# American Institute of Mathematical Sciences

August  2011, 5(3): 731-744. doi: 10.3934/ipi.2011.5.731

## Solving an inverse problem for the wave equation by using a minimization algorithm and time-reversed measurements

 1 Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68, 00014 Helsinki, Finland

Received  January 2011 Revised  April 2011 Published  August 2011

We consider the inverse problem for the wave equation on a compact Riemannian manifold or on a bounded domain of $\mathbb{R}^n$, and generalize the concept of domain of influence. We present an efficient minimization algorithm to compute the volume of a domain of influence using boundary measurements and time-reversed boundary measurements. Moreover, we show that if the manifold is simple, then the volumes of the domains of influence determine the manifold. For a continuous real valued function $\tau$ on the boundary of the manifold, the domain of influence is the set of those points on the manifold from which the travel time to some boundary point $y$ is less than $\tau(y)$.
Citation: Lauri Oksanen. Solving an inverse problem for the wave equation by using a minimization algorithm and time-reversed measurements. Inverse Problems and Imaging, 2011, 5 (3) : 731-744. doi: 10.3934/ipi.2011.5.731
##### References:
 [1] R. Alexander and S. Alexander, Geodesics in Riemannian manifolds-with-boundary, Indiana Univ. Math. J., 30 (1981), 481-488. [2] M. Anderson, A. Katsuda, Y. Kurylev, M. Lassas and M. Taylor, Boundary regularity for the Ricci equation, geometric convergence, and Gel'fand's inverse boundary problem, Invent. Math., 158 (2004), 261-321. doi: 10.1007/s00222-004-0371-6. [3] C. Bardos, A mathematical and deterministic analysis of the time-reversal mirror, in "Inside out: Inverse Problems and Applications," Math. Sci. Res. Inst. Publ., 47, Cambridge Univ. Press, (2003), 381-400. [4] C. Bardos and M. Fink, Mathematical foundations of the time reversal mirror, Asymptot. Anal., 29 (2002), 157-182. [5] M. Belishev, An approach to multidimensional inverse problems for the wave equation, (Russian) Dokl. Akad. Nauk SSSR, 297 (1987), 524-527, translation in Soviet Math. Dokl., 36 (1988), 481-484. [6] M. Belishev, Wave bases in multidimensional inverse problems, (Russian) Mat. Sb., 180 (1989), 584-602, 720, translation in Math. USSR-Sb., 67 (1990), 23-42. [7] M. Belishev, Boundary control in reconstruction of manifolds and metrics (the BC method), Inverse Problems, 13 (1997), R1-R45. doi: 10.1088/0266-5611/13/5/002. [8] M. Belishev and Y. Kurylev, To the reconstruction of a Riemannian manifold via its spectral data (BC-method), Comm. Partial Differential Equations, 17 (1992), 767-804. [9] K. Bingham, Y. Kurylev, M. Lassas and S. Siltanen, Iterative time reversal control for inverse problems, Inverse Problems and Imaging, 2 (2008), 63-81. doi: 10.3934/ipi.2008.2.63. [10] A. S. Blagoveščenskiĭ, "The Inverse Problem of the Theory of Seismic Wave Propagation," (Russian) Problems of