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 & 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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[4]

C. Clason and M. Klibanov, The quasi-reversibility method for thermoacoustic tomography in a heterogeneous medium,, SIAM J. Sci. Comput., 30 (2009), 1.   Google Scholar

[5]

R. F. Curtain and G. Weiss, Exponential stabilization of well-posed systems by colocated feedback,, SIAM J. Control Optim., 45 (2006), 273.   Google Scholar

[6]

B. Gebauer and O. Scherzer, Impedance-acoustic tomography,, SIAM J. Appl. Math., 69 (2008), 565.   Google Scholar

[7]

L. F. Ho, Observabilité frontière de l'équation des ondes,, C. R. Acad. Sci. Paris Sér. I Math., 302 (1986), 443.   Google Scholar

[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).  doi: doi:10.1088/0266-5611/24/5/055006.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[11]

V. Komornik, On the exact internal controllability of a Petrowsky system,, J. Math. Pures Appl., 71 (1992), 331.   Google Scholar

[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, 131 (2009).  doi: doi:10.1115/1.3023128.  Google Scholar

[13]

P. Kuchment and L. Kunyansky, On the exact internal controllability of a Petrowsky system,, European J. Appl. Math., 19 (2008), 191.   Google Scholar

[14]

K. Liu, Locally distributed control and damping for the conservative systems,, SIAM J. Control Optim., 35 (1997), 1574.   Google Scholar

[15]

K. D. Phung and X. Zhang, Time reversal focusing of the initial state for kirchhoff plate,, SIAM J. Appl. Math., 68 (2008), 1535.   Google Scholar

[16]

K. Ramdani, M. Tucsnak and G. Weiss, Recovering the initial state of an infinite-dimensional system using observers,, Automatica, 46 (2010), 1616.   Google Scholar

[17]

J. J. Teng, G. Zhang and S. X. Huang, Some theoretical problems on variational data assimilation,, Appl. Math. Mech., 28 (2007), 581.   Google Scholar

[18]

M. Tucsnak and G. Weiss, "Observation and Control for Operator Semigroups,", Birkäuser, (2009).   Google Scholar

[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.   Google Scholar

show all references

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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[4]

C. Clason and M. Klibanov, The quasi-reversibility method for thermoacoustic tomography in a heterogeneous medium,, SIAM J. Sci. Comput., 30 (2009), 1.   Google Scholar

[5]

R. F. Curtain and G. Weiss, Exponential stabilization of well-posed systems by colocated feedback,, SIAM J. Control Optim., 45 (2006), 273.   Google Scholar

[6]

B. Gebauer and O. Scherzer, Impedance-acoustic tomography,, SIAM J. Appl. Math., 69 (2008), 565.   Google Scholar

[7]

L. F. Ho, Observabilité frontière de l'équation des ondes,, C. R. Acad. Sci. Paris Sér. I Math., 302 (1986), 443.   Google Scholar

[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).  doi: doi:10.1088/0266-5611/24/5/055006.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[11]

V. Komornik, On the exact internal controllability of a Petrowsky system,, J. Math. Pures Appl., 71 (1992), 331.   Google Scholar

[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, 131 (2009).  doi: doi:10.1115/1.3023128.  Google Scholar

[13]

P. Kuchment and L. Kunyansky, On the exact internal controllability of a Petrowsky system,, European J. Appl. Math., 19 (2008), 191.   Google Scholar

[14]

K. Liu, Locally distributed control and damping for the conservative systems,, SIAM J. Control Optim., 35 (1997), 1574.   Google Scholar

[15]

K. D. Phung and X. Zhang, Time reversal focusing of the initial state for kirchhoff plate,, SIAM J. Appl. Math., 68 (2008), 1535.   Google Scholar

[16]

K. Ramdani, M. Tucsnak and G. Weiss, Recovering the initial state of an infinite-dimensional system using observers,, Automatica, 46 (2010), 1616.   Google Scholar

[17]

J. J. Teng, G. Zhang and S. X. Huang, Some theoretical problems on variational data assimilation,, Appl. Math. Mech., 28 (2007), 581.   Google Scholar

[18]

M. Tucsnak and G. Weiss, "Observation and Control for Operator Semigroups,", Birkäuser, (2009).   Google Scholar

[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.   Google Scholar

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