Inverse scattering via Heisenberg's uncertainty principle

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Abstract
We present a stable method for the fully nonlinear inverse scattering problem of the Helmholtz equation in two dimensions. The new approach is based on the observation that ill-posedness of the inverse problem can be beneficially used to solve it. This means that not all equations of the nonlinear problem are strongly nonlinear due to the ill-posedness, and that when solved recursively in a proper order, they can be reduced to a collection of essentially linear problems. The algorithm requires multi-frequency scattering data, and the recursive linearization is acheived by a standard perturbational analysis on the wavenumber k. At each frequency k, the algorithm determines a forward model which produces the prescribed scattering data. It first solves nearly linear equations at the lowest k to obtain low-frequency modes of the scatterer. The approximation is used to linearize equations at the next higher k to produce a better approximation which contains more modes of the scatterer, and so on, until a sufficiently high wavenumber k where the dominant modes of the scatterer are essentially recovered. The robustness of the procedure is demonstrated by several numerical examples.

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