Applying acceleration to Krylov subspace eigenvalue solvers
This work provides an incremental improvement for researchers and practitioners in numerical linear algebra by enhancing the efficiency of eigenvalue solvers for specific clustered eigenvalue scenarios.
The paper tackles the problem of accelerating Krylov subspace eigenvalue solvers for symmetric generalized eigenvalue problems, showing that applying Nesterov and Polyak acceleration methods reduces outer iterations compared to existing methods like LOBPCG, with numerical results demonstrating fewer iterations for clustered eigenvalues in small dimensions.
In this paper, we apply acceleration to the inverse-free preconditioned Krylov subspace method introduced by Golub and Ye, which solves the symmetric generalized eigenvalue problem for the algebraically smallest eigenvalue. As the method is an improvement on steepest descent, we consider acceleration based on Nesterov accelerated steepest descent and Polyak's heavy-ball method. We extend acceleration to the block version of the Krylov subspace method and prove convergence for a more generalized choice of subspace. We present numerical results demonstrating the effect of fixed and safeguarded-adaptive choice of the momentum parameter, which show convergence in fewer outer iterations compared with LOBPCG with the same subspace size and generally fewer iterations than the base method when solving for multiple clustered eigenvalues with small dimension size. We also provide an explanation for the acceleration seen from implementing Polyak's heavy-ball method, including justifying the given parameter range.