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 & 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. Google Scholar

[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. doi: 10.1007/s00222-004-0371-6. Google Scholar

[3]

C. Bardos, A mathematical and deterministic analysis of the time-reversal mirror,, in, 47 (2003), 381. Google Scholar

[4]

C. Bardos and M. Fink, Mathematical foundations of the time reversal mirror,, Asymptot. Anal., 29 (2002), 157. Google Scholar

[5]

M. Belishev, An approach to multidimensional inverse problems for the wave equation,, (Russian) Dokl. Akad. Nauk SSSR, 297 (1987), 524. Google Scholar

[6]

M. Belishev, Wave bases in multidimensional inverse problems,, (Russian) Mat. Sb., 180 (1989), 584. Google Scholar

[7]

M. Belishev, Boundary control in reconstruction of manifolds and metrics (the BC method),, Inverse Problems, 13 (1997). doi: 10.1088/0266-5611/13/5/002. Google Scholar

[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. Google Scholar

[9]

K. Bingham, Y. Kurylev, M. Lassas and S. Siltanen, Iterative time reversal control for inverse problems,, Inverse Problems and Imaging, 2 (2008), 63. doi: 10.3934/ipi.2008.2.63. Google Scholar

[10]

A. S. Blagoveščenskiĭ, "The Inverse Problem of the Theory of Seismic Wave Propagation,", (Russian) Problems of Mathematical Physics, (1966), 68. Google Scholar

[11]

L. Borcea, G. Papanicolaou, C. Tsogka and J. Berryman, Imaging and time reversal in random media,, Inverse Problems, 18 (2002), 1247. doi: 10.1088/0266-5611/18/5/303. Google Scholar

[12]

M. Cheney, D. Isaacson and M. Lassas, Optimal acoustic measurements,, SIAM J. Appl. Math., 61 (2001), 1628. Google Scholar

[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. Google Scholar

[14]

M. Fink, Time reversal mirrors,, J. Phys. D: Appl.Phys., 26 (1993), 1333. doi: 10.1088/0022-3727/26/9/001. Google Scholar

[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. doi: 10.1088/0034-4885/63/12/202. Google Scholar

[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). Google Scholar

[17]

A. Katchalov and Y. Kurylev, Multidimensional inverse problem with incomplete boundary spectral data,, Comm. Part. Diff. Equations, 23 (1998), 55. Google Scholar

[18]

A. Katchalov, Y. Kurylev and M. Lassas, "Inverse Boundary Spectral Problems,", Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 123 (2001). Google Scholar

[19]

A. Katchalov, Y. Kurylev and M. Lassas, Energy measurements and equivalence of boundary data for inverse problems on non-compact manifolds,, in, 137 (2004), 183. Google Scholar

[20]

M. Klibanov and A. Timonov, On the mathematical treatment of time reversal,, Inverse Problems, 19 (2003), 1299. doi: 10.1088/0266-5611/19/6/005. Google Scholar

[21]

Y. Kurylev, Multidimensional Gel'fand inverse problem and boundary distance map,, in, (1997), 1. Google Scholar

[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. Google Scholar

[23]

S. Miyatake, Mixed problem for hyperbolic equation of second order,, J. Math. Kyoto Univ., 13 (1973), 435. Google Scholar

[24]

L. Pestov, V. Bolgova and O. Kazarina, Numerical recovering of a density by the BC-method,, Inverse Problems and Imaging, 4 (2010), 703. doi: 10.3934/ipi.2010.4.703. Google Scholar

[25]

C. Pugh, "Real Mathematical Analysis,", Undergraduate Texts in Mathematics, (2002). Google Scholar

[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. Google Scholar

[27]

D. Tataru, Unique continuation for operators with partially analytic coefficients,, J. Math. Pures Appl. (9), 78 (1999), 505. Google Scholar

show all references

References:
[1]

R. Alexander and S. Alexander, Geodesics in Riemannian manifolds-with-boundary,, Indiana Univ. Math. J., 30 (1981), 481. Google Scholar

[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. doi: 10.1007/s00222-004-0371-6. Google Scholar

[3]

C. Bardos, A mathematical and deterministic analysis of the time-reversal mirror,, in, 47 (2003), 381. Google Scholar

[4]

