The efficient solution of large, sparse linear systems on high-performance multiprocessors continues to be the subject of research. There is no single method that is consistently superior across application domains and computing platforms. For example, for linear systems associated with the numerical solutions of elliptic partial differential equations, domain decomposition methods provide a naturally parallel, efficient formulation. However, domain-decomposition relies on close tie-ins to partial-differential-equation discretization and mesh formulation. Consequently, it is not suitable as a general-purpose "black-box" sparse solver given only the sparse matrix and no other information. Both Krylov subspace iterative (KSP) solvers and direct solvers can be used as "black-box" solver. But, once again, both classes of methods have serious limitations. Direct solvers (based on some form of matrix factorization) are not memory scalable; memory requirement grows nonlinearly with problem size when original zeroes fill-in. KSP iterative solvers avoid the memory problem but their convergence can be very slow or fail altogether depending on the spectral properties of the sparse matrix. Robust, scalable sparse solver, suitable for a variety of large-scale applications on high-performance multiprocessors, require a spectrum of methods that range from pure iterative to pure direct methods. This project will develop parallel, "flexible," hybrid solvers based on KSP with matrix factorization preconditioners. The hybrid solvers developed will combine the inherent scalability and parallelism of KSP iterative solvers with robust preconditioners obtained using data-structures, algorithms and graph-techniques from sparse direct solver. The project will work on two fronts: (1) developing scalable, hybrid solvers by extending current technology, i.e., incomplete factorization preconditioners, and , (2) developing new improved "structurally enhanced" matrix factor preconditioners by using a combination of structural and numerical information.
