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

Xiaoping Fang; Youjun Deng; Wing-Yan Tsui; Zaiyun Zhang; Xiaoping Fang; Youjun Deng; Wing-Yan Tsui; Zaiyun Zhang

doi:10.3934/era.2020068

Electronic Research Archive

2020, Volume 28, Issue 3: 1239-1255. doi: 10.3934/era.2020068

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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

Received: 01 May 2020 Revised: 01 June 2020
Primary: 35Q60, 35J05, 31B10; Secondary: 35R30, 78A40

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.
- Wave scattering,
- geomagnetism,
- magnetized anomalies,
- inverse problem,
- uniqueness,
- frequency domain,
- asymptotic analysis
Citation: Xiaoping Fang, Youjun Deng, Wing-Yan Tsui, Zaiyun Zhang. On simultaneous recovery of sources/obstacles and surrounding mediums by boundary measurements[J]. Electronic Research Archive, 2020, 28(3): 1239-1255. doi: 10.3934/era.2020068

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Abstract

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.

References

[1]	Surface plasmon resonance of nanoparticles and applications in imaging. Arch. Ration. Mech. Anal. (2016) 220: 109-153.
[2]	(1996) Foundations of Geomagnetism.Cambridge University Press.
[3]	Simultaneously recovering potentials and embedded obstacles for anisotropic fractional Schrödinger operators. Inverse Probl. Imaging (2019) 13: 197-210.
[4]	X. Cao and H. Liu, Determining a fractional Helmholtz system with unknown source and medium parameter, preprint, arXiv: 1803.09538.
[5]	A three-dimensional spherical nonlinear interface dynamo. Astrophys. J. (2003) 596: 663-679.
[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
[7]	On identifying magnetized anomalies using geomagnetic monitoring. Arch. Ration. Mech. Anal. (2019) 231: 153-187.
[8]	On identifying magnetized anomalies using geomagnetic monitoring within a magnetohydrodynamic model. Arch. Ration. Mech. Anal. (2020) 235: 691-721.
[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.
[10]	Recovery of an embedded obstacle and the surrounding medium for Maxwell's system. J. Differential Equations (2019) 267: 2192-2209.
[11]	On an inverse boundary problem arising in brain imaging. J. Differential Equations (2019) 267: 2471-2502.
[12]	Uniqueness on recovery of piecewise constant conductivity and inner core with one measurement. Inverse Probl. Imaging (2018) 12: 733-743.
[13]	A geodynamo model incorporating a finitely conducting inner core. Phys. Earth Planet. Inter. (1993) 75: 317-327.
[14]	Determining a random Schrödinger equation with unknown source and potential. SIAM J. Math. Anal. (2019) 51: 3465-3491.
[15]	J. Li, H. Liu and S. Ma, Determining a random Schrödinger operator: Both potential and source are random, preprint, arXiv: 1906.01240
[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
[17]	H. Liu and S. Ma, Determining a random source in a Schrödinger equation involving an unknown potential, preprint, arXiv: 2005.04984
[18]	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) (2019) 21: 2945-2993.
[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
[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
[21]	An inverse boundary value problem in electrodynamics. Duke Math. J. (1993) 70: 617-653.
[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

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