Fast spectral methods for the shape identification problem of a perfectly conducting obstacle

We are concerned with fast methods for the numerical implementation of the direct and inverse scattering problems for a perfectly conducting obstacle. The scattering problem is usually reduced to a single uniquely solvable modified combined-field integral equation (MCFIE). For the numerical solution of the M-CFIE we propose a new high-order spectral algorithm by transporting this equation on the unit sphere via the Piola transform. The inverse problem is formulated as a nonlinear least squares problem for which the iteratively regularized Gauss-Newton method is applied to recover an approximate solution. Numerical experiments are presented in the special case of star-shaped obstacles.