OPINS: An Orthogonally Projected Implicit Null-Space Method for Singular and Nonsingular Saddle-Point Systems∗

Saddle-point systems appear in many scientific and engineering applications. The systems can be sparse, symmetric or nonsymmetric, and possibly singular. In many of these applications, the number of constraints is relatively small compared to the number of unknowns. The traditional null-space method is inefficient for these systems, because it is expensive to find the null space explicitly. Some alternatives, notably constraintpreconditioned or projected Krylov methods, are relatively efficient, but they can suffer from numerical instability and even nonconvergence. In addition, most existing methods require the system to be nonsingular or be reducible to a nonsingular system. In this paper, we propose a new method, called OPINS, for singular and nonsingular saddle-point systems. OPINS is equivalent to the null-space method with an orthogonal projector, without forming the orthogonal basis of the null space explicitly. OPINS can not only solve for the unique solution for nonsingular saddlepoint problems, but also find the minimum-norm solution in terms of the solution variables for singular systems. The method is efficient and easy to implement using existing Krylov solvers for singular systems. At the same time, it is more stable than the other alternatives, such as projected Krylov methods. We present some preconditioners to accelerate the convergence of OPINS for nonsingular systems, and compare OPINS against some present state-of-the-art methods for various types of singular and nonsingular systems. 1This work was supported by DoD-ARO under contract #W911NF0910306. The third author is also supported by a subcontract to Stony Brook University from Argonne National Laboratory under Contract DE-AC02-06CH11357 for the SciDAC program funded by the Office of Science, Advanced Scientific Computing Research of the U.S. Department of Energy. 2Dept. of Applied Math. & Stat., Stony Brook University, Stony Brook, NY 11794, USA. 3Corresponding author. Email: [email protected]. 1 ar X iv :1 51 1. 06 84 5v 1 [ m at h. N A ] 2 1 N ov 2 01 5 ORTHOGONALLY PROJECTED IMPLICIT NULL-SPACE METHOD 2