Algorithms for Lattice QCD with Dynamical Fermions

1
Algorithms for Lattice QCD with Dynamical
Fermions

• A D Kennedy
• University of Edinburgh

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Testimonial

• The lattice conferences have had a steady diet
  of spectroscopy and matrix element calculations
  from these projects. But this year there seemed
  to be a pause. I think many people have decided
  that they just cannot push the quark masses down
  far enough to be interesting, and have gone back
  to studying algorithms.
• Tom DeGrand hep-ph/0312241

(I expect to get a lot of
unhappy mail about this sentence.)
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Numerical Simulations under Battlefield
Conditions I
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Numerical Simulations under Battlefield
Conditions II

• No continuum limit
• Two lattice spacings or fewer
• Autocorrelations ignored
• Depend on the physics anyhow
• Naïve volume scaling
• V for R algorithm
• V5/4 for HMC
• Dynamical quarks
• No valence mass
• Two or three flavours
• CAVEAT EMPTOR!

• S Gottlieb (MILC) hep-lat/0402030
• K Jansen (tmlQCD and ?LF)
• R Mawhinney (Columbia and RBRC)
• CP-PACS hep-lat/0404014
• CP-PACS and JLQCD Nucl. Phys. B106 (2002) 195-196

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Numerical Simulations under Battlefield
Conditions III

• Wilson (Clover)
• Very expensive
• Dirac spectrum not bounded
• ASQTAD (KS/Staggered)
• Relatively cheap
• Dirac spectrum bounded
• Twisted Mass (QCD)
• Relatively cheap
• Dirac spectrum bounded
• Domain Wall (GW/Overlap)
• Chiral limit for non-vanishing lattice spacing
• How much chiral symmetry is wanted?
• 10100 times ASQTAD
• At todays parameters
• Dirac spectrum bounded

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Why Locality?

• If a QFT is local then
• Cluster decomposition
• (Perturbative) Renormalisability
• Power counting
• Universality
• Improvement
• otherwise
• Who knows?

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Locality
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Is M1/2 Local?
Finite Volume
Improvement
Lattice Spacing

• 1 B Bunk, M Della Morte, K Jansen F Knechtli
  hep-lat/0403022
• 2,3 A Hart and E Müller hep-lat/0406030
• S Dürr C Hoebling hep-lat/0311002

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Hidden Locality

• It is easy to transform a manifestly local theory
  into an equivalent but non-manifestly local form
• We use this freedom to replace fermion fields
  with a non-manifestly local fermion determinant
• To introduce a pseudofermion representation
• To simulate the square root of a determinant
• but in general a non-local theory has no reason
  to be equivalent to a local one

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Fewer Staggered Tastes

• Staggered quarks come in four degenerate tastes
• det(M) in functional integral
• det(M)1/2 for two tastes
• Generate gauge configurations with square root
  weight using R, PHMC, or RHMC algorithms
• Is this a local field theory?
• ? local M such that det(M) det(M1/2)?
• Must use M for valence quarks too for a
  consistent unitary theory
• Mixed action computations?
• 4 taste valence 2 taste sea
• Unitarity? (same disease as quenched)
• Which are the physical valence states?

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Why Fat Links?

• Construct good sources sinks
• Better overlap with ground state
• Spatial smearing
• Suppress UV fluctuations
• Improved actions for dynamical computations

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Fat Links Buyers Guide

• DBW2
• HYP
• APE
• Stout
• FLIC
• Lüscher-Weiss
• Iwasaki
• Symanzik

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Fat Links Dynamical FLIC

• W Kamleh, D B Leinweber A G Williams
  hep-lat/0403019

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Stout Links

• C Morningstar M Peardon hep-lat/0311018

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Does Stoutness Pay?

• These methods can be applied iteratively to
  produce differentiable links of arbitrary obesity

• Stout links seem to be about as good as ordinary
  projected links, but require more tuning

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Schwartz Alternating Procedure

• M Lüscher hep-lat/0304007

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SAP Preconditioner

• 2 Preconditioner for linear solver
• Use GCR or FGMRES solver
• Accurate block solves not required
• 4 block MR steps, 5 Schwarz cycles
• Parallelises easily
• Especially on coarse-grained architectures such
  as PC clusters
• Reduces condition number by preconditioning high
  frequency modes

• M Lüscher hep-lat/0310048

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R Algorithm

• Inexact algorithm
• Distribution has errors of O(?? 2)
• Clever combination of non-reversibility and area
  non-preservation
• Asymptotic expansion in ??
• Results for large ?? do not just correspond to a
  renormalisation of the parameters
• C.f., perturbation theory is also asymptotic (or
  worse)

• C.f., improvement expansion in the cut-off a
  (lattice spacing) is also asymptotic
• Scaling for highly-improved theories breaks down
  at smaller values of a
• ?? independent of volume
• But probably so for HMC too, because of
  instability

• S Gottlieb, W Liu, D Toussaint, R Renken, R
  Sugar Phys.Rev.D352531-2542,1987

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PHMC and RHMC

• Use polynomial or rational function approximation
  for action
• Approximate action suffices for MD
• Accurate action need for acceptance
• Functions on matrices
• Defined for a Hermitian matrix by diagonalisation
• H UDU -1
• f(H) f(UDU -1) U f(D)U -1
• Polynomials and rational functions do not require
  diagonalisation
• ? Hm ? Hn U(? Dm ? Dn) U -1
• H -1 U D -1U 1

