Mathematical Sciences: Preconditioned Parallel Methods for Large Symmetric Eigenproblems

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9501507 Knyazev This project will study preconditioned iterative methods for computing several extreme eigenpairs of generalized eigenvalue problems, with very large symmetric matrices. Such problems are of major importance in many grand challenge scientific and engineering applications: structural dynamics and buckling, ocean modeling, quantum chemistry, and magnetohydrodynamics. They arise typically from the discretization of continuous models, described by systems of partial differential equations. Matrices may be so large that the standard numerical methods become unsatisfactory and cannot be implemented even on the most powerful modern supercomputers as they are too slow or require too much memory. Preconditioned iterative methods were specially designed for problems of that kind and now they are well understood for solving large ill-conditioned linear algebraic systems of equations. For eigenproblems the theory is still poor, and these methods are rarely used. Ideally, the methods would compute well-separated clusters of eigenvalues and corresponding eigenspaces at the same order of computational cost as that for solution of the corresponding linear algebraic system. Such processes can be effective and parallelizable. Domain decomposition and iterative substructuring methods are particularly promising. The ultimate goal of the research is to develop a complete theory of the preconditioned methods now known for symmetric eigenproblems and to find new fast, accurate, and robust iterative methods. This would provide a basis for the development of advanced software and algorithms on high performance parallel computing systems. It is expected that a research monograph on preconditioned iterative methods for large symmetric eigenvalue problems will be prepared to summarize the results of the project.