Recovery of a source term or a speed with one measurement and applications
HTML articles powered by AMS MathViewer
- by Plamen Stefanov and Gunther Uhlmann PDF
- Trans. Amer. Math. Soc. 365 (2013), 5737-5758 Request permission
Abstract:
We study the problem of recovery of the source $a(t,x)F(x)$ in the wave equation in anisotropic medium with $a$ known so that $a(0,x)\not =0$, with a single measurement. We use Carleman estimates combined with geometric arguments and give sharp conditions for uniqueness. We also study the non-linear problem of recovery of the sound speed in the equation $u_{tt} -c^2(x)\Delta u =0$ with one measurement. We give sharp conditions for stability as well. An application to thermoacoustic tomography is also presented.References
- Claude Bardos, Gilles Lebeau, and Jeffrey Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim. 30 (1992), no. 5, 1024–1065. MR 1178650, DOI 10.1137/0330055
- A. L. Bukhgeĭm and M. V. Klibanov, Uniqueness in the large of a class of multidimensional inverse problems, Dokl. Akad. Nauk SSSR 260 (1981), no. 2, 269–272 (Russian). MR 630135
- David Finch, Sarah K. Patch, and Rakesh, Determining a function from its mean values over a family of spheres, SIAM J. Math. Anal. 35 (2004), no. 5, 1213–1240. MR 2050199, DOI 10.1137/S0036141002417814
- D. Finch and Rakesh. Recovering a function from its spherical mean values in two and three dimensions. in: Photoacoustic Imaging and Spectroscopy, CRC Press, 2009.
- Yulia Hristova, Time reversal in thermoacoustic tomography—an error estimate, Inverse Problems 25 (2009), no. 5, 055008, 14. MR 2501026, DOI 10.1088/0266-5611/25/5/055008
- Yulia Hristova, Peter Kuchment, and Linh Nguyen, Reconstruction and time reversal in thermoacoustic tomography in acoustically homogeneous and inhomogeneous media, Inverse Problems 24 (2008), no. 5, 055006, 25. MR 2438941, DOI 10.1088/0266-5611/24/5/055006
- Oleg Yu. Imanuvilov and Masahiro Yamamoto, Global Lipschitz stability in an inverse hyperbolic problem by interior observations, Inverse Problems 17 (2001), no. 4, 717–728. Special issue to celebrate Pierre Sabatier’s 65th birthday (Montpellier, 2000). MR 1861478, DOI 10.1088/0266-5611/17/4/310
- Oleg Yu. Imanuvilov and Masahiro Yamamoto, Global uniqueness and stability in determining coefficients of wave equations, Comm. Partial Differential Equations 26 (2001), no. 7-8, 1409–1425. MR 1855284, DOI 10.1081/PDE-100106139
- Oleg Yu. Imanuvilov and Masahiro Yamamoto, Determination of a coefficient in an acoustic equation with a single measurement, Inverse Problems 19 (2003), no. 1, 157–171. MR 1964256, DOI 10.1088/0266-5611/19/1/309
- Victor Isakov, Carleman type estimates and their applications, New analytic and geometric methods in inverse problems, Springer, Berlin, 2004, pp. 93–125. MR 2053418
- Victor Isakov, Inverse problems for partial differential equations, 2nd ed., Applied Mathematical Sciences, vol. 127, Springer, New York, 2006. MR 2193218
- Alexander Katchalov, Yaroslav Kurylev, and Matti Lassas, Inverse boundary spectral problems, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, vol. 123, Chapman & Hall/CRC, Boca Raton, FL, 2001. MR 1889089, DOI 10.1201/9781420036220
- Peter Kuchment and Leonid Kunyansky, Mathematics of thermoacoustic tomography, European J. Appl. Math. 19 (2008), no. 2, 191–224. MR 2400720, DOI 10.1017/S0956792508007353
- I. Lasiecka, J.-L. Lions, and R. Triggiani, Nonhomogeneous boundary value problems for second order hyperbolic operators, J. Math. Pures Appl. (9) 65 (1986), no. 2, 149–192. MR 867669
- I. Lasiecka, R. Triggiani, and P. F. Yao, Exact controllability for second-order hyperbolic equations with variable coefficient-principal part and first-order terms, Proceedings of the Second World Congress of Nonlinear Analysts, Part 1 (Athens, 1996), 1997, pp. 111–122. MR 1489773, DOI 10.1016/S0362-546X(97)00004-7
- S. K. Patch. Thermoacoustic tomography – consistency conditions and the partial scan problem. Physics in Medicine and Biology, 49(11):2305–2315, 2004.
