The nonlinear Landweber method applied to an inverse scattering problem for sound-soft obstacles in 3D Original Research Article

The inverse scattering problem for sound-soft obstacles is considered for both smooth and piecewise smooth surfaces in 3D. The nonlinear and ill-posed integral equation of the first kind is solved by the nonlinear Landweber method. It is an iterative regularization scheme to obtain approximations for the unknown boundary of the obstacle. It is stable with respect to noise and essentially no extra work is required to incorporate several incident waves. So far, it has only been applied to the two dimensional case. Two different integral equations are presented to obtain far-field data. Furthermore, the domain derivative and its adjoint are characterized. The integral equations of the second kind are approximated by a boundary element collocation method. The two-grid method is used to solve the large and dense linear systems. Numerical examples are illustrated to show that both smooth and piecewise smooth obstacles can be reconstructed with this method, where the latter case has not yet been reported.