Polynomial Filtered Hybrid Monte Carlo

Transcription

1 Polynomial Filtered Hybrid Monte Carlo Waseem Kamleh and Michael J. Peardon CSSM & U NIVERSITY OF A DELAIDE QCDNA VII, July 4th 6th, 2012

2 Introduction Generating large, light quark dynamical gauge field configurations is expensive. Hybrid Monte Carlo (HMC) is still the most used algorithm for generating dynamical configurations. There have been many improvements to the basic HMC algorithm developed over the years. A partial list of multi-scale type algorithmic improvements: Domain Decomposition method (Luscher). Mass Preconditioning (Hasenbuch). Polynomial Filtering (Peardon & Sexton, WK & Peardon). We begin by quickly reviewing the basics of the HMC algorithm.

3 Hybrid Monte Carlo Review The lattice is embedded in a Hamiltonian system by the addition of a fictitious simulation time τ, along with an additional fictitious field P which are the conjugate momenta to U, H(P, U) = x,µ 1 2 Tr P µ(x) 2 + S[U]. The conjugate momenta P µ (x) are drawn from a Gaussian distribution. In this way, the Hamiltonian H is constructed so that after path integration the expectation values of observables are unaltered. Where are the fermions?

4 Fermion Determinant The fermion fields ψ and ψ are Grassmannian, hence to perform simulations we need to integrate them out. We have that det D w = D ψdψ e d 4 x ψ(x)d w ψ(x). Define the effective action as S eff [U] = S g [U] ln det D w [U]. We can write the action for full QCD in terms of bosonic fields φ using the identity det M = 1 det M 1 = Dφ Dφ e d 4 x φ (x)m 1 φ(x).

5 Pseudofermions The convergence of the Gaussian integral in φ is only guaranteed for Hermitian positive definite (Hpd) M. For Wilson-type fermions, D w is a complex matrix, but det D w is real and positive. So, we can define M = D wd w, and M will be Hpd with det M = det D 2 f. Then the pseudofermionic effective action for full QCD with two flavours of degenerate quarks is S eff [U] = S g [U] + d 4 x φ (x)m 1 [U]φ(x). The pseudofermion fields φ(x) are drawn from a Gaussian distribution.

6 Hybrid Monte Carlo The Hybrid Monte Carlo algorithm creates a Markov chain by alternately performing two steps: A Molecular Dynamics (MD) integration to generate a new configuration (U U, P P ). A Metropolis accept/reject step on the proposed configuration (U, P ). The accept/reject step is based upon the change in the Hamiltonian ρ(u U, P P ) e H. The Molecular Dynamics integration takes place along a trajectory which for sufficiently small integration step sizes h approximately conserves H, hence yielding high acceptance rates.

7 Molecular Dynamics Start by requiring that the Hamiltonian be conserved along the trajectory dh dτ = 0. Then derive the discretised equations of motion, U µ (x, τ + h) = U µ (x, τ) exp ( ihp µ (x, τ) ), δs P µ (x, τ + h) = P µ (x, τ) U µ (x, τ) δu µ (x, τ). The derivative of the action with respect to the gauge fields is known as the force term, F µ (x) = δs δu µ (x).

8 Splitting The Action Lets split our action into its gauge field and pseudofermion field components, S = S g + S pf, where 1 S g = β x,µ<ν 3 Re Tr(1 U µν(x)), S pf = φ (x)(d wd w ) 1 φ(x), x and D w is the Wilson fermion matrix. Each of the terms in the action induces a force term. The size (variance?) of the force term is the dominant factor in determining what step size h is need for a given acceptance probability ρ acc. As the quark mass becomes lighter, the size of the pseudofermion force term increases.

9 Leapfrog Integration The MD equations of motion induce corresponding time evolution operators, V T (h) : {U(τ), P(τ)} {U(τ + h), P(τ)}, V S (h) : {U(τ), P(τ)} {U(τ), P(τ + h)}. The simplest MD integration scheme is the leapfrog V(h) = V S ( h 2 )V T(h)V S ( h 2 ). MD Integration trajectories typically have unit length, and hence as the step size h decreases, the number of integration steps increases.

10 Pseudofermion Force Each time we act with V S (h) we need to evaluate the pseudofermion force term, F pf = δs pf δu. This involves inverting the fermion matrix, and hence is expensive! However, for split actions S = S 1 + S 2 we can use a multiple time scale integration scheme (nested leapfrog), V(h) = V 2 ( h 2 ) [V 1 ( h m ) ] m V 2 ( h 2 ), V 1 (h) = V S1 ( h 2 )V T(h)V S1 ( h 2 ), V 2 = V S2 (h).

11 Multiple Time Scales Multiple time scale integration is effective when the force term F 1 due to S 1 is cheap to evaluate compared to F 2 (that of S 2 ). However, as the step-size for S 2 is larger, we also require that the size of the force term for S 2 is relatively small compared to that of S 1. The gauge force F g is cheap compared to the pseudofermion force F pf, and at heavy quark masses F g > F pf, but at light quark masses the UV fluctuations in the pseudo fermion force become too large for multiple time scales to be effective. This is where polynomial filtering steps in.

