[Paper Review] Comparison of Domain Wall Fermion Multigrid Methods

Peter Boyle, Azusa Yamaguchi|arXiv (Cornell University)|Mar 8, 2021

Quantum Chromodynamics and Particle Interactions27 references4 citations

TL;DR

This paper introduces a novel recursive Chebyshev polynomial-based multigrid setup scheme for Domain Wall Fermions, demonstrating that both setup and solution of twice the standard Shamir domain wall fermion system can be performed faster than a single red-black preconditioned Conjugate Gradient solve on large volumes using modern GPUs. The method achieves significant speedups over conventional solvers, particularly when setup costs are amortized across multiple Hasenbusch determinant factors, marking a key step toward practical HMC acceleration for chiral fermions.

ABSTRACT

We present a detailed comparison of several recent and new approaches to multigrid solver algorithms suitable for the solution of 5d chiral fermion actions such as Domain Wall fermions in the Shamir formulation, and also for the Partial Fraction and Continued Fraction overlap. Our focus is on the acceleration of gauge configuration sampling, and a compact nearest neighbour stencil is required to limit the calculational cost of obtaining a coarse operator. This necessitates the coarsening of a nearest neighbour operator to preserve sparsity in coarsened grids, unlike HDCG. We compare the approaches of HDCR and the Multigrid algorithm and also several new hybrid schemes. In this work we introduce a new recursive Chebyshev polynomial based setup scheme. We find that the HDCR approach, can both setup, and solve standard Shamir Domain Wall Fermions faster than a single solve with red-black preconditioned Conjugate Gradients on large volumes and for modern GPU systems such as the Summit supercomputer. This is promising for the acceleration of HMC, particularly if setup costs are shared across multiple Hasenbusch determinant factors. The setup scheme is likely generally applicable to other Fermion actions.

Motivation & Objective

• To accelerate gauge configuration sampling in lattice QCD by improving multigrid solvers for chiral fermion actions, particularly Domain Wall Fermions.
• To address the high computational cost of multigrid setup and coarse operator construction in 5D chiral fermion systems.
• To enable efficient, scalable multigrid solvers with compact nearest-neighbor stencils to limit computational overhead in large-volume simulations.
• To evaluate and compare multiple multigrid approaches, including hybrid schemes, for 5D domain wall and overlap fermion actions.
• To explore the feasibility of amortizing multigrid setup costs across multiple Hasenbusch determinant factors in HMC.

Proposed method

• Proposes a recursive Chebyshev polynomial-based multigrid setup scheme to generate coarse-space subspaces efficiently, reducing setup cost while preserving convergence properties.
• Applies two main approaches: one based on coarsening the hermitian domain wall operator (Yamaguchi:2016kop) and another based on coarsening the Wilson operator (Brower:2020xmc), both enabling nearest-neighbor coarsening to preserve sparsity.
• Introduces a hybrid coarsening scheme combining one level of Wilson operator-based coarsening with a second level of 5D domain wall operator coarsening to reduce coarse space cost.
• Employs a compact 5D stencil for the coarse operator to maintain computational efficiency, avoiding the high cost of non-local stencils from squared operators.
• Uses the GCR algorithm as a smoother in the multigrid cycle, showing superior performance over BiCGSTAB for the $M^{ar{ar{M}}}$ and hybrid schemes.
• Implements the multigrid framework on the Grid library and benchmarks on the Summit supercomputer using $16^3 \times 32$ and $48^3 \times 96$ test volumes.

Experimental results

Research questions

• RQ1 Can a recursive Chebyshev-based multigrid setup scheme reduce the total cost of solving Domain Wall Fermion systems below that of red-black preconditioned Conjugate Gradient on large lattices?
• RQ2 How does the performance of the Yamaguchi:2016kop and Brower:2020xmc approaches compare when extended to 4D $SU(3)$ gauge theory with nearest-neighbor coarsening?
• RQ3 To what extent can the multigrid setup cost be amortized across multiple Hasenbusch determinant factors in HMC, enabling net performance gains?
• RQ4 Why do non-hermitian multigrid approaches (e.g., MG-PV) fail to outperform standard solvers, and what modifications could make them viable?
• RQ5 Can hybrid coarsening schemes combining 4D and 5D coarsening reduce the coarse space cost while maintaining solver efficiency?

Key findings

• On the $48^3 \times 96$ volume, the proposed method both sets up and solves twice the standard Shamir domain wall fermion system faster than a single red-black preconditioned Conjugate Gradient solve.
• The hybrid coarsening scheme (Algorithm F) reduces the number of fine matrix multiplies from 19,082 to 2,744, significantly improving efficiency over non-red-black solvers.
• The GCR algorithm proves more effective than BiCGSTAB as a smoother in the multigrid cycle, especially for the $M^{ar{ar{M}}}$ and hybrid schemes.
• The setup cost is proportional to the four-dimensional lattice volume rather than the five-dimensional system, which reduces the scaling penalty of chiral fermions.
• The method achieves a net performance gain over red-black CG when setup costs are amortized across multiple Hasenbusch determinant factors, suggesting strong potential for HMC acceleration.
• Non-hermitian multigrid approaches (MG-PV) remain ineffective due to high coarse space costs, indicating that further development is needed for such schemes to be competitive.