Solving Hermitian positive definite systems using indefinite incomplete factorizations

Incomplete LDL∗LDL∗ factorizations sometimes produce an indefinite preconditioner even when the input matrix is Hermitian positive definite. The two most popular iterative solvers for symmetric systems, CG and MINRES, cannot use such preconditioners; they require a positive definite preconditioner. One approach, that has been extensively studied to address this problem is to force positive definiteness by modifying the factorization process. We explore a different approach: use the incomplete factorization with a Krylov method that can accept an indefinite preconditioner. The conventional wisdom has been that long recurrence methods (like GMRES), or alternatively non-optimal short recurrence methods (like symmetric QMR and BiCGStab) must be used if the preconditioner is indefinite. We explore the performance of these methods when used with an incomplete factorization, but also explore a less known Krylov method called PCG-ODIR that is both optimal and uses a short recurrence and can use an indefinite preconditioner. Furthermore, we propose another optimal short recurrence method called IP-MINRES that can use an indefinite preconditioner, and a variant of PCG-ODIR, which we call IP-CG, that is more numerically stable and usually requires fewer iterations.