Mathematical Physics, No. 1: Spectral Theory and Wave Processes, (errata insert) Izdat. Leningrad. Univ., Leningrad, (1966), 68-81. [11] L. Borcea, G. Papanicolaou, C. Tsogka and J. Berryman, Imaging and time reversal in random media, Inverse Problems, 18 (2002), 1247-1279. doi: 10.1088/0266-5611/18/5/303. [12] M. Cheney, D. Isaacson and M. Lassas, Optimal acoustic measurements, SIAM J. Appl. Math., 61 (2001), 1628-1647. [13] M. Dahl, A. Kirpichnikova and M. Lassas, Focusing waves in unknown media by modified time reversal iteration, SIAM J. Control Optim., 48 (2009), 839-858. [14] M. Fink, Time reversal mirrors, J. Phys. D: Appl.Phys., 26 (1993), 1333-1350. doi: 10.1088/0022-3727/26/9/001. [15] M. Fink, D. Cassereau, A. Derode, C. Prada, P. Roux, M. Tanter, J.-L. Thomas and F. Wu, Time-reversed acoustics, Rep. Prog. Phys., 63 (2000), 1933-1995. doi: 10.1088/0034-4885/63/12/202. [16] L. Gårding, Le problème de la dérivée oblique pour l'équation des ondes, (French) C. R. Acad. Sci. Paris Sér. A-B, 285 (1977), A773-A775. [17] A. Katchalov and Y. Kurylev, Multidimensional inverse problem with incomplete boundary spectral data, Comm. Part. Diff. Equations, 23 (1998), 55-95. [18] A. Katchalov, Y. Kurylev and M. Lassas, "Inverse Boundary Spectral Problems," Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 123, Chapman & Hall/CRC, Boca Raton, FL, 2001. [19] A. Katchalov, Y. Kurylev and M. Lassas, Energy measurements and equivalence of boundary data for inverse problems on non-compact manifolds, in "Geometric Methods in Inverse Problems and PDE Control" (eds. C. Croke, I. Lasiecka, G. Uhlmann and M. Vogelius), IMA volumes in Mathematics and Applications, 137, Springer, New York, (2004), 183-213. [20] M. Klibanov and A. Timonov, On the mathematical treatment of time reversal, Inverse Problems, 19 (2003), 1299-1318. doi: 10.1088/0266-5611/19/6/005. [21] Y. Kurylev, Multidimensional Gel'fand inverse problem and boundary distance map, in "Inverse Problems Related with Geometry" (ed. H. Soga), Proceedings of the Symposium at Tokyo Metropolitan University, (1997), 1-15. [22] I. Lasiecka and R. Triggiani, Regularity theory of hyperbolic equations with nonhomogeneous Neumann boundary conditions. II. General boundary data, J. Differential Equations, 94 (1991), 112-164. [23] S. Miyatake, Mixed problem for hyperbolic equation of second order, J. Math. Kyoto Univ., 13 (1973), 435-487. [24] L. Pestov, V. Bolgova and O. Kazarina, Numerical recovering of a density by the BC-method, Inverse Problems and Imaging, 4 (2010), 703-712. doi: 10.3934/ipi.2010.4.703. [25] C. Pugh, "Real Mathematical Analysis," Undergraduate Texts in Mathematics, Springer-Verlag, New York, 2002. [26] D. Tataru, Unique continuation for solutions to PDE's; between Hörmander's theorem and Holmgren's theorem, Comm. Partial Differential Equations, 20 (1995), 855-884. [27] D. Tataru, Unique continuation for operators with partially analytic coefficients, J. Math. Pures Appl. (9), 78 (1999), 505-521.

