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Algorithms for Lattice QCD with Dynamical Fermions

Algorithms for Lattice QCD with Dynamical Fermions . A D Kennedy University of Edinburgh. Testimonial.

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Algorithms for Lattice QCD with Dynamical Fermions

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  1. Algorithms for Lattice QCD with Dynamical Fermions A D Kennedy University of Edinburgh Lattice 2004, Fermilab

  2. 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.) Algorithms for Lattice QCD

  3. Numerical Simulations under Battlefield Conditions: I Algorithms for Lattice QCD

  4. 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 χLF) • R Mawhinney (Columbia and RBRC) • CP-PACS hep-lat/0404014 • CP-PACS and JLQCD Nucl. Phys. B106 (2002) 195-196 Algorithms for Lattice QCD

  5. 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? • 10—100 times ASQTAD • At today’s parameters • Dirac spectrum bounded Algorithms for Lattice QCD

  6. Why Locality? • If a QFT is local then… • Cluster decomposition • (Perturbative) Renormalisability • Power counting • Universality • Improvement • …otherwise • Who knows? Algorithms for Lattice QCD

  7. A “wavefunction” is obtained by applying a lattice Dirac operator to a point source • A QFT is ultralocal if the wavefunction has “compact support,” • A QFT is local if the wavefunction has “fast decrease,” Locality Algorithms for Lattice QCD

  8. Is M1/2 Local? Finite Volume Improvement Lattice Spacing • [1] B Bunk, M Della Morte, K Jansen & F Knechtlihep-lat/0403022 • [2,3] A Hart and E Müller hep-lat/0406030 • S Dürr & C Hoeblinghep-lat/0311002 Algorithms for Lattice QCD

  9. 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 Algorithms for Lattice QCD

  10. 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? Algorithms for Lattice QCD

  11. Why Fat Links? • Construct good sources & sinks • Better overlap with ground state • Spatial smearing • Suppress UV fluctuations • Improved actions for dynamical computations Algorithms for Lattice QCD

  12. Fat Links: Buyer’s Guide • DBW2 • HYP • APE • Stout • FLIC • Lüscher-Weiss • Iwasaki • Symanzik Algorithms for Lattice QCD

  13. Molecular dynamics with fat links • Fat links appear to work better when projected back onto the SU(3) group manifold • Let V be the usual APE smeared link • Iterative projection to U satisfying is not differentiable • Define instead • This is now (almost) differentiable Fat Links: Dynamical FLIC • W Kamleh, D B Leinweber & A G Williamshep-lat/0403019 Algorithms for Lattice QCD

  14. Even more differentiable • Let V be a suitably smeared link • Define , where T means the traceless antihermitian part (i.e., projection onto the Lie algebra) • This is now differentiable • But it does not look too much like projection (except for ) Stout Links • C Morningstar & M Peardon hep-lat/0311018 Algorithms for Lattice QCD

  15. 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 Algorithms for Lattice QCD

  16. #1: Dynamical Fermion MC algorithm • Split lattice into blocks (64) • Alternate even & odd block updates • Only update links that do not affect neighbouring pseudofermions • Painful for fatter fermion actions? • Shift blocks to ensure all links get updated • Factorize quark determinant (Block LU/Schur) • Allows larger step size for light modes (block interaction updates)? Schwartz Alternating Procedure • M Lüscherhep-lat/0304007 Algorithms for Lattice QCD

  17. 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 Lüscherhep-lat/0310048 Algorithms for Lattice QCD

  18. 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.D35:2531-2542,1987 Algorithms for Lattice QCD

  19. 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 Algorithms for Lattice QCD

  20. sn(z/M,λ) sn(z,k) Чебышев Approximation • Theory of optimal L∞ (Чебышев) approximation is well understood • Equal alternating error maxima • Ремез algorithm to find coefficients • Золотарев analytic solution for sgn(x)and x±1/2 • Rational approximations to sgn(x)of degree (20,21) • Чебышев/Золотарев for 10-6 < |x| < 1 • tanh[20 tanh-1(x)] • Чебышевpolynomials • Tn(x) ≡ cos[n cos-1(x)] • Give optimal approximation to higher degree polynomials • Not optimal approximation in general • Polynomial or Rational? • Maximum L∞ [-1,1] error for |x| proportional to • Rational: exp(n/ln ε) • Polynomial: 1/n • Polynomial approximation to 1/x correspondsto matrix inversion • Basically Jacobi method • Compare with Крылов solvers • Partial fraction expansion and multi-shift Крылов solver to apply rational function Algorithms for Lattice QCD

