# American Institute of Mathematical Sciences

June  2011, 4(3): 641-652. doi: 10.3934/dcdss.2011.4.641

## A time reversal based algorithm for solving initial data inverse problems

 1 Department of Mathematics, Box 8205, North Carolina State University, Raleigh, NC 27695-8205, United States 2 INRIA Nancy Grand-Est (CORIDA), 615 rue du Jardin Botanique, 54600, Villers-lès-Nancy, France 3 Institut Elie Cartan Nancy, Université Henri Poincaré, B.P. 70239, 54506 Vandoeuvre-lès-Nancy Cedex, France

Received  April 2009 Revised  November 2009 Published  November 2010

We propose an iterative algorithm to solve initial data inverse problems for a class of linear evolution equations, including the wave, the plate, the Schrödinger and the Maxwell equations in a bounded domain $\Omega$. We assume that the only available information is a distributed observation (i.e. partial observation of the solution on a sub-domain $\omega$ during a finite time interval $(0,\tau)$). Under some quite natural assumptions (essentially : the exact observability of the system for some time $\tau_{obs}>0$, $\tau\ge \tau_{obs}$ and the existence of a time-reversal operator for the problem), an iterative algorithm based on a Neumann series expansion is proposed. Numerical examples are presented to show the efficiency of the method.
Citation: Kazufumi Ito, Karim Ramdani, Marius Tucsnak. A time reversal based algorithm for solving initial data inverse problems. Discrete and Continuous Dynamical Systems - S, 2011, 4 (3) : 641-652. doi: 10.3934/dcdss.2011.4.641
##### References:
 [1] C. Alves, A. L. Silvestre, T. Takahashi and M. Tucsnak, Solving inverse source problems using observability. Applications to the Euler-Bernoulli plate equation, SIAM J. Control Optim, 48 (2009), 1632-1659. [2] D. Auroux and J. Blum, A nudging-based data assimilation method: The Back and Forth Nudging (BFN) algorithm, Nonlin. Proc. Geophys., 15 (2008), 305-319. [3] C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary, SIAM J. Control. and Optim., 30 (1992), 1024-1065. [4] C. Clason and M. Klibanov, The quasi-reversibility method for thermoacoustic tomography in a heterogeneous medium, SIAM J. Sci. Comput., 30 (2009), 1-23. [5] R. F. Curtain and G. Weiss, Exponential stabilization of well-posed systems by colocated feedback, SIAM J. Control Optim., 45 (2006), 273-297. [6] B. Gebauer and O. Scherzer, Impedance-acoustic tomography, SIAM J. Appl. Math., 69 (2008), 565-576. [7] L. F. Ho, Observabilité frontière de l'équation des ondes, C. R. Acad. Sci. Paris Sér. I Math., 302 (1986), 443-446. [8] Y. Hristova, P. Kuchment and L. Nguyen, Reconstruction and time reversal in thermoacoustic tomography in acoustically homogeneous and inhomogeneous media, Inverse Problems, 24 (2008), 055006. doi: doi:10.1088/0266-5611/24/5/055006. [9] B. L. G. Jonsson, M. Gustafsson, V. H. Weston and M. V. de Hoop, Retrofocusing of acoustic wave fields by iterated time reversal, SIAM J. Appl. Math., 64 (2004), 1954-1986. [10] F.-X. Le Dimet, V. Shutyaev and I. Gejadze, On optimal solution error in variational data assimilation: Theoretical aspects, Russian J. Numer. Anal. Math. Modelling, 21 (2006), 139-152. [11] V. Komornik, On the exact internal controllability of a Petrowsky system, J. Math. Pures Appl., 71 (1992), 331-342. [12] M. Krstic, L. Magnis and R. Vazquez, Nonlinear control of the viscous burgers equation: Trajectory generation, tracking, and observer design, Journal of Dynamic Systems, Measurement, and Control, 131 (2009), 021012. doi: doi:10.1115/1.3023128. [13] P. Kuchment and L. Kunyansky, On the exact internal controllability of a Petrowsky system, European J. Appl. Math., 19 (2008), 191-224. [14] K. Liu, Locally distributed control and damping for the conservative systems, SIAM J. Control Optim., 35 (1997), 1574-1590. [15] K. D. Phung and X. Zhang, Time reversal focusing of the initial state for kirchhoff plate, SIAM J. Appl. Math., 68 (2008), 1535-1556. [16] K. Ramdani, M. Tucsnak and G. Weiss, Recovering the initial state of an infinite-dimensional system using observers, Automatica, 46 (2010), 1616-1625. [17] J. J. Teng, G. Zhang and S. X. Huang, Some theoretical problems on variational data assimilation, Appl. Math. Mech., 28 (2007), 581-591. [18] M. Tucsnak and G. Weiss, "Observation and Control for Operator Semigroups," Birkäuser, Basel, 2009. [19] X. Zou, I.-M. Navon and F.-X. Le Dimet, An optimal nudging data assimilation scheme using parameter estimation, Quart. J. Roy. Met. Soc., 118 (1992), 1193-1186.

