M. Nool et A. Van Der Ploeg, A parallel Jacobi-Davidson-type method for solving large generalized eigenvalue problems in magnetohydrodynamics, SIAM J SC C, 22(1), 2000, pp. 95-112
We study the solution of generalized eigenproblems generated by a model whi
ch is used for stability investigation of tokamak plasmas. The eigenvalue p
roblems are of the form Ax = lambda Bx, in which the complex matrices A and
B are block-tridiagonal, and B is Hermitian positive definite. The Jacobi
Davidson method appears to be an excellent method for parallel computation
of a few selected eigenvalues because the basic ingredients are matrix vect
or products, vector updates, and inner products. The method is based on sol
ving projected eigenproblems of order typically less than 30.
We apply a complete block LU decomposition in which reordering strategies b
ased on a combination of block cyclic reduction and domain decomposition re
sult in a well-parallelizable algorithm. One decomposition can be used for
the calculation of several eigenvalues. Spectral transformations are presen
ted to compute certain interior eigenvalues and their associated eigenvecto
rs. The convergence behavior of several variants of the Jacobi Davidson alg
orithm is examined. Special attention is paid to the parallel performance,
memory requirements, and prediction of the speed-up. Numerical results obta
ined on a distributed memory Cray T3E are shown.