# Multiple right hand side multigrid for domain wall fermions with a multigrid preconditioned block conjugate gradient algorithm

arXiv: https://arxiv.org/abs/2409.03904 · HTML: https://mlsanity.com/paper/2409.03904/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Sep 2024.

## Built on by (most cited first, 1 of 1)

- [Matrix-free Neural Preconditioner for the Dirac Operator in Lattice Gauge Theory](https://mlsanity.com/paper/2509.10378/) (2025)

## Abstract (arXiv)

We introduce a class of efficient multiple right-hand side multigrid algorithm for domain wall fermions. The simultaneous solution for a modest number of right hand sides concurrently allows for a significant reduction in the time spent solving the coarse grid operator in a multigrid preconditioner. We introduce a preconditioned block conjuate gradient with a multigrid preconditioner, giving additional algorithmic benefit from the multiple right hand sides. There is also a very significant additional to computation rate benefit to multiple right hand sides. This both increases the arithmetic intensity in the coarse space and increases the amount of work being performed in each subroutine call, leading to excellent performance on modern GPU architectures. Further, the software implementation makes use of vendor linear algebra routines (batched GEMM) that can make use of high throughput tensor hardware on recent Nvidia, AMD and Intel GPUs. The cost of the coarse space is made sub-dominant in this algorithm, and benchmarks from the Frontier supercomputer system show up to a factor of twenty speed up over the standard red-black preconditioned conjugate gradient algorithm on a large system with physical quark masses.