A block MINRES algorithm based on the banded Lanczos method

TL;DR

This paper introduces a novel block MINRES algorithm based on the band Lanczos method, enhancing efficiency and breakdown detection for symmetric indefinite matrices, with applications in high-performance computing.

Contribution

It develops a new block MINRES algorithm utilizing the band Lanczos process, improving breakdown handling and communication efficiency in large-scale computations.

Findings

Effective detection of breakdowns via scalar values.

Improved communication efficiency through block operations.

Numerical experiments demonstrate performance benefits.

Abstract

We develop a block minimum residual (MINRES) algorithm for symmetric indefinite matrices. This version is built upon the band Lanczos method that generates one basis vector of the block Krylov subspace per iteration rather than a whole block as in the block Lanczos process. However, we modify the method such that the most expensive operations are still performed in a block fashion. The benefit of using the band Lanczos method is that one can detect breakdowns from scalar values arising in the computation, allowing for a handling of breakdown which is straightforward to implement. We derive a progressive formulation of the MINRES method based on the band Lanczos process and give some implementation details. Specifically, a simple reordering of the steps allows us to perform many of the operations at the block level in order to take advantage of communication efficiencies offered by the block Lanczos process. This is an important concern in the context of next-generation super computing applications. We also present a technique allowing us to maintain the block size by replacing dependent Lanczos vectors with pregenerated random vectors whose orthogonality against all Lanczos vectors is maintained. Numerical results illustrate the performance on some sample problems. We present experiments that show how the relationship between right-hand sides can effect the performance of the method.