We consider the inverse problem of the Helmholtz equation at fixed wave number. First we study the Born approximation. Using the limiting absorption principle we give an explicit error estimate, showing that the condition for the Born approximation to hold is that the phase shift is small. We consider also the determination of the support of a scatterer from a single incoming wave. Second we derive an error estimate for the eikonal approximation that is also based on limiting absorption. Third we study an adjoint method that makes use of initial value techniques for the Helmholtz equation. We investigate the resolution and present numerical results.
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