Bootstrapped Block Lanczos for large-dimension eigenvalue problems

TL;DR

This paper introduces a bootstrapped block Lanczos method that uses an initial approximate eigenvector-based pivot to significantly reduce the number of matrix multiplications needed for large eigenvalue problems, improving efficiency.

Contribution

The paper presents a novel bootstrapped pivot strategy for block Lanczos, which accelerates convergence by leveraging approximate eigenvectors, outperforming traditional methods.

Findings

Reduces time-to-solution by up to a factor of ten.

Requires the pivot to have overlap with final eigenvectors.

Effective in nuclear structure computational code.

Abstract

The Lanczos algorithm has proven itself to be a valuable matrix eigensolver for problems with large dimensions, up to hundreds of millions or even tens of billions. The computational cost of using any Lanczos algorithm is dominated by the number of sparse matrix-vector multiplications until suitable convergence is reached. Block Lanczos replaces sparse matrix-vector multiplication with sparse matrix-matrix multiplication, which is more efficient, but for a randomly chosen starting block (or pivot), more multiplications are required to reach convergence. We find that a bootstrapped pivot block, that is, an initial block constructed from approximate eigenvectors computed in a truncated space, leads to a dramatically reduced number of multiplications, significantly outperforming both standard vector Lanczos and block Lanczos with a random pivot. A key condition for speed-up is that the pivot block have a non-trivial overlap with the final converged vectors. We implement this approach in a configuration-interaction code for nuclear structure, and find a reduction in time-to-solution by a factor of two or more, up to a factor of ten.