TRPL+K: Thick-Restart Preconditioned Lanczos+K Method for Large Symmetric Eigenvalue Problems

Abstract

The Lanczos method is one of the standard approaches for computing a few eigenpairs of a large, sparse, symmetric matrix. It is typically used with restarting to avoid unbounded growth of memory and computational requirements. Thick-restart Lanczos is a popular restarted variant because of its simplicity and numerically robustness. However, convergence can be slow for highly clustered eigenvalues so more effective restarting techniques and the use of preconditioning is needed. In this paper, we present a thick-restart preconditioned Lanczos method, TRPL+K, that combines the power of locally optimal restarting (+K) and preconditioning techniques with the efficiency of the thick-restart Lanczos method. TRPL+K employs an inner-outer scheme where the inner loop applies Lanczos on a preconditioned operator while the outer loop augments the resulting Lanczos subspace with certain vectors from the previous restart cycle before it restarts. We first identify the differences from various relevant methods in the literature. Then, based on an optimization perspective, we show an asymptotic global quasi-optimality of a simplified TRPL+K method compared to an unrestarted global optimal method. The theory applies also to LOBPCG as a special case. Finally, we present extensive experiments showing that TRPL+K either outperforms or matches other state-of-the-art eigenmethods in both matrix-vector multiplications and computational time.

Convergence rates of individual Ritz values in block preconditioned gradient-type eigensolvers

“…It is remarkable that BPG as two-block iterations are not necessarily cluster robust for small block sizes. This drawback can be overcome by three(or more)-block iterations such as LOBPCG and restarted Davidson methods [19,20,21]. Extending our analysis of BPG to more powerful eigensolvers is desirable in our future research.…”

Convergence rates of individual Ritz values in block preconditioned gradient-type eigensolvers

“…It is remarkable that BPG as two-block iterations are not necessarily cluster robust for small block sizes. This drawback can be overcome by three(or more)-block iterations such as LOBPCG and restarted Davidson methods [17,18,19]. Extending our analysis of BPG to more powerful eigensolvers is desirable in our future research.…”

Preconditioners for Krylov subspace methods: An overview

“…We have focused here on preconditioning the inner solves within iterative eigenvalue solvers. However, for some Krylov-based eigenvalue solvers, for example certain Lanczos-based methods, it is possible to precondition the eigenvalue solver itself to accelerate convergence to the desired eigenvalues [360][361][362]. • Symmetrization: As we saw in Section 2, MINRES is only applicable when a real coefficient matrix is symmetric, while CG additionally requires positive definiteness.…”

“…This idea has been analyzed for nonsymmetric generalized eigenvalue problems [356], and applied in inverse subspace iteration [357] and Arnoldi‐type eigensolvers [358,359].We have focused here on preconditioning the inner solves within iterative eigenvalue solvers. However, for some Krylov‐based eigenvalue solvers, for example certain Lanczos‐based methods, it is possible to precondition the eigenvalue solver itself to accelerate convergence to the desired eigenvalues [360‐362]. Symmetrization: As we saw in Section 2, MINRES is only applicable when a real coefficient matrix is symmetric, while CG additionally requires positive definiteness. It is somewhat surprising, therefore, that at least in theory any linear system with a real nonsymmetric coefficient matrix , can be solved using MINRES (or possibly CG) after appropriate preconditioning.…”

“…We have focused here on preconditioning the inner solves within iterative eigenvalue solvers. However, for some Krylov‐based eigenvalue solvers, for example certain Lanczos‐based methods, it is possible to precondition the eigenvalue solver itself to accelerate convergence to the desired eigenvalues [360‐362].…”