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##### References:
 [1] R. Alexander and S. Alexander, Geodesics in Riemannian manifolds-with-boundary, Indiana Univ. Math. J., 30 (1981), 481-488. [2] M. Anderson, A. Katsuda, Y. Kurylev, M. Lassas and M. Taylor, Boundary regularity for the Ricci equation, geometric convergence, and Gel'fand's inverse boundary problem, Invent. Math., 158 (2004), 261-321. doi: 10.1007/s00222-004-0371-6. [3] C. Bardos, A mathematical and deterministic analysis of the time-reversal mirror, in "Inside out: Inverse Problems and Applications," Math. Sci. Res. Inst. Publ., 47, Cambridge Univ. Press, (2003), 381-400. [4] C. Bardos and M. Fink, Mathematical foundations of the time reversal mirror, Asymptot. Anal., 29 (2002), 157-182. [5] M. Belishev, An approach to multidimensional inverse problems for the wave equation, (Russian) Dokl. Akad. Nauk SSSR, 297 (1987), 524-527, translation in Soviet Math. Dokl., 36 (1988), 481-484. [6] M. Belishev, Wave bases in multidimensional inverse problems, (Russian) Mat. Sb., 180 (1989), 584-602, 720, translation in Math. USSR-Sb., 67 (1990), 23-42. [7] M. Belishev, Boundary control in reconstruction of manifolds and metrics (the BC method), Inverse Problems, 13 (1997), R1-R45. doi: 10.1088/0266-5611/13/5/002. [8] M. Belishev and Y. Kurylev, To the reconstruction of a Riemannian manifold via its spectral data (BC-method), Comm. Partial Differential Equations, 17 (1992), 767-804. [9] K. Bingham, Y. Kurylev, M. Lassas and S. Siltanen, Iterative time reversal control for inverse problems, Inverse Problems and Imaging, 2 (2008), 63-81. doi: 10.3934/ipi.2008.2.63. [10] A. S. Blagoveščenskiĭ, "The Inverse Problem of the Theory of Seismic Wave Propagation," (Russian) Problems of Mathematical Physics, No. 1: Spectral Theory and Wave Processes, (errata insert) Izdat. Leningrad. Univ., Leningrad, (1966), 68-81. [11] L. Borcea, G. Papanicolaou, C. Tsogka and J. Berryman, Imaging and time reversal in random media, Inverse Problems, 18 (2002), 1247-1279. doi: 10.1088/0266-5611/18/5/303. [12] M. Cheney, D. Isaacson and M. Lassas, Optimal acoustic measurements, SIAM J. Appl. Math., 61 (2001), 1628-1647. [13] M. Dahl, A. Kirpichnikova and M. Lassas, Focusing waves in unknown media by modified time reversal iteration, SIAM J. Control Optim., 48 (2009), 839-858. [14] M. Fink, Time reversal mirrors, J. Phys. D: Appl.Phys., 26 (1993), 1333-1350. doi: 10.1088/0022-3727/26/9/001. [15] M. Fink, D. Cassereau, A. Derode, C. Prada, P. Roux, M. Tanter, J.-L. Thomas and F. Wu, Time-reversed acoustics, Rep. Prog. Phys., 63 (2000), 1933-1995. doi: 10.1088/0034-4885/63/12/202. [16] L. Gårding, Le problème de la dérivée oblique pour l'équation des ondes, (French) C. R. Acad. Sci. Paris Sér. A-B, 285 (1977), A773-A775. [17] A. Katchalov and Y. Kurylev, Multidimensional inverse problem with incomplete boundary spectral data, Comm. Part. Diff. Equations, 23 (1998), 55-95. [18] A. Katchalov, Y. Kurylev and M. Lassas, "Inverse Boundary Spectral Problems," Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 123, Chapman & Hall/CRC, Boca Raton, FL, 2001. [19] A. Katchalov, Y. Kurylev and M. Lassas, Energy measurements and equivalence of boundary data for inverse problems on non-compact manifolds, in "Geometric Methods in Inverse Problems and PDE Control" (eds. C. Croke, I. Lasiecka, G. Uhlmann and M. Vogelius), IMA volumes in Mathematics and Applications, 137, Springer, New York, (2004), 183-213. [20] M. Klibanov and A. Timonov, On the mathematical treatment of time reversal, Inverse Problems, 19 (2003), 1299-1318. doi: 10.1088/0266-5611/19/6/005. [21] Y. Kurylev, Multidimensional Gel'fand inverse problem and boundary distance map, in "Inverse Problems Related with Geometry" (ed. H. Soga), Proceedings of the Symposium at Tokyo Metropolitan University, (1997), 1-15. [22] I. Lasiecka and R. Triggiani, Regularity theory of hyperbolic equations with nonhomogeneous Neumann boundary conditions. II. General boundary data, J. Differential Equations, 94 (1991), 112-164. [23] S. Miyatake, Mixed problem for hyperbolic equation of second order, J. Math. Kyoto Univ., 13 (1973), 435-487. [24] L. Pestov, V. Bolgova and O. Kazarina, Numerical recovering of a density by the BC-method, Inverse Problems and Imaging, 4 (2010), 703-712. doi: 10.3934/ipi.2010.4.703. [25] C. Pugh, "Real Mathematical Analysis," Undergraduate Texts in Mathematics, Springer-Verlag, New York, 2002. [26] D. Tataru, Unique continuation for solutions to PDE's; between Hörmander's theorem and Holmgren's theorem, Comm. Partial Differential Equations, 20 (1995), 855-884. [27] D. Tataru, Unique continuation for operators with partially analytic coefficients, J. Math. Pures Appl. (9), 78 (1999), 505-521.
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