C. Bardos and M. Fink, Mathematical foundations of the time reversal mirror,, Asymptot. Anal., 29 (2002), 157. Google Scholar

[5]

M. Belishev, An approach to multidimensional inverse problems for the wave equation,, (Russian) Dokl. Akad. Nauk SSSR, 297 (1987), 524. Google Scholar

[6]

M. Belishev, Wave bases in multidimensional inverse problems,, (Russian) Mat. Sb., 180 (1989), 584. Google Scholar

[7]

M. Belishev, Boundary control in reconstruction of manifolds and metrics (the BC method),, Inverse Problems, 13 (1997). doi: 10.1088/0266-5611/13/5/002. Google Scholar

[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. Google Scholar

[9]

K. Bingham, Y. Kurylev, M. Lassas and S. Siltanen, Iterative time reversal control for inverse problems,, Inverse Problems and Imaging, 2 (2008), 63. doi: 10.3934/ipi.2008.2.63. Google Scholar

[10]

A. S. Blagoveščenskiĭ, "The Inverse Problem of the Theory of Seismic Wave Propagation,", (Russian) Problems of Mathematical Physics, (1966), 68. Google Scholar

[11]

L. Borcea, G. Papanicolaou, C. Tsogka and J. Berryman, Imaging and time reversal in random media,, Inverse Problems, 18 (2002), 1247. doi: 10.1088/0266-5611/18/5/303. Google Scholar

[12]

M. Cheney, D. Isaacson and M. Lassas, Optimal acoustic measurements,, SIAM J. Appl. Math., 61 (2001), 1628. Google Scholar

[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. Google Scholar

[14]

M. Fink, Time reversal mirrors,, J. Phys. D: Appl.Phys., 26 (1993), 1333. doi: 10.1088/0022-3727/26/9/001. Google Scholar

[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. doi: 10.1088/0034-4885/63/12/202. Google Scholar

[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). Google Scholar

[17]

A. Katchalov and Y. Kurylev, Multidimensional inverse problem with incomplete boundary spectral data,, Comm. Part. Diff. Equations, 23 (1998), 55. Google Scholar

[18]

A. Katchalov, Y. Kurylev and M. Lassas, "Inverse Boundary Spectral Problems,", Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 123 (2001). Google Scholar

[19]

A. Katchalov, Y. Kurylev and M. Lassas, Energy measurements and equivalence of boundary data for inverse problems on non-compact manifolds,, in, 137 (2004), 183. Google Scholar

[20]

M. Klibanov and A. Timonov, On the mathematical treatment of time reversal,, Inverse Problems, 19 (2003), 1299. doi: 10.1088/0266-5611/19/6/005. Google Scholar

[21]

Y. Kurylev, Multidimensional Gel'fand inverse problem and boundary distance map,, in, (1997), 1. Google Scholar

[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. Google Scholar

[23]

S. Miyatake, Mixed problem for hyperbolic equation of second order,, J. Math. Kyoto Univ., 13 (1973), 435. Google Scholar

[24]

L. Pestov, V. Bolgova and O. Kazarina, Numerical recovering of a density by the BC-method,, Inverse Problems and Imaging, 4 (2010), 703. doi: 10.3934/ipi.2010.4.703. Google Scholar

[25]

C. Pugh, "Real Mathematical Analysis,", Undergraduate Texts in Mathematics, (2002). Google Scholar

[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. Google Scholar

[27]

D. Tataru, Unique continuation for operators with partially analytic coefficients,, J. Math. Pures Appl. (9), 78 (1999), 505. Google Scholar

[1]

Kenrick Bingham, Yaroslav Kurylev, Matti Lassas, Samuli Siltanen. Iterative time-reversal control for inverse problems. Inverse Problems & Imaging, 2008, 2 (1) : 63-81. doi: 10.3934/ipi.2008.2.63

[2]

Kazufumi Ito, Karim Ramdani, Marius Tucsnak. A time reversal based algorithm for solving initial data inverse problems. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 641-652. doi: 10.3934/dcdss.2011.4.641

[3]

Anna Doubova, Enrique Fernández-Cara. Some geometric inverse problems for the linear wave equation. Inverse Problems & Imaging, 2015, 9 (2) : 371-393. doi: 10.3934/ipi.2015.9.371

