Spectral Projection - Robustness and Orthogonality Considerations

This work deals with a special incarnation of subspace iteration—spectral projection— in order to solve Eigenproblems of standard or generalized form, given by Hermitian matrices or definite matrix pairs. After establishing the general theory, a selection of possible approximations of the ideal filtering function to obtain an approximate projector for a specified spectral target interval is highlighted, and many aspects of the convergence behavior of the resulting methods and its pathological peculiarities are analyzed experimentally. The work touches on implementation aspects and briefly introduces an accompanying software framework for parallel solution of said eigenproblems on large hybrid parallel supercomputers with many cores. In the remainder of the work, the major crucial implication of a parallel implementation that aims to subdivide the target interval for independent computation, the maintenance or reestablishment of orthogonality among the computed eigenvectors of all subintervals, are examined. Many aspects of possible approaches are presented and their feasibility is evaluated.