- Jean-Pierre Puel and Masahiro Yamamoto, On a global estimate in a linear inverse hyperbolic problem, Inverse Problems 12 (1996), no. 6, 995–1002. MR 1421661, DOI 10.1088/0266-5611/12/6/013
- Jianliang Qian, Plamen Stefanov, Gunther Uhlmann, and Hongkai Zhao, An efficient Neumann series-based algorithm for thermoacoustic and photoacoustic tomography with variable sound speed, SIAM J. Imaging Sci. 4 (2011), no. 3, 850–883. MR 2836390, DOI 10.1137/100817280
- V. G. Romanov, Carleman estimates for a second-order hyperbolic equation, Sibirsk. Mat. Zh. 47 (2006), no. 1, 169–187 (Russian, with Russian summary); English transl., Siberian Math. J. 47 (2006), no. 1, 135–151. MR 2215303, DOI 10.1007/s11202-006-0014-9
- V. A. Sharafutdinov, Integral geometry of tensor fields, Inverse and Ill-posed Problems Series, VSP, Utrecht, 1994. MR 1374572, DOI 10.1515/9783110900095
- Plamen Stefanov and Gunther Uhlmann, Stability estimates for the hyperbolic Dirichlet to Neumann map in anisotropic media, J. Funct. Anal. 154 (1998), no. 2, 330–358. MR 1612709, DOI 10.1006/jfan.1997.3188
- Plamen Stefanov and Gunther Uhlmann, Boundary rigidity and stability for generic simple metrics, J. Amer. Math. Soc. 18 (2005), no. 4, 975–1003. MR 2163868, DOI 10.1090/S0894-0347-05-00494-7
- Plamen Stefanov and Gunther Uhlmann, Thermoacoustic tomography with variable sound speed, Inverse Problems 25 (2009), no. 7, 075011, 16. MR 2519863, DOI 10.1088/0266-5611/25/7/075011
- Plamen Stefanov and Gunther Uhlmann, Thermoacoustic tomography arising in brain imaging, Inverse Problems 27 (2011), no. 4, 045004, 26. MR 2781028, DOI 10.1088/0266-5611/27/4/045004
- Daniel Tataru, On the regularity of boundary traces for the wave equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 26 (1998), no. 1, 185–206. MR 1633000
- Daniel Tataru, Unique continuation for operators with partially analytic coefficients, J. Math. Pures Appl. (9) 78 (1999), no. 5, 505–521. MR 1697040, DOI 10.1016/S0021-7824(99)00016-1
- Daniel Tataru, Unique continuation problems for partial differential equations, Geometric methods in inverse problems and PDE control, IMA Vol. Math. Appl., vol. 137, Springer, New York, 2004, pp. 239–255. MR 2169906, DOI 10.1007/978-1-4684-9375-7_{8}
- Michael E. Taylor, Partial differential equations. I, Applied Mathematical Sciences, vol. 115, Springer-Verlag, New York, 1996. Basic theory. MR 1395148, DOI 10.1007/978-1-4684-9320-7
- Roberto Triggiani and P. F. Yao, Carleman estimates with no lower-order terms for general Riemann wave equations. Global uniqueness and observability in one shot, Appl. Math. Optim. 46 (2002), no. 2-3, 331–375. Special issue dedicated to the memory of Jacques-Louis Lions. MR 1944764, DOI 10.1007/s00245-002-0751-5
- Peng-Fei Yao, On the observability inequalities for exact controllability of wave equations with variable coefficients, SIAM J. Control Optim. 37 (1999), no. 5, 1568–1599. MR 1710233, DOI 10.1137/S0363012997331482
Additional Information
- Plamen Stefanov
- Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
- MR Author ID: 166695
- Gunther Uhlmann
- Affiliation: Department of Mathematics, University of Washington, Seattle, Washington 98195
- MR Author ID: 175790
- Received by editor(s): March 13, 2011
- Received by editor(s) in revised form: August 10, 2011
- Published electronically: April 25, 2013
- Additional Notes: The first author was partially supported by an NSF Grant DMS-0800428 and a Simons Visiting Professorship
The second author was partially supported by an NSF, a Senior Clay Award and Chancellor Professorship at the University of California, Berkeley - © Copyright 2013 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 365 (2013), 5737-5758
- MSC (2010): Primary 35L05, 35R30
- DOI: https://doi.org/10.1090/S0002-9947-2013-05703-0
- MathSciNet review: 3091263