On the Convergence of Ritz Pairs and Refined Ritz Vectors for Quadratic Eigenvalue Problems

For a given subspace, the Rayleigh-Ritz method projects the large quadratic eigenvalue problem (QEP) onto it and produces a small sized dense QEP. Similar to the Rayleigh-Ritz method for the linear eigenvalue problem, the Rayleigh-Ritz method defines the Ritz values and the Ritz vectors of the QEP with respect to the projection subspace. We analyze the convergence of the method when the angle between the subspace and the desired eigenvector converges to zero. We prove that there is a Ritz value that converges to the desired eigenvalue unconditionally but the Ritz vector converges conditionally and may fail to converge. To remedy the drawback of possible non-convergence of the Ritz vector, we propose a refined Ritz vector that is mathematically different from the Ritz vector and is proved to converge unconditionally. We construct examples to illustrate our theory.Comment: 20 page

Hybrid preconditioning for iterative diagonalization of ill-conditioned generalized eigenvalue problems in electronic structure calculations

The iterative diagonalization of a sequence of large ill-conditioned generalized eigenvalue problems is a computational bottleneck in quantum mechanical methods employing a nonorthogonal basis for {\em ab initio} electronic structure calculations. We propose a hybrid preconditioning scheme to effectively combine global and locally accelerated preconditioners for rapid iterative diagonalization of such eigenvalue problems. In partition-of-unity finite-element (PUFE) pseudopotential density-functional calculations, employing a nonorthogonal basis, we show that the hybrid preconditioned block steepest descent method is a cost-effective eigensolver, outperforming current state-of-the-art global preconditioning schemes, and comparably efficient for the ill-conditioned generalized eigenvalue problems produced by PUFE as the locally optimal block preconditioned conjugate-gradient method for the well-conditioned standard eigenvalue problems produced by planewave methods

Some Preconditioning Techniques for Saddle Point Problems

Saddle point problems arise frequently in many applications in science and engineering, including constrained optimization, mixed finite element formulations of partial differential equations, circuit analysis, and so forth. Indeed the formulation of most problems with constraints gives rise to saddle point systems. This paper provides a concise overview of iterative approaches for the solution of such systems which are of particular importance in the context of large scale computation. In particular we describe some of the most useful preconditioning techniques for Krylov subspace solvers applied to saddle point problems, including block and constrained preconditioners.\ud \ud The work of Michele Benzi was supported in part by the National Science Foundation grant DMS-0511336

A robust and efficient implementation of LOBPCG

Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) is widely used to compute eigenvalues of large sparse symmetric matrices. The algorithm can suffer from numerical instability if it is not implemented with care. This is especially problematic when the number of eigenpairs to be computed is relatively large. In this paper we propose an improved basis selection strategy based on earlier work by Hetmaniuk and Lehoucq as well as a robust convergence criterion which is backward stable to enhance the robustness. We also suggest several algorithmic optimizations that improve performance of practical LOBPCG implementations. Numerical examples confirm that our approach consistently and significantly outperforms previous competing approaches in both stability and speed