A comparison of eigensolvers for large-scale 3D modal analysis using AMG-preconditioned iterative methods

Abstract

SUMMARYThe goal of our paper is to compare a number of algorithms for computing a large number of eigenvectors of the generalized symmetric eigenvalue problem arising from a modal analysis of elastic structures. The shift-invert Lanczos algorithm has emerged as the workhorse for the solution of this generalized eigenvalue problem; however, a sparse direct factorization is required for the resulting set of linear equations. Instead, our paper considers the use of preconditioned iterative methods. We present a brief review of available preconditioned eigensolvers followed by a numerical comparison on three problems using a scalable algebraic multigrid (AMG) preconditioner.

Accelerated Line-search and Trust-region Methods

“…The algorithm in [BAG07], where the "Jacobi" step consists of solving approximately a certain trust-region-like problem, shows promising numerical results even without using a "Davidson" step. In locally optimal CG methods, the dimension of the acceleration subspace is kept low and the emphasis is laid on the use of efficient preconditioners [AHLT05,LAOK05]. At the other extreme, the principle of the Jacobi-Davidson method [SV96] is to let the Davidson step compensate for a crude approximation of the Jacobi update.…”

Computing the Optical Properties of Large Systems

“…For large cell volumes, it can become prohibitive to construct, store and diagonaliseĤ G,G The above problems can be avoided by using an iterative eigensolver. Since solving for a subset of eigenvalues of large matrices is a standard problem in computational science, a variety of different algorithms exist that vary in convergence properties and memory requirements [51]. Among the most popular in electronic structure theory are the Davidson [52] and Conjugate Gradients algorithms [53], which are based on minimising the Rayleigh quotient…”

Salinas : theory manual.

“…We refer you to an excellent comparison report for a few of the iterative methods available (see 6). Direct methods such as the QR algorithm or Jacobi transformations are not scalable to very large systems.…”

“…While various methods are available for solving the generalized, linear eigenvalue problem, 6 solution of the quadratic eigenvalue problem is more challenging. The approach followed here is to transform the problem into a reduced space, solve the corresponding dense matrix system completely, and project back out to the original space.…”