12 Polynomial Filtering We can use a polynomial filter P = P(M) to separate the ultraviolet and infrared physics in the pseudofermion force, S poly = χ Pχ, S pf = φ (MP) 1 φ. Recall M = D wd w is Hermitian positive definite. As S poly is fast to evaluate compared to S pf we split the action in the following way, S 1 = S g + S poly, S 2 = S pf. If P 1/z then it will capture the short-distance physics in S poly and act as a UV filter in S pf.

13 Fermionic Determinant Note that the fermionic determinant is unchanged by the introduction of the polynomial filter. To see this we note that Dφ DφDχ Dχ e d 4 x χ Pχ+φ (x)(mp) 1 φ(x) = det P det(mp) = 1 = det M det M 1 In the limit as the order of the polynomial n we have MP 1 and P(M) M 1. In this way the polynomial term can reproduce as much or as little of the pseudofermion term as we want.

14 Chebyshev Filter An effective UV filter is the n th order Hermitian Chebyshev polynomial approximation to 1/z, P n (z) = a n n k=1 (z z k ) 1 z, where we set θ k = 2πk n+1 to obtain the roots z k = λ[ 1 2 (1 + ɛ)(1 cos θ k) i ɛ sin θ k ]. The normalisation is defined by z 0 = 1 2 (1 + ɛ), with a n = 1 z 0 n k=1 (z 0 z k ). The approximation is good between [ɛ, 1], so we rescale with λ = 1 + 8κ.

15 Chebyshev Roots n = 24

16 Intermediate Filter Using this property of the Chebyshev polynomials, we can add an intermediate filter to our action as a bridge between the high and low energy scales, e.g. S 1 = S g S 2 = χ P m χ, S 3 = χ P m n χ, S 4 = φ (MP n ) 1 φ. P n denotes the Chebyshev polynomial of order n. P m n is defined so that P n = P m P m n. This allows us to perform fermion matrix inversions even less frequently. This may(?) be more efficient than a single filter algorithm. Note: filter implementation makes use of multi-shift linear solvers for efficiency.

17 Integration Scheme With a second filter, we have four different scales, five (or more) if we are doing a 2+1 flavour simulation. Nested leapfrog is too cumbersome for fine tuning these scales. We make use of the fact that V i = V Si all commute to introduce a generalised integration scheme. If N i is the number of integration steps per trajectory, then nested leapfrog requires N i N i 1 i The new integration scheme requires only that N i N 1 i

18 Generalised Leapfrog Set N = N 1 and set n i = N 1 /N i The generalised leapfrog algorithm is then: 1 Perform an initial half-step V i ( 1 2 h i) updating P for all i. 2 Loop over j = 1 to N 1 Apply V T (h) to update U. If {0 j mod N i } apply V i (h i ) to update P 3 Apply V T (h) to update U. 4 Perform a final half-step V i ( 1 2 h i) updating P for all i. Can show using BCH that it has errors of O[(h) 2 ]. Reduces to nested leapfrog for N i N i 1.

19 Integrator Error Analysis Given a Hamiltonian H the evolution operator with stepsize h for our system is exp hĥ. Ĥ is the linear operator on the vector space of functions f on phase space (p, q) defined by the Poisson bracket Ĥf = {H, f } = i Write the Hamiltonian as ( H f H ) f. p i q i q i p i H = T + S 1 + S 2 + S 3 + S

20 Integrator Error Analysis For each term in the Hamiltonian we can define a linear operator using the Poisson bracket relation above. We make use of the Baker-Campbell-Hausdorff result, e λâ e λ ˆB e λâ = exp (λ(2â + ˆB) + λ3 ) 6 ([[Â, ˆB], Â] + [[Â, ˆB], ˆB]) + O(λ 4 ) Apply this to the generalised leapfrog integrator: Use the simplest non-trivial case, H = T + S 1 + S 2. Number of integration steps N T = 6, N 1 = 3 and N 2 = 2.

21 Integrator Error Analysis The integrator for this simplest non-trivial case is V(h) = e h 4 Ŝ2 e h 6 Ŝ1 e h 3 ˆT e h 3 Ŝ1 e h 6 ˆT e h 2 Ŝ2 e h 6 ˆT e h 3 Ŝ1 e h 3 ˆT e h 6 Ŝ1 e h 4 Ŝ2. Repeatedly apply BCH result to obtain ( V(h) = exp hĥ + h 3( 1 48 [[Ŝ 2, ˆT], ˆT] [[Ŝ 2, ˆT], Ŝ 2 ] [[Ŝ 1, ˆT], Ŝ 1 ] [[Ŝ 1, ˆT], ˆT] [[Ŝ 1, ˆT], Ŝ 2 ] )) The error in the generalised integrator relative to the leading term is O(h 2 ), just as for the regular leapfrog.

