A projection method with regularization for Cauchy problem of the time-harmonic Maxwell equations

We develop a projection method with regularization for reconstructing the radiation electromagnetic field in the exterior of a bounded domain from the knowledge of Cauchy data. The method is divided into two parts. We first solve the complete tangential component of the electrical field on the boundary of that domain from Cauchy data. The radiation electromagnetic field is then recovered from the complete tangential component of the electrical field. For the first part, we transform the Cauchy problem into a compact operator equation by means of the electric-to-magnetic Calderón operator and propose a projection method with regularization to solve that compact operator equation. Meanwhile, we analyze the asymptotic behavior of the singular values of the corresponding compact operator. For the second part, we expend the radiation electromagnetic field to the vector spherical harmonics. Numerical examples are finally presented to demonstrate the computational efficiency of the proposed method.