Preconditioning Techniques for Linear Systems of Equations

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NSF Proposal Number: 0431068Title: Preconditioning techniques for linear systems of equationsPI: Vivek Sarin, Dept. of Computer Science, Texas A&M UniversityAbstract:The solution of linear systems of equations is a fundamental problem thatmust be tackled in various areas of science and engineering. Iterativemethods are used to solve large sparse systems from partial differentialequations as well as large dense systems from integral equations.Preconditioning techniques are necessary to accelerate the rate ofconvergence of these solvers. An ideal preconditioner should be robust,effective, parallelizable, and inexpensive to compute and apply to thelinear system. When solving dense systems, the unavailability of thecoefficient matrix often limits the choice of preconditioners. The goal ofthe project is to develop novel preconditioning techniques for sparse anddense linear systems. These techniques will use a sequence of lineartransformations to obtain a preconditioned system from the original one.Techniques will be developed to analyze the preconditioned system, andschemes will be developed to improve the effectiveness of thepreconditioner. The resulting preconditioners will be robust, effective,and inexpensive. These preconditioning techniques will be implemented,analyzed, and tested on a variety of problems. These efforts will beintegrated with interdisciplinary collaborative research activities toensure a greater impact on application areas.