Sparse symmetric preconditioners for dense linear systems in electromagnetism

We consider symmetric preconditioning strategies for the iterative solution of dense complex symmetric non-Hermitian systems arising in computational electromagnetics. In particular we report on the numerical behaviour of the classical Incomplete Cholesky factorization as well as some of its recent variants and consider also well known factorized approximate inverses. We illustrate the difficulties that those techniques encounter on the linear systems under consideration and give some clues to explain their disappointing behaviour. We propose two symmetric preconditioners based on Frobenius-norm minimization that use a prescribed sparsity pattern. The numerical and computational efficiency of the proposed preconditioners are illustrated on a set of model problems arising both from academic and from industrial applications.