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Transactions of the American Mathematical Society

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Recovery of a source term or a speed with one measurement and applications

Authors: Plamen Stefanov and Gunther Uhlmann
Journal: Trans. Amer. Math. Soc. 365 (2013), 5737-5758
MSC (2010): Primary 35L05, 35R30
Published electronically: April 25, 2013
MathSciNet review: 3091263
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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.

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  • 1. C. Bardos, G. Lebeau, and J. Rauch.
    Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary.
    SIAM J. Control Optim., 30(5):1024-1065, 1992. MR 1178650 (94b:93067)
  • 2. A. L. Bukhgeim and M. V. Klibanov.
    Uniqueness in the large of a class of multidimensional inverse problems.
    Dokl. Akad. Nauk SSSR, 260(2):269-272, 1981.
    English translation: Soviet Math. Dokl. 24 (1981), no. 2, 244-247 (1982). MR 630135 (83b:35157)
  • 3. D. Finch, S. K. Patch, and Rakesh.
    Determining a function from its mean values over a family of spheres.
    SIAM J. Math. Anal., 35(5):1213-1240 (electronic), 2004. MR 2050199 (2005b:35290)
  • 4. 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.
  • 5. Y. Hristova.
    Time reversal in thermoacoustic tomography - an error estimate.
    Inverse Problems, 25(5):055008, 2009. MR 2501026 (2010d:78036)
  • 6. Y. Hristova, P. Kuchment, and L. Nguyen.
    Reconstruction and time reversal in thermoacoustic tomography in acoustically homogeneous and inhomogeneous media.
    Inverse Problems, 24:055006, 2008. MR 2438941 (2010c:65162)
  • 7. O. Y. Imanuvilov and M. Yamamoto.
    Global Lipschitz stability in an inverse hyperbolic problem by interior observations.
    Inverse Problems, 17(4):717-728, 2001.
    Special issue to celebrate Pierre Sabatier's 65th birthday (Montpellier, 2000). MR 1861478 (2002i:35204)
  • 8. O. Y. Imanuvilov and M. Yamamoto.
    Global uniqueness and stability in determining coefficients of wave equations.
    Comm. Partial Differential Equations, 26(7-8):1409-1425, 2001. MR 1855284 (2002j:35310)
  • 9. O. Y. Imanuvilov and M. Yamamoto.
    Determination of a coefficient in an acoustic equation with a single measurement.
    Inverse Problems, 19(1):157-171, 2003. MR 1964256 (2004c:35415)
  • 10. V. Isakov.
    Carleman type estimates and their applications.
    In New analytic and geometric methods in inverse problems, pages 93-125. Springer, Berlin, 2004. MR 2053418 (2006c:35023)
  • 11. V. Isakov.
    Inverse problems for partial differential equations, volume 127 of Applied Mathematical Sciences.
    Springer, New York, second edition, 2006. MR 2193218 (2006h:35279)
  • 12. A. Katchalov, Y. Kurylev, and M. Lassas.
    Inverse boundary spectral problems, volume 123 of Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics.
    Chapman & Hall/CRC, Boca Raton, FL, 2001. MR 1889089 (2003e:58045)
  • 13. P. Kuchment and L. Kunyansky.
    Mathematics of thermoacoustic tomography.
    European J. Appl. Math., 19(2):191-224, 2008. MR 2400720 (2009c:92026)
  • 14. I. Lasiecka, J.-L. Lions, and R. Triggiani.
    Nonhomogeneous boundary value problems for second order hyperbolic operators.
    J. Math. Pures Appl., 65(2):149-192, 1986. MR 867669 (88c:35092)
  • 15. I. Lasiecka, R. Triggiani, and P. F. Yao.
    Exact controllability for second-order hyperbolic equations with variable coefficient-principal part and first-order terms.
    In Proceedings of the Second World Congress of Nonlinear Analysts, Part 1 (Athens, 1996), volume 30, pages 111-122, 1997. MR 1489773 (99a:35158)
  • 16. S. K. Patch.
    Thermoacoustic tomography - consistency conditions and the partial scan problem.
    Physics in Medicine and Biology, 49(11):2305-2315, 2004.
  • 17. J.-P. Puel and M. Yamamoto.
    On a global estimate in a linear inverse hyperbolic problem.
    Inverse Problems, 12(6):995-1002, 1996. MR 1421661 (97k:35271)
  • 18. J. Qian, P. Stefanov, G. Uhlmann, and H. Zhao.
    An efficient neumann-series based algorithm for thermoacoustic and photoacoustic tomography with variable sound speed.
    SIAM J. Imaging Sciences 4 (2011), no. 3, 850-883. MR 2836390
  • 19. V. G. Romanov.
    Carleman estimates for a second-order hyperbolic equation.
    Sibirsk. Mat. Zh., 47(1):169-187, 2006. MR 2215303 (2006m:35216)
  • 20. V. A. Sharafutdinov.
    Integral geometry of tensor fields.
    Inverse and Ill-posed Problems Series. VSP, Utrecht, 1994. MR 1374572 (97h:53077)
  • 21. P. Stefanov and G. Uhlmann.
    Stability estimates for the hyperbolic Dirichlet to Neumann map in anisotropic media.
    J. Funct. Anal., 154(2):330-358, 1998. MR 1612709 (99f:35120)
  • 22. P. Stefanov and G. Uhlmann.
    Boundary rigidity and stability for generic simple metrics.
    J. Amer. Math. Soc., 18(4):975-1003, 2005. MR 2163868 (2006h:53031)
  • 23. P. Stefanov and G. Uhlmann.
    Thermoacoustic tomography with variable sound speed.
    Inverse Problems, 25(7):075011, 16, 2009. MR 2519863 (2010i:35439)
  • 24. P. Stefanov and G. Uhlmann.
    Thermoacoustic tomography arising in brain imaging.
    Inverse Problems, 27(4):045004, 26, 2011. MR 2781028
  • 25. D. Tataru.
    On the regularity of boundary traces for the wave equation.
    Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 26(1):185-206, 1998. MR 1633000 (99e:35129)
  • 26. D. Tataru.
    Unique continuation for operators with partially analytic coefficients.
    J. Math. Pures Appl. (9), 78(5):505-521, 1999. MR 1697040 (2000e:35005)
  • 27. D. Tataru.
    Unique continuation problems for partial differential equations.
    In Geometric methods in inverse problems and PDE control, volume 137 of IMA Vol. Math. Appl., pages 239-255. Springer, New York, 2004. MR 2169906
  • 28. M. E. Taylor.
    Partial differential equations. I, volume 115 of Applied Mathematical Sciences.
    Springer-Verlag, New York, 1996.
    Basic theory. MR 1395148 (98b:35002b)
  • 29. R. 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(2-3):331-375, 2002.
    Special issue dedicated to the memory of Jacques-Louis Lions. MR 1944764 (2003j:93042)
  • 30. P.-F. Yao.
    On the observability inequalities for exact controllability of wave equations with variable coefficients.
    SIAM J. Control Optim., 37(5):1568-1599 (electronic), 1999. MR 1710233 (2000m:93027)

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Additional Information

Plamen Stefanov
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907

Gunther Uhlmann
Affiliation: Department of Mathematics, University of Washington, Seattle, Washington 98195

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
Article copyright: © Copyright 2013 American Mathematical Society

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