Parallelization model for successive approximations to the Rayleigh-Ritz linear variational problem

Many of the differential equations arising in science and engineering can be recast in the form of a matrix eigenvalue problem. Solution of this equation within the context of the Rayleigh-Ritz variational method may be viewed as one of the fundamental tasks of numerical analysis. Successive approximation approaches to the Rayleigh-Ritz problem seek to improve eigenvectors and eigenfunctions by sequentially refining a trial function. Parallelization of successive approximation approaches has been demonstrated numerous times in the literature; these studies addressed either the successive approximations or the matrix diagonalization levels of the algorithm. It is shown in this paper that these two strategies may be applied independently of one another, and the advantages of applying both parallelization levels simultaneously to the problem are discussed. Performance estimates for a two-tiered parallelization strategy are obtained by extrapolating from existing published performance data for which the two levels of parallelization were applied separately.