Numerical Methods for the Nonlinear Eigenvalue Problems in High Performance Materials Design

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The objective of this project is to develop adaptive methods for the first principles prediction of the properties technologically important materials. These materials may be characterized as having complex structures and compositions, large numbers of particles ( 100) and little or no symmetry. The methods being investigated include a suite of iterative, acceleration, and minimization schemes for the nonlinear eigenvalue problem and accompanying elliptic problem arising in first principles prediction of material properties. This suite includes the eigenvalue iterative methods of Rayleigh-Ritz and Longsine and McCormick; minimization methods such us conjugate gradient, steepest descent, and the trace minimization method by Sameh and Wisniewski; and multilevel methods such as multigrid and fast multipole method. These methods are being used with adaptive grid to exploit the underlying dynamical, localized structure inherent in electronic wave functions. Modern parallel architectures are being used to reduce the computational time and memory costs. The various components of this project are being developed through the close collaboration of researchers from chemistry, physics, computer science, and numerical analysis.