Many extreme-scale simulation codes encompass multiphysics components
in multiple spatial and length scales.
The resulting discretized sparse linear systems can be highly indefinite,
nonsymmetric and extremely ill-conditioned. For such problems,
factorization based algorithms are often the most robust algorithmic
choices among many alternatives, either being used as direct solvers,
or as coarse-grid solvers in multigrid, or as preconditioners for
iterative solvers which otherwise rarely converge.
We present our recent research on novel factorization algorithms
that are efficient for solving such problems.
We incorporate data-sparse low-rank structures, such as
hierarchical matrix algebra, to achieve lower arithmetic and
communication complexity as well as robust preconditioner.
We will illustrate both theoretical and practical aspects of the methods,
and demonstrate their performance on newer parallel machines,
using a variety of real world problems.