Adaptive Projection Subspace Dimension for the Thick-Restart Lanczos Method

The Thick-Restart Lanczos (TRLan) method is an effective method for solving large-scale Hermitian eigenvalue problems. However, its performance strongly depends on the dimension of the projection subspace. In this paper, we propose an objective function to quantify the effectiveness of a chosen subspace dimension, and then introduce an adaptive scheme to dynamically adjust the dimension at each restart. An open-source software package, nu-TRLan, which implements the TRLan method with this adaptive projection subspace dimension is available in the public domain. The numerical results of synthetic eigenvalue problems are presented to demonstrate that nu-TRLan achieves speedups of between 0.9 and 5.1 over the static method using a default subspace dimension. To demonstrate the effectiveness of nu-TRLan in a real application, we apply it to the electronic structure calculations of quantum dots. We show that nu-TRLan can achieve speedups of greater than 1.69 over the state-of-the-art eigensolver for this application, which is based on the Conjugate Gradient method with a powerful preconditioner