多网格减缩GMRES优化晶格QCD求解：适应性方法与临界速度改善

138 浏览量更新于2024-09-05 收藏 285KB PDF 举报

本文主要探讨了在量子色动力学（Lattice QCD）的数值求解中，如何通过带多重网格的减缩GMRES算法来解决因小晶格间距和轻夸克质量导致的严重减速问题。在传统的矩阵特征值放缩方法中，晶格QCD求解器在处理精细网格时会遇到性能瓶颈。为了提高计算效率和缩放比例，作者们针对二维Schwinger模型设计了一种自适应多级网格策略。研究的重点集中在三个网格层次结构中，即细网格、中间网格和粗网格。研究者采用了带放气（deflation）的GMRES算法，特别是GMRES-DR（Deflated GMRES with deflation）和GMRES-Proj（GMRES with projected deflation）。放气技术旨在消除或削弱矩阵中的某些特征值，从而减少求解过程中的困难，特别是在粗网格级别上进行的计算。实验对比了放气和非放气方法在不同网格层次上的效果。研究发现，在中间网格上实施部分求解策略能够有效地配合粗网格上的放气，使得放气求解在降低容忍度的同时保持较好的性能。这种方法特别在接近临界质量的晶格尺寸下显示出良好的缩放效果，意味着随着网格分辨率的提高，计算复杂度的增长得到有效控制。此外，文章还提到了研究的局限性和未来可能的研究方向，包括进一步优化网格选择策略，以及在更高维度的QCD模型中应用这种多级网格与放气结合的技术。该研究不仅对于提升Lattice QCD的计算效率具有实际意义，也为数值线性代数在高精度物理模拟中的应用提供了新的解决方案。由于是开放访问资源，这项工作对于学术界和实践者来说都是一个重要的贡献，促进了理论物理学和数值计算技术的融合。

Physics Letters B 803 (2020) 135281

Contents lists available at ScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

Deﬂated GMRES with multigrid for lattice QCD

Travis Whyte

, Walter Wilcox

a,∗

, Ronald B. Morgan

Department of Physics, Baylor University, Waco, TX 76798-7316, United States of America

Department of Mathematics, Baylor University, Waco, TX 76798-7316, United States of America

a r t i c l e i n f o a b s t r a c t

Article history:

Received 5 December 2019

Accepted 4 February 2020

Available online 12 February 2020

Editor: B. Grinstein

Keywords:

Lattice QCD

Matrix deﬂation

Multigrid

Lattice QCD solvers encounter critical slowing down for ﬁne lattice spacings and small quark mass.

Traditional matrix eigenvalue deﬂation is one approach to mitigating this problem. However, to improve

scaling we study the effects of deﬂating on the coarse grid in a hierarchy of three grids for adaptive

multigrid applications of the two dimensional Schwinger model. We compare deﬂation at the ﬁne

and coarse levels with other non deﬂated methods. We ﬁnd the inclusion of a partial solve on the

intermediate grid allows for a low tolerance deﬂated solve on the coarse grid. We ﬁnd very good scaling

in lattice size near critical mass when we deﬂate at the coarse level using the GMRES-DR and GMRES-Proj

algorithms.

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

1. Introduction

The problem of critical slowing down and strong scaling is one

of the foremost problems facing modern Lattice Quantum Chro-

modynamics

(LQCD) simulations. Current simulations require ex-

tremely

ﬁne lattice spacings, necessitating the need for larger lat-

tice

volumes. This in turn creates larger Dirac operators needed for

linear systems calculations such as stochastic trace estimators and

fermionic forces in Hybrid Monte Carlo. Moreover, as the fermion

mass approaches physically relevant values, the Dirac operator be-

comes

extremely ill conditioned. This ill conditioning leads to ex-

ceptional

eigenvalues, which drastically slow convergence of linear

equations. Adaptive multigrid (MG) [1]is one method that deals

with both the strong scaling and critical slowing down at the

same time, and has been used successfully for the Wilson, overlap

and staggered fermion discretizations [2–4]. Adaptive MG creates a

hierarchy of coarsened operators from the original ﬁne Dirac oper-

ator

by exploiting its near null kernel. This shifts critical slowing

down to the coarsest level, where the components of the error at-

tributed

to the exceptional eigenvalues can be more easily dealt

with. However, the cost of the coarse grid solve can be very large

when cast in terms of ﬁne grid equivalence.

Deﬂation has long been used as a method of dealing with ex-

ceptional

eigenvalues in many ﬁelds, but is not yet heavily used

in modern LQCD simulations, partly because of eigenvector stor-

age

costs for large systems. Adaptive MG allows for deﬂation to

Corresponding author.

E-mail addresses: travis_whyte@baylor.edu (T. Whyte), walter_wilcox@baylor.edu

(W. Wilcox),

ronald_morgan@baylor.edu (R.B. Morgan).

be employed on the coarsest level, where storage requirements of

deﬂation are much smaller [5–7]. The preferred method of MG in

LQCD is to use it as a preconditioner for an outer Krylov solver [8].

Because every iteration of the outer Krylov solver represents a new

right hand side for the MG preconditioner, deﬂation with projec-

tion

methods [9,10]can be eﬃciently employed on the coarsest

level. We demonstrate the effect that deﬂation on the coarsest

level has by comparing to MG without coarse grid deﬂation, and

the effect that this deﬂation has for multiple right hand sides. We

observe that a partial solve on the intermediate grid in conjunction

with deﬂation and projection methods on the coarse grid allows

for a partial coarse grid solve. This partial solve on the interme-

diate

grid reduces the number of outer iterations for convergence,

and we observe no sign of critical slowing down resurgence on

the higher grid levels with the use of a deﬂated partial coarse grid

solve.

2. Methods

We work with the Wilson-Dirac operator in the two-dimensional

lattice Schwinger Model [11], which shares many physical charac-

teristics

with 4D LQCD, and as such is a good algorithmic test-

ing

ground. We created 10 gauge conﬁgurations within QCDLAB

1.0 [12]for lattices of size 64

, 128

and 256

at β = 6.0. All

values are averaged over separate solves for each conﬁguration.

The method of coarsening follows that of reference [4]. A hier-

archy

of three grids was created by solving the residual system

†

e =−DD

†

x, where x is a random vector, for 12 near null vec-

tors

on the ﬁne grids. This system was solved to a tolerance of

−4

, and the near null vectors were constructed using ψ = x + e.

https://doi.org/10.1016/j.physletb.2020.135281

SCOAP

下载后可阅读完整内容，剩余3页未读，立即下载

weixin_38726712

粉丝: 2
资源: 958

多网格减缩GMRES优化晶格QCD求解：适应性方法与临界速度改善

格点QCD中的Abelian和非Abelian Berry曲率

格点QCD观测值的机器学习估计器

带有格点QCD约束的核子横向的首次蒙特卡洛全局分析

具有交叉过渡的Hagedorn袋状模型符合格点QCD

使用六夸克和两重子插值器对H二重子进行格点QCD研究

格子QCD光谱用于强子性CP违规

QCD多重基，具有任意parton顺序

格子QCD和QCD和规则确定ηc，J /ψ和hc状态的衰减常数

n点QCD两环振幅

有限密度格QCD中的符号问题

最新资源