Using parallel banded linear system solvers in generalized eigenvalue problems

Abstract

Subspace iteration is a reliable and cost effective method for solving positive definite banded symmetric generalized eigenproblems, especially in the case of large scale problems. This paper discusses an algorithm that makes use of two parallel banded solvers in subspace iteration. A shift is introduced to decompose the banded linear systems into relatively independent subsystems and to accelerate the iterations. With this shift, an eigenproblem is mapped efficiently into the memories of a multiprocessor and a high speed-up is obtained for parallel implementations. An optimal shift is a shift that balances total computation and communication costs. Under certain conditions, we show how to estimate an optimal shift analytically using the decay rate for the inverse of a banded matrix, and how to improve this estimate. Computational results on iPSC/2 and iPSC/860 i,,ultipr Acessors art prcstnt.W Accesion For NTIS CRA&MDTIC TAB 1Unannounced [

Parallelized structural topology optimization for eigenvalue problems

“…Papadrakakis and Yakoumidakis (1987) proposed a parallel PCG solver to minimize the Rayleigh quotient and Zhang and Moss (1994) suggested the parallelization of the linear equation analysis appearing in the eigensolution analysis procedure. In this work, however, we employ the subspace iteration method and parallelize it in the framework of the domain decomposition.…”

Linearly scaling direct method for accurately inverting sparse banded matrices

“…There exist many works in the literature aiming at parallelizing the calculations needed to solve a banded system [1,10,[39][40][41][42][43][44][45][46][47][48][49][50]. The decision about which one to choose, and, in particular, which one to apply to the algorithms presented in this work depends, of course, on the architecture of the machine in which the calculations are going to be performed.…”