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In this work we shall review the (phased) inverse scattering problem and thenpursue the phaseless reconstruction from far-field data with the help of theconcept of scattering coefficients. We perform sensitivity, resolution andstability analysis of both phased and phaseless problems and compare the degreeof ill-posedness of the phased and phaseless reconstructions. The phaselessreconstruction is highly nonlinear and much more severely ill-posed. Algorithmsare provided to solve both the phased and phaseless reconstructions in thelinearized case. Stability is studied by estimating the condition number of theinversion process for both the phased and phaseless cases. An optimal strategyis suggested to attain the infimum of the condition numbers of the phaselessreconstruction, which may provide an important guidance for efficient phaselessmeasurements in practical applications. To the best of our knowledge, thestability analysis in terms of condition numbers are new for the phased andphaseless inverse scattering problems, and are very important to help usunderstand the degree of ill-posedness of these inverse problems. Numericalexperiments are provided to illustrate the theoretical asymptotic behavior, aswell as the effectiveness and robustness of the phaseless reconstructionalgorithm.

Phased and phaseless domain reconstruction in inverse scattering problem via scattering coefficients