Deflating Eigenvalues for Linear Equations in QCD

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Efficient deflated iterative methods will be developed for solving linear equations with multiple right hand sides. One technique involves a deflated GMRES method that both solves the linear equations and computes eigenvectors. These eigenvectors then are used to assist in the solution of subsequent right-hand sides. This technique will be analyzed and evaluated for efficiency. For nonrestarted iterative methods such as BiCGStab and QMR, a better projection than minimum residual will be developed using both right and left eigenvectors. Simultaneous solution of multiply shifted systems will also be analyzed. Systems of linear equations arise in many areas of science, and it is common for these systems to have multiple right-hand sides and slightly changed parameters. One important example is the calculation of a set of interactions which involve simultaneously created quark-antiquark pairs. These co-called disconnected diagrams present a significant obstacle to progress in understanding the physics of the strong interaction (QCD), a theory which is fundamental to our knowledge of how the universe is built. In addition, new types of more realistic simulations of quarks in QCD are making a large demand on scientific computing. The goal of this interdisplinary math/physics project is to make a major advance in the iterative solution of linear equations in QCD and other scientific areas. Basic algorithmic improvements will be proposed and evaluated which will advance progress in these fields.