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We discuss algorithms for domain wall fermions focussing on accelerating Hybrid Monte Carlo sampling of gauge configurations. Firstly a new multigrid algorithm for domain wall solvers and secondly a domain decomposed hybrid monte carlo approach applied to large subvolumes and optimised for GPU accelerated nodes. We propose a formulation of DD-RHMC that is suitable for the simulation of odd numbers of fermions., Comment: 9 pages

The search for new physics requires a joint experimental and theoretical effort. Lattice QCD is already an essential tool for obtaining precise model-free theoretical predictions of the hadronic processes underlying many key experimental searches, such as those involving heavy flavor physics, the anomalous magnetic moment of the muon, nucleon-neutrino scattering, and rare, second-order electroweak processes. As experimental measurements become more precise over the next decade, lattice QCD will play an increasing role in providing the needed matching theoretical precision. Achieving the needed precision requires simulations with lattices with substantially increased resolution. As we push to finer lattice spacing we encounter an array of new challenges. They include algorithmic and software-engineering challenges, challenges in computer technology and design, and challenges in maintaining the necessary human resources. In this white paper we describe those challenges and discuss ways they are being dealt with. Overcoming them is key to supporting the community effort required to deliver the needed theoretical support for experiments in the coming decade., Comment: Contribution to Snowmass 2021. 22 pages

We discuss a substantial update to the Grid software library for Lattice QCD, enabling it to port to multiple GPU architectures while retaining CPU vectorisation and SIMD execution within OpenMP threads. The GPU environments supported include vendor specific Nvidia CUDA and AMD HIP environments and a (mostly) standards based SYCL implementation. This is performed by an internal abstraction interface giving single source cross-platform performance portability across all number of planned Exascale architectures, and all those planned by the US Department of Energy., Comment: 10 pages, 6 figures

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.

We present a progress report on a new class of multigrid solver algorithm suitable for the solution of 5d chiral fermions such as Domain Wall fermions and the Continued Fraction overlap. Unlike HDCG \cite{Boyle:2014rwa}, the algorithm works directly on a nearest neighbour fine operator. The fine operator used is Hermitian indefinite, for example $\Gamma_5 D_{dwf}$, and convergence is achieved with an indefinite matrix solver such as outer iteration based on conjugate residual. As a result coarse space representations of the operator remain nearest neighbour, giving an 8 point stencil rather than the 81 point stencil used in HDCG. It is hoped this may make it viable to recalculate the matrix elements of the little Dirac operator in an HMC evolution., Comment: 2 figures, 6 pages

In this proceedings we discuss the motivation, implementation details, and performance of a new physics code base called Grid. It is intended to be more performant, more general, but similar in spirit to QDP++\cite{QDP}. Our approach is to engineer the basic type system to be consistently fast, rather than bolt on a few optimised routines, and we are attempt to write all our optimised routines directly in the Grid framework. It is hoped this will deliver best known practice performance across the next generation of supercomputers, which will provide programming challenges to traditional scalar codes. We illustrate the programming patterns used to implement our goals, and advances in productivity that have been enabled by using new features in C++11., Comment: 14 pages, Lattice 2015

We present results for the dependence of the residual mass of domain wall fermions (DWF) on the size of the fifth dimension and its relation to the density and localization properties of low-lying eigenvectors of the corresponding hermitian Wilson Dirac operator relevant to simulations of 2+1 flavor domain wall QCD. Using the DBW2 and Iwasaki gauge actions, we generate ensembles of configurations with a $16^3\times 32$ space-time volume and an extent of 8 in the fifth dimension for the sea quarks. We demonstrate the existence of a regime where the degree of locality, the size of chiral symmetry breaking and the rate of topology change can be acceptable for inverse lattice spacings $a^{-1} \ge 1.6$ GeV., Comment: 59 Pages, 23 figures, 1 MPG linked

We study the chiral condensate, $<\bar\psi\psi>$, and various quark bilinear vertex functions for domain wall fermions at different lattice scales, with both the Wilson and DBW2 gauge actions, in both quenched and dynamical fermion simulations. We use the vertex functions to calculate renormalization factors within a non-perturbative scheme., Comment: 3 pages 4 figures: Lattice2003(chiral)

