 1986.

[15]

R. G. Mukhometov, The reconstruction problem of a two-dimensional Riemannian metric and integral geometry, Dokl. Akad. Nauk SSSR, 232 (1977), 32-35. 

[16]

R. G. Mukhometov and V. G. Romanov, On the problem of determining an isotropic Riemannian metric in $n$-dimensional space, Dokl. Akad. Nauk SSSR, 243 (1978), 41-44. 

[17]

R. G. Newton, Inverse Schrödinger Scattering in Three Dimensions, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1989. doi: 10.1007/978-3-642-83671-8.

[18]

R. G. Novikov, Formulas for phase recovering from phaseless scattering data at fixed frequency, Bulletin des Sciences Mathématiques, 139 (2015), 923-936.  doi: 10.1016/j.bulsci.2015.04.005.

[19]

R. G. Novikov, Phaseless inverse scattering in the one-dimensional case, Eurasian J. of Math. and Comp. Appl., 3 (2015), 64-70. 

[20]

R. G. Novikov, Explicit formulas and global uniqueness for phaseless inverse scattering in multidimensions, J. Geometrical Analysis, 26 (2016), 346-359.  doi: 10.1007/s12220-014-9553-7.

[21]

V. G. Romanov, Reconstructing a function by means of integrals along a family of curves, Siberian Math. J., 8 (1967), 923-925. 

[22]

V. G. Romanov, Integral Geometry and Inverse Problems for Hyperbolic Equations, Springer-Verlag, Springers Tracts in Natural Philosophy, Vol. 26, Berlin, 1974.

[23]

V. G. Romanov, Integral geometry on geodesics of the isotropic Riemannian metric, Soviet Math. Dokl., 19 (1978), 847-851. 

[24]

V. G. Romanov, Inverse Problems of Mathematical Physics, VNU Science Press, Utrecht, 1987.

[25]

V. G. Romanov, Investigation Methods for Inverse Problems, VSP, Utrecht, 2002. doi: 10.1515/9783110943849.

[26]

V. G. Romanov, The problem of recovering the permittivity coefficient from the modulus of the scattered electromagnetic field, Siberian Math. J., 58 (2017), 711-717.  doi: 10.1134/s0037446617040176.

[27]

V. G. Romanov, Problem of determining the permittivity in the stationary system of Maxwell equations, Doklady Math., 95 (2017), 230-234.  doi: 10.1134/s1064562417030164.

[28]

B. R. Vainberg Asymptotic Methods in Equations of Mathematical Physics, Gordon and Breach Science Publishers, New York, 1989.

show all references

References:
[1]

J. N. Bernstein and M. L. Gerver, On the problem of integral geometry for a family of geodesics and the inverse kinematics seismic problem, Dokl. Akad. Nauk SSSR, 243 (1978), 302-305 (in Russian).

[2]

K. Chadan and P. S. Sabatier, Inverse Problems in Quantum Scattering Theory, Texts and Monographs in Physics, Springer-Verlag, New York - Berlin, 1977.

[3]

M. V. Klibanov, Phaseless inverse scattering problems in three dimensions, SIAM J. Appl. Math., 74 (2014), 392-410.  doi: 10.1137/130926250.

[4]

M. V. Klibanov, On the first solution of a long standing problem: Uniqueness of the phaseless quantum inverse scattering problem in 3-d, Applied Mathematics Letters, 37 (2014), 82-85.  doi: 10.1016/j.aml.2014.06.005.

[5]

M. V. Klibanov, Uniqueness of two phaseless non-overdetermined inverse acoustics problems in 3-d, Applicable Analysis, 93 (2014), 1135-1149.  doi: 10.1080/00036811.2013.818136.

[6]

M. V. Klibanov, A phaseless inverse scattering problem for the 3-D Helmholtz equation, Inverse Problems and Imaging, 11 (2017), 263-276.  doi: 10.3934/ipi.2017013.

[7]

M. V. Klibanov and V. G. Romanov, The first solution of a long standing problem: Reconstruction formula for a 3-d phaseless inverse scattering problem for the Schrödinger equation, J. Inverse and Ill-Posed Problems, 23 (2015), 415-428.  doi: 10.1515/jiip-2015-0025.

