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A time reversal based algorithm for solving initial data inverse problems. (English) Zbl 1209.35153

Summary: 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 \(\tau_{obs}>0, \tau \geq \tau_{obs}\) 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.

MSC:

35R30 Inverse problems for PDEs
35L50 Initial-boundary value problems for first-order hyperbolic systems
35Q93 PDEs in connection with control and optimization
35Q55 NLS equations (nonlinear Schrödinger equations)
93B07 Observability
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