Innovative Algorithms and Techniques for Large Scale Simulations

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This project aims to develop a variety of robust, effective, and parallelizable solvers and preconditioners for sparse linear systems arising from finite element modeling and dense linear systems arising from boundary element modeling approaches. As is evidenced from our preliminary results, a careful cross-breeding of techniques from these two areas is capable of yielding the breakthrough needed in designing effective parallel preconditioners and solvers. The techniques proposed for sparse linear systems are based on a hierarchy of representations using block low-rank matrices and multipole approximations of the inverse. The project will study the implications of these techniques for compression schemes used in approximate dense linear system solvers as well. The project will also investigate algebraic multilevel techniques for sparse linear systems in which hiearchical representations are used to obtain an alternate preconditioned system. Parallel formulations of all these techniques will be developed along with theoretical estimates for convergence and parallel performance. In addition to these, this study will investigate projection-based techniques that affect preconditioning implicitly through a set of concurrent projections onto appropriate subspaces. Such techniques include row-projection methods and are robust forms of approximate inverse preconditioners.