CAREER: Preconditioning Large, Sparse Linear Systems: Theory and Practice

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Preconditioners comprise a range of techniques for transforming a system oflinear equations Ax=b into a related system that is somehow ``better''. Forexample, a solver may take less time to compute the solution to the new system,it may compute a more accurate solution, or it may do both. Although a goodpreconditioner can be critical to computing an accurate solution efficiently,choosing a good preconditioner from the many that are available can bedifficult.The goal of this project is the development of tools to help users choose fromlarge families of preconditioners one that is likely to be well-suited to theirparticular application; the initial focus will be on preconditioners that arebased on reordering the rows and/or columns of a matrix, and those that computean incomplete factorization of the matrix. In the process of developing thesetools, the PI, undergraduate researchers, and other collaborators will work ondeveloping better heuristics, improving theoretical knowledge about specificpreconditioners, creating an extensible framework for evaluating differentpreconditioners, and making the results of extensive experimentation availablein a user-friendly form. All work will help lead to a better understanding ofthe performance of specific preconditioners on sparse systems from specificapplications.As a professor at a liberal arts college, the PI's primary plans for broaderimpact include: supervising undergraduate and postbaccalaureate research,seeking opportunities for students to present their work outside of the localcommunity and more generally preparing them for graduate work; outreach togroups traditionally underrepresented in computer science through mentoring andthe formation of support networks; and the development, refinement, anddissemination of courses that emphasize the role of mathematics in computerscience, as well as of courses that give a basic understanding of computing tostudents who are not computer science majors but whose education willnevertheless be enhanced by such knowledge.