Computational Techniques for Differential Equation Eigenvalue Problems on Vector Processors

Research Report MRR99-032

Computational Techniques for Differential Equation Eigenvalue Problems on Vector Processors

D.L.Harrar II and M.R.Osborne

Abstract: Examples are presented which illustrate techniques for discretizing differential equation eigenvalue problems so that the resulting algebraic eigenvalue problems have block bidiagonal form and hence are amenable to solution using highly vectorized solvers based on wrap-around partitioning. Problems arising in the study of ocean acoustics, chemical reactions, and hydrodynamic stability are considered. In each case the problem is formulated as a generalized eigenvalue problem in which the matrices have (conformal) block bidiagonal form. These are solved using Newton's method in conjunction with inverse iteration, and a form of multiplicative Wielandt deflation. The methods are particularly suitable for use with continuation techniques in order to follow an eigenvalue as a function of an auxilliary parameter. An impressive convergence rate of 3.56 is attainable. A limiting case of multiplicative Wielandt deflation corresponds to successive (bi)orthogonalisation of equation right hand sides to previously computed eigenvectors. It has proved remarkably stable even when large numbers of eigenvectors have been computed; it makes economical use of the eigenvector information generated by the inverse iteration algorithm; and it preserves block bidiagonal sparsity. The problems are solved on Fujitsu VPP300 processors and the efficient vectorization via wrap-around partitioning techniques is evident based on extended performance experiments.

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