Conditionally stable unique continuation and applications to thermoacoustic tomography

Plamen Stefanov; Plamen Stefanov

doi:10.3934/mine.2019.4.789

Mathematics in Engineering

2019, Volume 1, Issue 4: 789-799. doi: 10.3934/mine.2019.4.789

Previous Article Next Article

Research article Special Issues

Conditionally stable unique continuation and applications to thermoacoustic tomography

Plamen Stefanov ^,

Department of Mathematics, Purdue University, West Lafayette, IN 47907
† This contribution is part of the Special Issue: Inverse problems in imaging and engineering science

Received: 02 March 2019 Accepted: 17 July 2019 Published: 18 September 2019

We prove a conditional Hölder stability estimate for the Cauchy problem on the lateral boundary for the wave equation under a strictly convex foliation condition. We apply this estimate for the problem in multiwave tomography with partial data.
- unique continuation,
- stability,
- thermoacoustic tomography
Citation: Plamen Stefanov. Conditionally stable unique continuation and applications to thermoacoustic tomography[J]. Mathematics in Engineering, 2019, 1(4): 789-799. doi: 10.3934/mine.2019.4.789

Related Papers:

Abstract

We prove a conditional Hölder stability estimate for the Cauchy problem on the lateral boundary for the wave equation under a strictly convex foliation condition. We apply this estimate for the problem in multiwave tomography with partial data.

References

[1]	Bardos C, Lebeau G, Rauch J (1992) Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary. SIAM J Control Optim 30: 1024-1065. doi: 10.1137/0330055
[2]	Bosi R, Kurylev Y, Lassas M (2016) Stability of the unique continuation for the wave operator via Tataru inequality and applications. J Differ Equations 260: 6451-6492. doi: 10.1016/j.jde.2015.12.043
[3]	Bosi R, Kurylev Y, Lassas M (2018) Stability of the unique continuation for the wave operator via Tataru inequality: The local case. J Anal Math 134: 157-199. doi: 10.1007/s11854-018-0006-2
[4]	Chervova O, Oksanen L (2016) Time reversal method with stabilizing boundary conditions for photoacoustic tomography. Inverse Probl 32: 125004. doi: 10.1088/0266-5611/32/12/125004
[5]	Cox BT, Arridge SR, Beard PC (2007) Photoacoustic tomography with a limited-aperture planar sensor and a reverberant cavity. Inverse Probl 23: S95. doi: 10.1088/0266-5611/23/6/S08
[6]	De Hoop MV, Kepley P, Oksanen L (2018) An exact redatuming procedure for the inverse boundary value problem for the wave equation. SIAM J Appl Math 78: 171-192. doi: 10.1137/16M1106729
[7]	Eller M, Isakov V, Nakamura G, et al. (2002) Uniqueness and stability in the Cauchy problem for Maxwell and elasticity systems. In: Nonlinear Partial Differential Equations and Their Applications: College de France Seminar 14: 329-349. Elsevier.
[8]	Finch D, Patch SK, Rakesh(2004) Determining a function from its mean values over a family of spheres. SIAM J Math Anal 35: 1213-1240.
[9]	Holman B, Kunyansky L (2015) Gradual time reversal in thermo-and photo-acoustic tomography within a resonant cavity. Inverse Probl 31: 035008. doi: 10.1088/0266-5611/31/3/035008
[10]	Isakov V(2006) Inverse problems for partial differential equations. New York: Springer, 127.
[11]	Kunyansky L, Holman B, Cox BT (2013) Photoacoustic tomography in a rectangular reflecting cavity. Inverse Probl 29: 125010. doi: 10.1088/0266-5611/29/12/125010
[12]	Nguyen LV, Kunyansky LA (2016) A dissipative time reversal technique for photoacoustic tomography in a cavity. SIAM J Imaging Sci 9: 748-769. doi: 10.1137/15M1049683
[13]	Paternain GP, Salo M, Uhlmann G, et al. (2016) The geodesic X-ray transform with matrix weights. arXiv:1605.07894.
[14]	Stefanov P, Uhlmann G(2009) Linearizing non-linear inverse problems and an application to inverse backscattering. J Funct Anal 256: 2842-2866.
[15]	Stefanov P, Uhlmann G (2009) Thermoacoustic tomography with variable sound speed. Inverse Probl 25: 075011. doi: 10.1088/0266-5611/25/7/075011
[16]	Stefanov P, Uhlmann G (2011) Thermoacoustic tomography arising in brain imaging. Inverse Probl 27:045004. doi: 10.1088/0266-5611/27/4/045004
[17]	Stefanov P, Uhlmann G (2013) Recovery of a source term or a speed with one measurement and applications. Trans Amer Math Soc 365: 5737-5758. doi: 10.1090/S0002-9947-2013-05703-0
[18]	Stefanov P, Uhlmann G, Vasy A (2016) Boundary rigidity with partial data. J Amer Math Soc 29: 299-332.
[19]	Stefanov P, Uhlmann G, Vasy A (2017) Local and global boundary rigidity and the geodesic X-ray transform in the normal gauge. arXiv:1702.03638.
[20]	Stefanov P, Uhlmann G, Vasy A (2018) Inverting the local geodesic X-ray transform on tensors. J Anal Math 136: 151-208. doi: 10.1007/s11854-018-0058-3
[21]	Stefanov P, Yang Y (2015) Multiwave tomography in a closed domain: averaged sharp time reversal. Inverse Probl 31: 065007. doi: 10.1088/0266-5611/31/6/065007
[22]	Stefanov P, Yang Y (2017) Multiwave tomography with reflectors: Landweber's iteration. arXiv:1603.07045.
[23]	Tataru D (1995) Unique continuation for solutions to PDE's; between Hörmander's theorem and Holmgren's theorem. Commun Part Diff Eq 20: 855-884. doi: 10.1080/03605309508821117
[24]	Tataru D (1999) Unique continuation for operators with partially analytic coefficients. J Math Pures Appl 78: 505-521. doi: 10.1016/S0021-7824(99)00016-1
[25]	Tataru D (2004) Unique continuation problems for partial differential equations. In: Geometric Methods in Inverse Problems and PDE Control, 239-255. Springer, New York, NY.
[26]	Triggiani R, Yao PF (2002) Carleman estimates with no lower-order terms for general Riemann wave equations. Global uniqueness and observability in one shot. Appl Math Optim 46: 331-375.
[27]	Uhlmann G, Vasy A (2016) The inverse problem for the local geodesic ray transform. Invent Math 205: 83-120. doi: 10.1007/s00222-015-0631-7

Reader Comments

Your name:*

Email:*
© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)