Stability estimates and reconstructions in inverse acoustic scattering using singular sources

The problem of stability for the reconstruction of the scattered field us in the exterior of a scatterer D from far field data u∞ and for the reconstruction of a sound-soft or sound-hard scatterer D from the far field pattern u∞(x̂,d), x̂,d∈Ω for all incident plane waves is investigated. In particular, we show how stability estimates are linked to the behaviour of the scattered field Φs for incident point-sources and to special minimum norm solutions for the Herglotz wave operator. The results can also be used to formulate a method of singular sources for the reconstruction of ∂D which is new in inverse scattering. The reconstruction method is described and numerical examples are provided.