[4]

Laurent Bourgeois, Dmitry Ponomarev, Jérémi Dardé. An inverse obstacle problem for the wave equation in a finite time domain. Inverse Problems & Imaging, 2019, 13 (2) : 377-400. doi: 10.3934/ipi.2019019

[5]

Sergei A. Avdonin, Sergei A. Ivanov, Jun-Min Wang. Inverse problems for the heat equation with memory. Inverse Problems & Imaging, 2019, 13 (1) : 31-38. doi: 10.3934/ipi.2019002

[6]

Rodrigo I. Brevis, Jaime H. Ortega, David Pardo. A source time reversal method for seismicity induced by mining. Inverse Problems & Imaging, 2017, 11 (1) : 25-45. doi: 10.3934/ipi.2017002

[7]

Gen Nakamura, Michiyuki Watanabe. An inverse boundary value problem for a nonlinear wave equation. Inverse Problems & Imaging, 2008, 2 (1) : 121-131. doi: 10.3934/ipi.2008.2.121

[8]

Albert Fannjiang, Knut Solna. Time reversal of parabolic waves and two-frequency Wigner distribution. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 783-802. doi: 10.3934/dcdsb.2006.6.783

[9]

Fabrice Planchon, John G. Stalker, A. Shadi Tahvildar-Zadeh. $L^p$ Estimates for the wave equation with the inverse-square potential. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 427-442. doi: 10.3934/dcds.2003.9.427

[10]

Fabrice Planchon, John G. Stalker, A. Shadi Tahvildar-Zadeh. Dispersive estimate for the wave equation with the inverse-square potential. Discrete & Continuous Dynamical Systems - A, 2003, 9 (6) : 1387-1400. doi: 10.3934/dcds.2003.9.1387

[11]

Lucie Baudouin, Emmanuelle Crépeau, Julie Valein. Global Carleman estimate on a network for the wave equation and application to an inverse problem. Mathematical Control & Related Fields, 2011, 1 (3) : 307-330. doi: 10.3934/mcrf.2011.1.307

[12]

Mohammed Al Horani, Angelo Favini, Hiroki Tanabe. Inverse problems for evolution equations with time dependent operator-coefficients. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 737-744. doi: 10.3934/dcdss.2016025

[13]

Frank Natterer. Incomplete data problems in wave equation imaging. Inverse Problems & Imaging, 2010, 4 (4) : 685-691. doi: 10.3934/ipi.2010.4.685

[14]

Kazuhiro Ishige, Asato Mukai. Large time behavior of solutions of the heat equation with inverse square potential. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 4041-4069. doi: 10.3934/dcds.2018176

[15]

Jaan Janno, Kairi Kasemets. Uniqueness for an inverse problem for a semilinear time-fractional diffusion equation. Inverse Problems & Imaging, 2017, 11 (1) : 125-149. doi: 10.3934/ipi.2017007

[16]

Chuang Zheng. Inverse problems for the fourth order Schrödinger equation on a finite domain. Mathematical Control & Related Fields, 2015, 5 (1) : 177-189. doi: 10.3934/mcrf.2015.5.177

[17]

Masaru Ikehata. The enclosure method for inverse obstacle scattering using a single electromagnetic wave in time domain. Inverse Problems & Imaging, 2016, 10 (1) : 131-163. doi: 10.3934/ipi.2016.10.131

[18]

Corinna Burkard, Aurelia Minut, Karim Ramdani. Far field model for time reversal and application to selective focusing on small dielectric inhomogeneities. Inverse Problems & Imaging, 2013, 7 (2) : 445-470. doi: 10.3934/ipi.2013.7.445

[19]

Jussi Korpela, Matti Lassas, Lauri Oksanen. Discrete regularization and convergence of the inverse problem for 1+1 dimensional wave equation. Inverse Problems & Imaging, 2019, 13 (3) : 575-596. doi: 10.3934/ipi.2019027

[20]

Patrick Martinez, Judith Vancostenoble. Exact controllability in "arbitrarily short time" of the semilinear wave equation. Discrete & Continuous Dynamical Systems - A, 2003, 9 (4) : 901-924. doi: 10.3934/dcds.2003.9.901

2018 Impact Factor: 1.469

Metrics

  • PDF downloads (5)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]