ABSTRACT:

Searching for the optimal eigensolver, we describe the Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) Method for symmetric eigenvalue problems, based on a local Rayleigh-Ritz optimization of a three-term recurrence. LOBPCG can be viewed as a nonlinear conjugate gradient method of minimization of the Rayleigh quotient, which takes advantage of the optimality of the Rayleigh-Ritz procedure.

Numerical results establish that our LOBPCG Method is practically as efficient as the best possible algorithm on the whole class of preconditioned eigensolvers. We discuss several competitors, namely, some inner-outer iterative preconditioned eigensolvers. Direct numerical comparisons with one of them, the inexact Jacobi-Davidson method, show that our LOBPCG method is more robust and converges almost two times faster.

A MATLAB code of the LOBPCG method and the Preconditioned Eigensolvers Benchmarking are available at http://math.ucdenver.edu/~aknyazev/software/CG/

The talk is mostly based on the papers:

Andrew Knyazev, "Toward the Optimal Preconditioned Eigensolver: Locally Optimal Block Preconditioned Conjugate Gradient Method." Published as a technical report UCD-CCM 149, 2000, at the Center for Computational Mathematics, University of Colorado Denver, see
http://math.ucdenver.edu/ccmreports/rep149.ps.gz.
A revised version accepted to SIAM SISC.

Andrew Knyazev and Klaus Neymeyr, "A geometric theory for preconditioned inverse iteration. III: A short and sharp convergence estimate for generalized eigenvalue problems." Published as a technical report UCD-CCM 173, 2000, at the Center for Computational Mathematics, University of Colorado Denver, see http://math.ucdenver.edu/ccmreports/rep173.ps.gz and as a technical report SFB 382, Tuebingen, Report 161, April 2001. Submitted to Linear Algebra and Its Applications, 2001.