September  2020, 28(3): 1239-1255. doi: 10.3934/era.2020068

On simultaneous recovery of sources/obstacles and surrounding mediums by boundary measurements

1. 

School of Mathematics and Statistics, Key Laboratory of Hunan Province for Statistical Learning and Intelligent Computation, Hunan University of Technology and Business, Changsha 410205, China

2. 

School of Mathematics and Statistics, Central South University, Changsha, Hunan, China

3. 

Department of Mathematics, Hong Kong Baptist University, Kowloon, Hong Kong SAR, China

4. 

School of Mathematics, Hunan Institute of Science and Technology, Yueyang 414006, Hunan, China

* Corresponding author: youjundeng@csu.edu.cn, dengyijun_001@163.com

Received  May 2020 Revised  June 2020 Published  July 2020

Fund Project: The work of X. Fang was supported by Humanities and Social Sciences Foundation of the Ministry of Education no. 20YJC910005, Major Project for National Natural Science Foundation of China no. 71991465, PSCF of Hunan No. 18YBQ077 and RFEB of Hunan No. 18B337. The work of Y. Deng was supported by NSF grant of China No. 11971487 and NSF grant of Hunan No.2020JJ2038 and No.2017JJ3432. The work of Z. Zhang was supported by Scientific Research Fund of Hunan Provincial Education Department No.18A325, Open project of Hainan Key Laboratory of Computing Science and Application No.JSKX201905, NNSF of China Grant Nos.11671101. Also, this work was partially supported by Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering of Changsha University of Science and Technology Grant No. 2018MMAEZD05

We consider a particular type of inverse problems where an unknown source embedded in an inhomogeneous medium, and one intends to recover the source and/or the medium by knowledge of the wave field (generated by the unknown source) outside the medium. This type of inverse problems arises in many applications of practical importance, including photoacoustic and thermoacoustic tomography, brain imaging and geomagnetic anomaly detections. We survey the recent mathematical developments on this type of inverse problems. We discuss the mathematical tools developed for effectively tackling this type of inverse problems. We also discuss a related inverse problem of recovering an embedded obstacle and its surrounding medium by active measurements.

Citation: Xiaoping Fang, Youjun Deng, Wing-Yan Tsui, Zaiyun Zhang. On simultaneous recovery of sources/obstacles and surrounding mediums by boundary measurements. Electronic Research Archive, 2020, 28 (3) : 1239-1255. doi: 10.3934/era.2020068
References:
[1]

H. AmmariY. Deng and P. Millien, Surface plasmon resonance of nanoparticles and applications in imaging, Arch. Ration. Mech. Anal., 220 (2016), 109-153.  doi: 10.1007/s00205-015-0928-0.  Google Scholar

[2] G. BackusR. Parker and C. Constable, Foundations of Geomagnetism, Cambridge University Press, 1996.   Google Scholar
[3]

X. CaoY.-H. Lin and H. Liu, Simultaneously recovering potentials and embedded obstacles for anisotropic fractional Schrödinger operators, Inverse Probl. Imaging, 13 (2019), 197-210.  doi: 10.3934/ipi.2019011.  Google Scholar

[4]

X. Cao and H. Liu, Determining a fractional Helmholtz system with unknown source and medium parameter, preprint, arXiv: 1803.09538.  Google Scholar

[5]

K. ChanK. ZhangX. LiaoG. Schubert and J. Zou, A three-dimensional spherical nonlinear interface dynamo, Astrophys. J., 596 (2003), 663-679.   Google Scholar

[6]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Applied Mathematical Sciences, 93, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-03537-5.  Google Scholar

[7]

Y. DengJ. Li and H. Liu, On identifying magnetized anomalies using geomagnetic monitoring, Arch. Ration. Mech. Anal., 231 (2019), 153-187.  doi: 10.1007/s00205-018-1276-7.  Google Scholar

[8]

Y. DengJ. Li and H. Liu, On identifying magnetized anomalies using geomagnetic monitoring within a magnetohydrodynamic model, Arch. Ration. Mech. Anal., 235 (2020), 691-721.  doi: 10.1007/s00205-019-01429-x.  Google Scholar