Critical conjectures on confinement in the Landau gauge is numerically tested in focus to Gribov copy effects. One of the subjects is of the Kugo-Ojima confinement criterion and the other is of various viewpoints in the Gribov-Zwanziger theory. We use the smearing gauge as a reference gauge free of Gribov copy, and performed three types of simulations, log U, U-linear and log U in the smearing gauge. It is found that Gribov copy effect on the Kugo-Ojima parameter is small. log U and U-linear simulations yield only global scale factor difference in gluon propagator and in ghost propagator, and about 10% difference in Kugo-Ojima parameter. The horizon function defined by Zwanziger is evaluated in three types of gauge field and compared. All data show the negative horizon function as expected., Comment: 4 pages, 2 eps figures, espcrc2.sty included, Lattice 2000 contribution(Confinement and Strings)

The Kugo-Ojima color confinement criterion, which is based on the BRST symmetry of the continuum QCD is numerically tested by the lattice Landau gauge simulation. We first discuss the Gribov copy problem and the BRST symmetry on the lattice. The lattice Landau gauge can be formulated with options of the gauge field definition, U(link)-linear type or log U type. The Kugo-Ojima parameter u^a_b which is expected to be -1^a_b in the continuum theory is found to be -0.7*1^a_b in the strong coupling region, and the magnitude is a little less in the weak coupling region in log U type simulation. Those values are weakened even further in U-linear type. The horizon function defined by Zwanziger is evaluated in both types of gauge field and compared. The horizon function in the log U version is larger than the other, but in the weak coupling region, the expectation value of the horizon function is suggested to be zero or negative., Comment: 17 pages, 2 figures, ICHEP2000 contributed paper

A class of algorithms for the Landau gauge fixing is proposed, which makes the steepest ascent (SA) method be more efficient by concepts of genetic algorithm. Main concern is how to incorporate random gauge transformation (RGT) %, mutation in genetic algorithm (GA) terminology, to gain higher achievement of the minimal Landau gauge fixing, and to keep lower time consumption. One of these algorithms uses the block RGT, and another uses RGT controlled by local fitness density, and the last uses RGT determined by Ising Monte Carlo process. We tested these algorithms on SU(2) lattice gauge theory in 4 dimension with small $\beta$s, 2.0, 1.75 and 1.5, and report improvements in hit rate and/or in time consumption, compared to other methods., Comment: 3 pages, 3 figures LATTICE'99(ALGORITHM)

Applying Genetic Algorithm for the Lattice Gauge Theory is formed to be an effective method to minimize the action of gauge field on a lattice. In 4 dimensions, the critical point and the Wilson loop behaviour of SU(2) lattice gauge theory as well as the phase transition of U(1) theory have been studied. The proper coding methodi has been developed in order to avoid the increase of necessary memory and the overload of calculation for Genetic Algorithm. How hichhikers toward equlibrium appear against kidnappers is clarified., Comment: 3 pages 4 figures

The contributions from the Ochanomizu CP Study Group to the KEK meetings on "CP violation and its origin" (1993-1997) are summarized on electroweak baryogenesis and anomalous W-boson couplings. We survey planned new experiments which could examine some aspects studied in our contributions. We also discuss several issues on baryogenesis. Ten problems are presented for further studies., Comment: 18 pages, To appear in the Proceedings of the KEK meetings on 'CP violation and its origin' (1993-1997). Addresses are corrected

The electroweak baryogenesis is studied in the charge transport mechanism with the vector-like quark model. Introducing an extra vector-like up-type quark and a singlet Higgs scalar with the mass of the order of a few hundred GeV, the baryon number generation from the bubble wall is estimated. We show that this scenario is consistent with the measurement of the present baryon to entropy ratio of our universe, if the parameters are in the right region., Comment: Latex file, 17pp, 1 Postscript figure

The baryogenesis is reanalyzed based on the model by A.G.Cohen et al., in which the lepton number, generated by the neutrinos' scattering from the bubble walls appearing in the development of the electroweak phase transition, is converted to the baryon number excess through the sphaleron transition. A formula obtained in this paper on the lepton number production rate is correct for the both thin and thick walls within the linear approximation. Investigation on the time-development of the first order phase transition is simulated, including the temporal change of the wall velocity as well as the fusion effect of the bubbles. The details of such phase transition dynamics are found to affect considerably the final value of the baryon number excess., Comment: OCHA-PP-44, (contribution paper to ICHEP94, Glasgow, 20 - 27 th July 1994)