Numerical solution of a non-linear ill-posed problem arising in inverse scattering

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Abstract
The author considers the problem of determining the shape of a two-dimensional obstacle from far-field scattering data. A model for sparse, discrete, error-contaminated data is introduced. This model has applications for both linear acoustic and electromagnetic inverse scattering problems. By parametrising the boundary of the obstacle, a one-dimensional problem, which is non-linear and ill-posed, is obtained. A numerical method is presented to deal with the ill-posedness, non-linearity and error in the data. This method yields a parametrised sequence of smooth approximate solutions. A statistical technique known as generalised cross validation is then used to determine an appropriate value of the smoothing parameter. Two numerical examples are given, showing that the method is quite robust.

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