22 Integrator Error Analysis Examine the individual leapfrog integrators H 1 = T + S 1, V 1 (h) = e h 6 Ŝ1 e h 3 ˆT e h 3 Ŝ1 e h 3 ˆT e h 3 Ŝ1 e h 3 ˆT e h 6 Ŝ1 H 2 = T + S 2, V 2 (h) = e h 4 Ŝ2 e h 2 ˆT e h 2 Ŝ2 e h 2 ˆT e h 4 Ŝ2. Using BCH we obtain ( V 1 (h) = exp hĥ 1 + h 3 ( 1 V 2 (h) = exp ) 216 [[Ŝ 1, ˆT], Ŝ 1 ]), ) 108 [[Ŝ 1, ˆT], ˆT] + 1 ( hĥ 2 + h 3 ( 1 48 [[Ŝ 2, ˆT], ˆT] [[Ŝ 2, ˆT], Ŝ 2 ]) The only difference is the cross term [[Ŝ 1, ˆT], Ŝ 2 ]..

23 Pseudofermion Force, Single Filter

24 Polynomial Force, Single Filter

25 ρ acc = erfc(h 2 F /τ2 0 ), Single Filter

26 Characteristic Scale, Single Filter

27 Pseudofermion Force, Double Filter

28 Polynomial Force, Double Filter

29 ρ acc = erfc(h 2 F /τ2 0 ), Double Filter

30 Characteristic Scale, Double Filter

31 Results κ = , m π = 665MeV n p n q 1/h g 1/h p 1/h q 1/h cg

32 Results κ = , m π = 665MeV

33 Pseudofermion Force, κ =

34 Polynomial Force, κ =

35 ρ acc = erfc(h 2 F /τ2 0 ), κ =

36 Characteristic Scale, Double Filter

37 Results κ = , m π 400MeV n p n q 1/h g 1/h p 1/h q 1/h cg

38 Results κ = , m π 400MeV

39

40 Single Flavour QCD Single fermion flavours can be simulated using a rational polynomial, R(M) = a i 1 M + b i M (or some other method e.g. polynomial approx. to 1/ M). Recall M = D wd w is Hermitian positive definite. Can we extend our polynomial filtering technique to single flavour simulations? Suppose we have a polynomial Q such that Q(M) 1 M E.g. numerically calculate coefficients for Chebyshev approximation to 1/ z, then calculate the roots...

41 Single Flavour Polynomial Filter Then we can write S poly1f = χ 1f Qχ 1f, S 1pf = φ 1f RQ 1 φ 1f. As before, the determinant is unaffected by the addition of the polynomial filter. Could add an intermediate filter for the single flavour as before but at the strange quark mass probably not worth it. Knowing roots of Q is needed to rewrite RQ 1 as a sum over poles. This allows the use of efficient linear multi-shift system solvers.

42 Alternative Approximation Note that RQ 1 1. May be advantageous to simply use the Remes algorithm to get a rational approximation to R(z) f (z) = 1 zq(z) and use that instead of the product RQ 1. What are the possible advantages? The rational approximation to f might have improved precision for a given order. Smallest shift for R expressed as a sum over poles might be bigger. = Less iterations to solve.

43 Another Variant Recall that we used the Hermitian Chebyshev approximation to 1/z in our two flavour polynomial filter. The non-hermitian Chebyshev approximation K(z) to 1/z has the same normalisation, but slightly different roots, y k = d(1 cos θ k ) + i d 2 c 2 sin θ k. Valid for an elliptical region in the complex plane. So long as the spectrum of the non-hermitian matrix D w is within this ellipse, we can write n/2 K(D w ) = a n (D w y k )(D w yk ) 1 k=1 D w

44 Factoring the Polynomial Now construct a polynomial using only half the roots (say those with positive imaginary parts), K + (D w ) = n/2 a n (D w y k ) k=1 We need the following two properties of determinants, det(ab) = det A det B det A = (det A) Using these we can deduce that det K +(D w )K + (D w ) = det K(D w ) (det D w ) 1 So long as det D w is real and positive, this is the correct weighting for a single fermion flavour.

45 The Action So we can the construct a polynomial filtered one-flavour action using K +, S poly1f = χ 1f K +K + χ 1f, S 1pf = φ 1f W (D w )W(D w )φ 1f. Here, W needs to be a rational polynomial approximation to {zk +(z)k + (z)} 1. Should be able to obtain this by factoring R(z) the Zolotarev approximation to 1/z and setting W(z) = R + (z)k 1 +. At this point, it might be more efficient to take large n polynomial limit for K +, split it into two and do filtered polynomial HMC for the single flavour...

46 Summary The use of a polynomial approximation to the inverse as a filter successfully separates the UV and IR pseudofermion dynamics. We successfully reduce the cost of dynamical simulations. Polynomial filtering is very simple. Minimal modification to existing code. Technique is not necessarily orthogonal to other improvements (e.g. DD).

47 Summary The generalised leapfrog algorithm is applicable to any multiple time scale integration scheme, far more flexible than nested leapfrog. Polynomial filtering can also be extended to single flavour simulations. Light quark mass results to come, plus comparison.