Iterative methods for solving integral equations

TL;DR: A number of iterative algorithms to solve integral equations arising in field problems are discussed and the essential features of the Neumann Series, overrelaxation methods, Krylov subspace methods, and the conjugate gradient technique are described.

Abstract: A number of iterative algorithms to solve integral equations arising in field problems are discussed. We describe the essential features of the Neumann Series, overrelaxation methods, Krylov subspace methods, and the conjugate gradient technique. Proofs of convergence of the conjugate gradient method are directly available when the underlying integral operator is self-adjoint, and in this case the method is equivalent to the Krylov method. However, for non-self-adjoint operators the conjugate gradient method requires an implicit symmetrization which results in poorer convergence than that obtained using the Krylov method. Some convergence results are also available for overrelaxation methods for both self-adjoint and non-self-adjoint operators. Relations between all of the methods will be described and numerical performance will be contrasted using a uniform square error criterion. All the methods are treated in the continuous operator form which is especially useful in using the physical setting to arrive at effective preconditioners.