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The problem considered is that of determining the shape of a three-dimensional scattering object, illuminated by an acoustic field, from a knowledge of scattered far-field data. The far-field data are the asymptotic form of the solution of an exterior transmission problem for the Helmholtz equation. The problem is reformulated as an optimisation problem, specifically, finding that surface, in a suitably restricted class, which minimises an appropriate functional of the far field generated by the surface through the solution of the direct problem. Through the use of complete families of solutions, the problem is further reduced to finding a surface which minimises error in satisfying the transmission conditions.

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We develop a new algorithm for solving inverse acoustic scattering problems. In particular we show that this algorithm can reproduce scattering shapes efficiently, using synthetic data, from only one incident wave in the acoustically hard case and using at most two incident waves for the acoustically soft problem. In order to test the inversion algorithm we generate synthetic data using a technique which combines the distributed source method and the fundamental solution of the Helmholtz equation in order to calculate the scattered field for each of these problems. Numerical results for three-dimensional axially symmetric shapes are compared with those obtained previously by other authors.

No abstract