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Conditionally stable unique continuation and applications to thermoacoustic tomography

Department of Mathematics, Purdue University, West Lafayette, IN 47907

This contribution is part of the Special Issue: Inverse problems in imaging and engineering science
Guest Editors: Lauri Oksanen; Mikko Salo
Link: https://www.aimspress.com/newsinfo/1270.html

Special Issues: Inverse problems in imaging and engineering science

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.
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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.    

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.    

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.    

4. Chervova O, Oksanen L (2016) Time reversal method with stabilizing boundary conditions for photoacoustic tomography. Inverse Probl 32: 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.    

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.    

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.    

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.    

12. Nguyen LV, Kunyansky LA (2016) A dissipative time reversal technique for photoacoustic tomography in a cavity. SIAM J Imaging Sci 9: 748-769.    

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.    

16. Stefanov P, Uhlmann G (2011) Thermoacoustic tomography arising in brain imaging. Inverse Probl 27:045004.    

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.    

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.    

21. Stefanov P, Yang Y (2015) Multiwave tomography in a closed domain: averaged sharp time reversal. Inverse Probl 31: 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.    

24. Tataru D (1999) Unique continuation for operators with partially analytic coefficients. J Math Pures Appl 78: 505-521.    

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.    

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

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