A time reversal based algorithm for solving initial data inverse problems

We propose an iterative algorithm to solve initial data inverse problems for a class of linear evolution equations, including the wave, the plate, the Schr{o}dinger 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.