# 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 & Continuous Dynamical Systems - S, 2011, 4 (3) : 641-652. doi: 10.3934/dcdss.2011.4.641
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