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##### References:
 [1] C. Alves, A. L. Silvestre, T. Takahashi and M. Tucsnak, Solving inverse source problems using observability. Applications to the Euler-Bernoulli plate equation, SIAM J. Control Optim, 48 (2009), 1632-1659. [2] D. Auroux and J. Blum, A nudging-based data assimilation method: The Back and Forth Nudging (BFN) algorithm, Nonlin. Proc. Geophys., 15 (2008), 305-319. [3] C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary, SIAM J. Control. and Optim., 30 (1992), 1024-1065. [4] C. Clason and M. Klibanov, The quasi-reversibility method for thermoacoustic tomography in a heterogeneous medium, SIAM J. Sci. Comput., 30 (2009), 1-23. [5] R. F. Curtain and G. Weiss, Exponential stabilization of well-posed systems by colocated feedback, SIAM J. Control Optim., 45 (2006), 273-297. [6] B. Gebauer and O. Scherzer, Impedance-acoustic tomography, SIAM J. Appl. Math., 69 (2008), 565-576. [7] L. F. Ho, Observabilité frontière de l'équation des ondes, C. R. Acad. Sci. Paris Sér. I Math., 302 (1986), 443-446. [8] Y. Hristova, P. Kuchment and L. Nguyen, Reconstruction and time reversal in thermoacoustic tomography in acoustically homogeneous and inhomogeneous media, Inverse Problems, 24 (2008), 055006. doi: doi:10.1088/0266-5611/24/5/055006. [9] B. L. G. Jonsson, M. Gustafsson, V. H. Weston and M. V. de Hoop, Retrofocusing of acoustic wave fields by iterated time reversal, SIAM J. Appl. Math., 64 (2004), 1954-1986. [10] F.-X. Le Dimet, V. Shutyaev and I. Gejadze, On optimal solution error in variational data assimilation: Theoretical aspects, Russian J. Numer. Anal. Math. Modelling, 21 (2006), 139-152. [11] V. Komornik, On the exact internal controllability of a Petrowsky system, J. Math. Pures Appl., 71 (1992), 331-342. [12] M. Krstic, L. Magnis and R. Vazquez, Nonlinear control of the viscous burgers equation: Trajectory generation, tracking, and observer design, Journal of Dynamic Systems, Measurement, and Control, 131 (2009), 021012. doi: doi:10.1115/1.3023128. [13] P. Kuchment and L. Kunyansky, On the exact internal controllability of a Petrowsky system, European J. Appl. Math., 19 (2008), 191-224. [14] K. Liu, Locally distributed control and damping for the conservative systems, SIAM J. Control Optim., 35 (1997), 1574-1590. [15] K. D. Phung and X. Zhang, Time reversal focusing of the initial state for kirchhoff plate, SIAM J. Appl. Math., 68 (2008), 1535-1556. [16] K. Ramdani, M. Tucsnak and G. Weiss, Recovering the initial state of an infinite-dimensional system using observers, Automatica, 46 (2010), 1616-1625. [17] J. J. Teng, G. Zhang and S. X. Huang, Some theoretical problems on variational data assimilation, Appl. Math. Mech., 28 (2007), 581-591. [18] M. Tucsnak and G. Weiss, "Observation and Control for Operator Semigroups," Birkäuser, Basel, 2009. [19] X. Zou, I.-M. Navon and F.-X. Le Dimet, An optimal nudging data assimilation scheme using parameter estimation, Quart. J. Roy. Met. Soc., 118 (1992), 1193-1186.
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