[9]

Y. Deng, H. Liu and W. Tsui, Identifying variations of magnetic anomalies using geomagnetic monitoring, Discrete & Continuous Dynamical Systems - A, 40 (2020), in press. Google Scholar

[10]

Y. DengH. Liu and X. Liu, Recovery of an embedded obstacle and the surrounding medium for Maxwell's system, J. Differential Equations, 267 (2019), 2192-2209.  doi: 10.1016/j.jde.2019.03.009.  Google Scholar

[11]

Y. DengH. Liu and G. Uhlmann, On an inverse boundary problem arising in brain imaging, J. Differential Equations, 267 (2019), 2471-2502.  doi: 10.1016/j.jde.2019.03.019.  Google Scholar

[12]

X. Fang and Y. Deng, Uniqueness on recovery of piecewise constant conductivity and inner core with one measurement, Inverse Probl. Imaging, 12 (2018), 733-743.  doi: 10.3934/ipi.2018031.  Google Scholar

[13]

R. Hollerbach and C. A. Jones, A geodynamo model incorporating a finitely conducting inner core, Phys. Earth Planet. Inter., 75 (1993), 317-327.  doi: 10.1016/0031-9201(93)90007-V.  Google Scholar

[14]

J. LiH. Liu and S. Ma, Determining a random Schrödinger equation with unknown source and potential, SIAM J. Math. Anal., 51 (2019), 3465-3491.  doi: 10.1137/18M1225276.  Google Scholar

[15]

J. Li, H. Liu and S. Ma, Determining a random Schrödinger operator: Both potential and source are random, preprint, arXiv: 1906.01240 Google Scholar

[16]

H. Liu and X. Liu, Recovery of an embedded obstacle and its surrounding medium from formally determined scattering data, Inverse Problems, 33 (2017), 20pp. doi: 10.1088/1361-6420/aa6770.  Google Scholar

[17]

H. Liu and S. Ma, Determining a random source in a Schrödinger equation involving an unknown potential, preprint, arXiv: 2005.04984 Google Scholar

[18]

H. LiuL. Rondi and J. Xiao, Mosco convergence for $H(curl)$ spaces, higher integrability for Maxwell's equations, and stability in direct and inverse EM scattering problems, J. Eur. Math. Soc. (JEMS), 21 (2019), 2945-2993.  doi: 10.4171/JEMS/895.  Google Scholar

[19]

H. Liu and G. Uhmann, Determining both sound speed and internal source in thermo- and photo-acoustic tomography, Inverse Problems, 31 (2015), 10pp. doi: 10.1088/0266-5611/31/10/105005.  Google Scholar

[20]

J.-C. Nédélec, Acoustic and Electromagnetic Equations. Integral Representations for Harmonic Problems, Applied Mathematical Sciences, 144, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4757-4393-7.  Google Scholar

[21]

P. OlaL. Päivärinta and E. Somersalo, An inverse boundary value problem in electrodynamics, Duke Math. J., 70 (1993), 617-653.  doi: 10.1215/S0012-7094-93-07014-7.  Google Scholar

[22]

R. H. Torres, Maxwell's equations and dielectric obstacles with Lipschitz boundaries, J. London Math. Soc. (2), 57 (1998), 157–169. doi: 10.1112/S0024610798005900.  Google Scholar

show all references

References:
[1]

H. AmmariY. Deng and P. Millien, Surface plasmon resonance of nanoparticles and applications in imaging, Arch. Ration. Mech. Anal., 220 (2016), 109-153.  doi: 10.1007/s00205-015-0928-0.  Google Scholar

[2] G. BackusR. Parker and C. Constable, Foundations of Geomagnetism, Cambridge University Press, 1996.   Google Scholar
[3]

X. CaoY.-H. Lin and H. Liu, Simultaneously recovering potentials and embedded obstacles for anisotropic fractional Schrödinger operators, Inverse Probl. Imaging, 13 (2019), 197-210.  doi: 10.3934/ipi.2019011.  Google Scholar

[4]

X. Cao and H. Liu, Determining a fractional Helmholtz system with unknown source and medium parameter, preprint, arXiv: 1803.09538.  Google Scholar