  21. Instability of Symplectic Integrators • Symmetric symplectic integrator • Leapfrog • Exactly reversible… • …up to rounding errors • Ляпунов exponent  • >0  • Chaotic equations of motion •    when  exceeds critical value c • Instability of integrator • cdepends on quark mass • C Liu, A Jaster, & K Jansenhep-lat/9708017 • R Edwards, I Horváth,& A D Kennedyhep-lat/9606004 • B Joó et al. (UKQCD) hep-lat/0005023 Algorithms for Lattice QCD

  22. We want to evaluate a functional integral including the fermionic determinant det M • We write this as a bosonic functional integral over a pseudofermion field with kernel M-1: • We are introducing extra noise into the system by using a single pseudofermion field to sample this functional integral • This noise manifests itself as fluctuations in the force exerted by the pseudofermions on the gauge fields • This increases the maximum fermion force • This triggers the integrator instability • This requires decreasing the integration step size • A better estimate is det M = [det M1/n]n Multipseudofermions Algorithms for Lattice QCD

  23. Reduction of Maximum Force • Hasenbusch trick • Wilson fermion action M=1-H • Introduce the quantity M’=1-’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 Algorithms for Lattice QCD

  24. RHMC Force Reduction • Use RHMC technique to implement nthroot for multipseudofermions • Use partial fractions & multishift for nthroot • 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 (eln) / • Cannot reduce exact (mean) force • A D Kennedy & M Clark Algorithms for Lattice QCD

  25. If • Follow trajectory U0 →U2with step size δτ/2 • If accept U2 with probability • Otherwise reject (stay at U0) • How much does this help if we are hitting the integrator instability? • Things only get worse if we are not yet in equilibrium Reducing δHfluctuations • M Lüscher& R Sommer Algorithms for Lattice QCD

  26. 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) Algorithms for Lattice QCD

  27. Conventions • We shall work in Euclidean space with Hermitian  matrices • We shall write • We shall take all Dirac operators to be 5Hermitian Dynamical Chiral Fermions Algorithms for Lattice QCD

  28. Such a transformation should be of the form (Lüscher) • For it to be a symmetry the Dirac operator must be invariant • For a small transformation this implies that • Which is the Ginsparg-Wilson relation 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? Algorithms for Lattice QCD

  29. We can find a solution of the Ginsparg-Wilson relation as follows • Let the lattice Dirac operator to be of the form • This satisfies the GW relation if • And it must also have the correct continuum limit • Both of these conditions are satisfied if we define (Neuberger) Neuberger’s operator: I Algorithms for Lattice QCD

  30. There are many other possible solutions • Use another Dirac operator within Neuberger’s operator • The discontinuity is necessary • We are only considering vector-like theories • Chiral theories with unpaired Weyl fermions can be discretised on the lattice, but getting the phase of the fermion measure correct is critical • Simulating such theories is a much harder problem Neuberger’s operator: II Algorithms for Lattice QCD

  31. 5D formulation of overlap • Continued fraction representation • Relation between Domain Wall & Overlap formulations • Gauge fields are only four dimensional • Construct fifth dimensional transfer matrix T • Integrate out bulk pseudofermion fields • Jacobian cancelled by Pauli-Villars fields • Construct effective four dimensional Hamiltonian (Cayley transform) • Gives Higham approximation to Neuberger operator • Truncated Overlap for which • Optimal Domain Wall using Золотаревapproximation 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 Algorithms for Lattice QCD

  32. 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 Algorithms for Lattice QCD

  33. 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 Eshofet al.hep-lat/0202025, hep-lat/0311025, hep-lat/0405003 Algorithms for Lattice QCD

  34. 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 Algorithms for Lattice QCD

  35. The Future: Faster Monte Carlo Algorithms for Lattice QCD

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