[8]

M. V. Klibanov and V. G. Romanov, Explicit solution of 3-D phaseless inverse scattering problems for the Schrödinger equation: The plane wave case, Eurasian J. of Math. and Comp. Appl., 3 (2015), 48-63. 

[9]

M. V. Klibanov and V. G. Romanov, Explicit formula for the solution of the phaseless inverse scattering problem of imaging of nano structures, J. of Inverse and Ill-Posed Problems, 23 (2015), 187-193.  doi: 10.1515/jiip-2015-0004.

[10]

M. V. Klibanov and V. G. Romanov, Two reconstruction procedures for a 3-d phaseless inverse scattering problem for the generalized Helmholtz equation, Inverse Problems, 32 (2016), 015005 (16pp). doi: 10.1088/0266-5611/32/1/015005.

[11]

M. V. Klibanov and V. G. Romanov, Reconstruction procedures for two inverse scattering problem without the phase information, SIAM J. Appl. Math., 76 (2016), 178-196.  doi: 10.1137/15M1022367.

[12]

M. V. Klibanov and V. G. Romanov, Uniqueness of a 3-D coefficient inverse scattering problem without the phase information, Inverse Problems, 33 (2017), 095007 (10 pp), https://doi.org/10.1088/1361-6420/aa7a18 doi: 10.1088/1361-6420/aa7a18.

[13]

M. M. Lavrent'ev and V. G. Romanov, On three linearized inverse problems for hyperbolic equations, Soviet Math. Dokl., 7 (1966), 1650-1652. 

[14]

M. M. Lavrent'ev, V. G. Romanov and S. P. Shishat·skiǐ, Ill-Posed Problems of Mathematical Physics and Analysis, Transl. of Math. Monographs, Vol. 64, AMS, Providence, Rhode Island, 1986.

[15]

R. G. Mukhometov, The reconstruction problem of a two-dimensional Riemannian metric and integral geometry, Dokl. Akad. Nauk SSSR, 232 (1977), 32-35. 

[16]

R. G. Mukhometov and V. G. Romanov, On the problem of determining an isotropic Riemannian metric in $n$-dimensional space, Dokl. Akad. Nauk SSSR, 243 (1978), 41-44. 

[17]

R. G. Newton, Inverse Schrödinger Scattering in Three Dimensions, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1989. doi: 10.1007/978-3-642-83671-8.

[18]

R. G. Novikov, Formulas for phase recovering from phaseless scattering data at fixed frequency, Bulletin des Sciences Mathématiques, 139 (2015), 923-936.  doi: 10.1016/j.bulsci.2015.04.005.

[19]

R. G. Novikov, Phaseless inverse scattering in the one-dimensional case, Eurasian J. of Math. and Comp. Appl., 3 (2015), 64-70. 

[20]

R. G. Novikov, Explicit formulas and global uniqueness for phaseless inverse scattering in multidimensions, J. Geometrical Analysis, 26 (2016), 346-359.  doi: 10.1007/s12220-014-9553-7.

[21]

V. G. Romanov, Reconstructing a function by means of integrals along a family of curves, Siberian Math. J., 8 (1967), 923-925. 

[22]

V. G. Romanov, Integral Geometry and Inverse Problems for Hyperbolic Equations, Springer-Verlag, Springers Tracts in Natural Philosophy, Vol. 26, Berlin, 1974.

[23]

V. G. Romanov, Integral geometry on geodesics of the isotropic Riemannian metric, Soviet Math. Dokl., 19 (1978), 847-851. 

[24]

V. G. Romanov, Inverse Problems of Mathematical Physics, VNU Science Press, Utrecht, 1987.

[25]

V. G. Romanov, Investigation Methods for Inverse Problems, VSP, Utrecht, 2002. doi: 10.1515/9783110943849.

[26]

V. G. Romanov, The problem of recovering the permittivity coefficient from the modulus of the scattered electromagnetic field, Siberian Math. J., 58 (2017), 711-717.  doi: 10.1134/s0037446617040176.

[27]

V. G. Romanov, Problem of determining the permittivity in the stationary system of Maxwell equations, Doklady Math., 95 (2017), 230-234.  doi: 10.1134/s1064562417030164.

[28]

B. R. Vainberg Asymptotic Methods in Equations of Mathematical Physics, Gordon and Breach Science Publishers, New York, 1989.

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