Ito, Kazufumi; Ramdani, Karim; Tucsnak, Marius A time reversal based algorithm for solving initial data inverse problems. (English) Zbl 1209.35153 Discrete Contin. Dyn. Syst., Ser. S 4, No. 3, 641-652 (2011). 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. Cited in 15 Documents 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 Keywords:time reversal; initial data inverse problems; exact observability; wave equation; Schrödinger equation PDFBibTeX XMLCite \textit{K. Ito} et al., Discrete Contin. Dyn. Syst., Ser. S 4, No. 3, 641--652 (2011; Zbl 1209.35153) Full Text: DOI