New Sequential and Scalable Parallel Algorithms for Incomplete Factor Preconditioning

The solution of large, sparse, linear systems of equations Ax = b is an important kernel, and the dominant term with regard to execution time, in many applications in scientific computing. The large size of the systems of equations being solved currently (millions of unknowns and equations) requires iterative solvers on parallel computers. Preconditioning, which is the process of translating a linear system into a related system that is easier to solve, is widely used to reduce solution time and is sometimes required to ensure convergence. Level-based preconditioning (ILU(e)) has long been used in serial contexts and is widely recognized as robust and effective for a wide range of problems. However, the method has long been regarded as an inherently sequential technique. Parallelism, it has been thought, can be achieved primarily at the expense of increased iterations. We dispute these claims. The first half of this dissertation takes an in-depth look at structurally based ILU(e) symbolic factorization. There are two definitions of fill level, “sum” and “max,” that have been proposed. Hitherto, these definitions have been cast in terms of matrix terminology. We develop a sequence of lemmas and theorems that provide graph theoretic characterizations of both definitions; these characterizations are based on the static graph of a matrix, G(A). Our Incomplete Fill Path Theorem characterizes fill levels per the sum definition; this is the definition that is used in most library implementations of the “classic” ILU(e) factorization algorithm. Our theorem leads to several new graph-search algorithms that compute factors identical, or nearly identical, to those computed by the “classic” algorithm. Our analyses shows that the new algorithms have lower run time complexity than that of the previously existing algorithms for certain classes of matrices that are commonly encountered in scientific applications. The second half of this dissertation presents a Parallel ILU algorithmic framework (PILU). This framework enables scalable parallel ILU preconditioning by combining concepts from domain decomposition and graph ordering. The framework can accommodate ILU(e) factorization as well as threshold-based ILUT methods. A model implementation of the framework, the Euclid library, was developed as part of this dissertation. This library was used to obtain experimental results for Poisson's equation, the Convection-Diffusion equation, and a nonlinear Radiative Transfer problem. The experiments, which were conducted on a variety of platforms with up to 400 CPUs, demonstrate that our approach is highly scalable for arbitrary ILU(e) fill levels.