Marching on in anything: solving electromagnetic field equations with a varying physical parameter

In this paper, we consider the determination of electromagnetic fields for a (large) number of values of a physical parameter. We restrict ourselves to the case where the linear system originates from one or more integral equations. We apply an iterative procedure based on the minimization of an integrated squared error, and start this procedure from an initial estimate that is a linear combination of the last few "final" results. When the coefficients in this extrapolation are determined by minimizing the integrated squared error for the actual value of the parameter, the built-in orthogonality in this type of scheme ensures that only a few iteration steps are required to obtain the solution. The paper is organized as follows. We first describe the general approach. Second, we give an overview of various practical applications. Third, the iterative procedure is illustrated for scattering by a two-dimensional dielectric cylinder in free space. For that example, finally, we outline the use of the algorithm in transient scattering, in linearized and nonlinear inverse-scattering algorithms, and in scattering by an object in a more general environment. Results for all four applications are available, but cannot be included because of space limitations.