• T Takaishi Ph de Forcrand hep-lat/9608093
• R Frezzotti K Jansen hep-lat/9702016
• A D Kennedy, I Horváth, S Sint hep-lat/9809092
• M Clark A D Kennedy hep-lat/0309084

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??????? Approximation

• Theory of optimal L8 (???????) approximation is
  well understood
• Equal alternating error maxima
• ????? algorithm to find coefficients
• ????????? analytic solution for sgn(x) and x1/2
• Rational approximations to sgn(x) of degree
  (20,21)
• ???????/????????? for 10-6 lt x lt 1
• tanh20 tanh-1(x)
• ??????? polynomials
• Tn(x) cosn cos-1(x)
• Give optimal approximation to higher degree
  polynomials
• Not optimal approximation in general
• Polynomial or Rational?
• Maximum L8 -1,1 error for x proportional to
• Rational exp(n/ln e)
• Polynomial 1/n
• Polynomial approximation to 1/x correspondsto
  matrix inversion
• Basically Jacobi method
• Compare with ?????? solvers
• Partial fraction expansion and multi-shift ??????
  solver to apply rational function

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Instability of Symplectic Integrators

• Symmetric symplectic integrator
• Leapfrog
• Exactly reversible
• up to rounding errors
• ??????? exponent ?
• ?gt0 ???
• Chaotic equations of motion
• ? ? ?? when ?? exceeds critical value ??c
• Instability of integrator
• ??c depends on quark mass

• C Liu, A Jaster, K Jansen hep-lat/9708017
• R Edwards, I Horváth, A D Kennedy
  hep-lat/9606004
• B Joó et al. (UKQCD) hep-lat/0005023

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Multipseudofermions
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Reduction of Maximum Force

• Hasenbusch trick
• Wilson fermion action M1-?H
• Introduce the quantity M1-?H
• Use the identity M M(M-1M)
• Write the fermion determinant asdet M det M
  det (M-1M)
• Separate pseudofermion for each determinant
• Tune ? to minimise the cost
• Easily generalises
• More than two pseudofermions
• Wilson-clover action

• M Hasenbusch hep-lat/0107019

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RHMC Force Reduction

• Use RHMC technique to implement nth root for
  multipseudofermions
• Use partial fractions multishift for nth root
• No tuning required
• Cost proportional to condition number ?(M)
• Maximum force reduction
• Condition number ?(r(M))?(M)1/n
• Force reduced by factor n?(M)(1/n)-1
• Increase step size to instability again
• Cost reduced by a factor of n?(M)(1/n)-1
• Optimal value nopt ? ln ?(M)
• So optimal cost reduction is (e ln?) /?
• Cannot reduce exact (mean) force

• A D Kennedy M Clark

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Reducing dH fluctuations

• M Lüscher R Sommer

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Old Integrator Tricks

• Sexton-Weingarten
• Split MD Hamiltonian into parts
• Boson and fermion actions
• Construct symmetric symplectic integrator with
  larger steps for more expensive (fermion) part
• Use BCH formula
• Helps if step size limited by cheaper (boson)
  part
• Unfortunately, becomes less useful as mQ ? 0

• D Weingarten J Sexton Nucl. Phys.Proc.Suppl.
  26, 613-616 (1992)

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Dynamical Chiral Fermions
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On-shell chiral symmetry

• Is it possible to have chiral symmetry on the
  lattice without doublers if we only insist that
  the symmetry holds on shell?

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Neubergers operator I
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Neubergers operator II
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Into Five Dimensions

• H Neuberger hep-lat/9806025
• A Boriçi, A D Kennedy, B Pendleton, U Wenger
  hep-lat/0110070
• A Boriçi hep-lat/9909057, hep-lat/9912040,
  hep-lat/0402035
• R Edwards U Heller hep-lat/0005002
• T-W Chiu hep-lat/0209153 , hep-lat/0211032,
  hep-lat/0303008

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Overlap Algorithms

• ? many formulations of overlap fermions
• All satisfy GW relation
• Equivalent in continuum limit
• Different lattice Dirac operators within sgn
  function
• Different approximations to sgn function
• Presumably also true for perfect actions
• Trade-off between speed and amount of chirality
• Choice of inversion algorithm
• Inner-outer Krylov iterations
• Various five dimensional formulations
• Several possible preconditioners

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Dynamical Overlap

• Fodor, Katz, Szabó dynamical overlap on
  ridiculously small lattices
• Reflection refraction
• Brower, Originos, Neff Interpolate between DW
  Truncated Overlap
• Topology change in chiral limit?
• van der Eshof et al. Preconditioners, flexible
  inverters
• NIC/DESY Inverter tests for QCD overlap

• Z Fodor, S Katz, and K Szabó hep-lat/0311010
• R Brower, K Originos, H Neff
• van der Eshof et al. hep-lat/0202025,
  hep-lat/0311025, hep-lat/0405003

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Other Topics

• Keh-Fei Liu Andrei Alexandru
• Noisy non-vanishing baryonnumber density on tiny
  lattices
• Shailesh Chandrasekharan
• Strong coupling algorithm for QCD
• Strong coupling, but chiral limit

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The Future Faster Monte Carlo