[5]

K. ChanK. ZhangX. LiaoG. Schubert and J. Zou, A three-dimensional spherical nonlinear interface dynamo, Astrophys. J., 596 (2003), 663-679.   Google Scholar

[6]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Applied Mathematical Sciences, 93, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-03537-5.  Google Scholar

[7]

Y. DengJ. Li and H. Liu, On identifying magnetized anomalies using geomagnetic monitoring, Arch. Ration. Mech. Anal., 231 (2019), 153-187.  doi: 10.1007/s00205-018-1276-7.  Google Scholar

[8]

Y. DengJ. Li and H. Liu, On identifying magnetized anomalies using geomagnetic monitoring within a magnetohydrodynamic model, Arch. Ration. Mech. Anal., 235 (2020), 691-721.  doi: 10.1007/s00205-019-01429-x.  Google Scholar

[9]

Y. Deng, H. Liu and W. Tsui, Identifying variations of magnetic anomalies using geomagnetic monitoring, Discrete & Continuous Dynamical Systems - A, 40 (2020), in press. Google Scholar

[10]

Y. DengH. Liu and X. Liu, Recovery of an embedded obstacle and the surrounding medium for Maxwell's system, J. Differential Equations, 267 (2019), 2192-2209.  doi: 10.1016/j.jde.2019.03.009.  Google Scholar

[11]

Y. DengH. Liu and G. Uhlmann, On an inverse boundary problem arising in brain imaging, J. Differential Equations, 267 (2019), 2471-2502.  doi: 10.1016/j.jde.2019.03.019.  Google Scholar

[12]

X. Fang and Y. Deng, Uniqueness on recovery of piecewise constant conductivity and inner core with one measurement, Inverse Probl. Imaging, 12 (2018), 733-743.  doi: 10.3934/ipi.2018031.  Google Scholar

[13]

R. Hollerbach and C. A. Jones, A geodynamo model incorporating a finitely conducting inner core, Phys. Earth Planet. Inter., 75 (1993), 317-327.  doi: 10.1016/0031-9201(93)90007-V.  Google Scholar

[14]

J. LiH. Liu and S. Ma, Determining a random Schrödinger equation with unknown source and potential, SIAM J. Math. Anal., 51 (2019), 3465-3491.  doi: 10.1137/18M1225276.  Google Scholar

[15]

J. Li, H. Liu and S. Ma, Determining a random Schrödinger operator: Both potential and source are random, preprint, arXiv: 1906.01240 Google Scholar

[16]

H. Liu and X. Liu, Recovery of an embedded obstacle and its surrounding medium from formally determined scattering data, Inverse Problems, 33 (2017), 20pp. doi: 10.1088/1361-6420/aa6770.  Google Scholar

[17]

H. Liu and S. Ma, Determining a random source in a Schrödinger equation involving an unknown potential, preprint, arXiv: 2005.04984 Google Scholar

[18]

H. LiuL. Rondi and J. Xiao, Mosco convergence for $H(curl)$ spaces, higher integrability for Maxwell's equations, and stability in direct and inverse EM scattering problems, J. Eur. Math. Soc. (JEMS), 21 (2019), 2945-2993.  doi: 10.4171/JEMS/895.  Google Scholar

[19]

H. Liu and G. Uhmann, Determining both sound speed and internal source in thermo- and photo-acoustic tomography, Inverse Problems, 31 (2015), 10pp. doi: 10.1088/0266-5611/31/10/105005.  Google Scholar

[20]

J.-C. Nédélec, Acoustic and Electromagnetic Equations. Integral Representations for Harmonic Problems, Applied Mathematical Sciences, 144, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4757-4393-7.  Google Scholar

[21]

P. OlaL. Päivärinta and E. Somersalo, An inverse boundary value problem in electrodynamics, Duke Math. J., 70 (1993), 617-653.  doi: 10.1215/S0012-7094-93-07014-7.  Google Scholar

[22]

R. H. Torres, Maxwell's equations and dielectric obstacles with Lipschitz boundaries, J. London Math. Soc. (2), 57 (1998), 157–169. doi: 10.1112/S0024610798005900.  Google Scholar

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