Inverse problems for acoustic waves using the penalised likelihood method

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Abstract
The authors consider the following inverse problem for scattering of time-harmonic waves by an impenetrable obstacle. Given data obtained from the scattered field at infinity, they determine the location of the boundary of the obstacle. To solve the forward scattering problem, they use the T matrix method, or null field approach. The ill-posedness of the inverse problem is discussed. To solve the inverse problem, an optimal control, or 'data fitting', approach is used, and stable approximate solutions are obtained using a technique known as the penalised likelihood method. Results of a numerical study are presented, showing the effects on the accuracy of the inversion of different types of far field data, various incident fields, and random error in the data. In two dimensions, several different obstacle shapes are considered. An example for an axially symmetric obstacle in three dimensions is also presented.