Multigrid for Chiral Lattice Fermions: Domain Wall

Abstract

Critical slowing down for the Krylov Dirac solver presents a major obstacle to further advances in lattice field theory as it approaches the continuum solution. We propose a new multi-grid approach for chiral fermions, applicable to both the 5-d domain wall or 4-d Overlap operator. The central idea is to directly coarsen the 4-d Wilson kernel, giving an effective domain wall or overlap operator on each level. We provide here an explicit construction for the Shamir domain wall formulation with numerical tests for the 2-d Schwinger prototype, demonstrating near ideal multi-grid scaling. The framework is designed for a natural extension to 4-d lattice QCD chiral fermions, such as the Möbius, Zolotarev or Borici domain wall discretizations or directly to a rational expansion of the 4-d Overlap operator. For the Shamir operator, the effective overlap operator is isolated by the use of a Pauli-Villars preconditioner in the spirit of the Kähler-Dirac spectral map used in a recent staggered MG algorithm [1].

arXiv comment39 pages, 13 figures

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Cited by in corpus (5)

• 50 Years of Quantum Chromodynamics
• Status of Lattice QCD Determination of Nucleon Form Factors and their Relevance for the Few-GeV Neutrino Program
• Neural-network preconditioners for solving the Dirac equation in lattice gauge theory
• Comparison of Domain Wall Fermion Multigrid Methods
• Optimizing Staggered Multigrid for Exascale performance