ALGORITHMS: Parallel Large-Scale Sparse Linear System Solvers: New Methods and Paradigms

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The investigators have developed a set of parallel algebraic multilevel techniques for solving distributed sparse linear systems of equations, leading to the ``parallel Algebraic Recursive Multilevel Solver'' (pARMS), a portable and general purpose library for solving sparse linear systems on parallel computers. One of their goals is to begin to address the `efficiency gap' between `special purpose' and `general purpose' methods. On one extreme of the spectrum of solvers available, lie sparse direct methods which are robust, general-purpose, but expensive. On the other extreme, are special purpose methods, such as multigrid, which utilize information about the underlying problem to tailor-design certain solution procedures. Such methods can be optimal but they aim at solving the original physical problem instead of the resulting linear system. In between these extremes are the preconditioned Krylov methods whose performance is variable.This research is characterized by a different vision of what a library of parallel iterative solvers should offer. The investigators strongly believe that a new paradigm is required where a solver is no longer a monolithic box comprising a set of preconditioners, but allows the user to input specific information, indeed even parts of the solution algorithm, in order to tailor the multilevel solution procedure. This approach, which is enabled by the modular design of pARMS, is in perfect agreement with the standard approach used in industry. The new paradigms and methods envisioned in this research will alsoinclude a number of other key issues which arise in a typical solution process. Thus, it is important to exploit the underlying context when solving nonlinear systems of equations. It is also important to ensure that the pARMS code provides a fall-back option to an efficient sparse direct solver for situations where the iterative solver fails. The investigators also plan to explore new questions which are starting to emerge with the advent of the Computational Grid. Finally, they will keep in mind architecture- and problem-dependent tuning of solution algorithms.The software developed by the investigators will continue to be made publicly available. Both investigators have major interactions with researchers in important applications areas. The pARMS-X code is also likely to have an impact on graduate education, as was the case with SPARSKIT in the past.