Numerical Time Integration in Hybrid Monte Carlo Schemes for Lattice Quantum Chromo Dynamics

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1 Numerical Time Integration in Hybrid Monte Carlo Schemes for Lattice Quantum Chromo Dynamics M. Günther and M. Wandelt Bergische Universität Wuppertal IMACM Dortmund, March 12, 2018

2 Contents Lattice QCD Hybrid Monte Carlo Molecular Dynamics Step Geometric Integration Fermionic fields M. Günther and M. Wandelt, HMC for Lattice QCD 1/30

3 Physical background Quantum Chromo Dynamics (QCD) QCD is the theory of the strong interaction (=color force) between quarks and gluons inside subatomic particles. Quark = elementary particle, fundamental constituent of matter, e.g., protons and neutrons. Gluons = exchange particles for the strong force between quarks. Lattice QCD approach to solve the theory of QCD investigation of elementary particles using computer simulations formulated on a lattice of points in space and time links U µ (x) SU(3) between points x and x + aˆµ M. Günther and M. Wandelt, HMC for Lattice QCD 2/30

4 Task of Lattice QCD Compuation of observables (particle mass, e.g.) observables given as expectation of certain operators O: O = O(U)Π(U)dU with probability space (Ω, σ(ω), F Π ) Ω = (SU(3)) N, N = lattice size Ω probability density Π(U) = exp( S(U))/Z with Z = Ω exp( S(U))dU, action S acting on U Ω. Problem: integral has very high dimension N = , for example M. Günther and M. Wandelt, HMC for Lattice QCD 3/30

5 Importance sampling Approximate integral by O 1 n n O(U i ) i=1 choose configurations U i according to the probability density exp( S(U))/Z prefer configurations that occur with a higher probability sequence is generated by a Markov chain is ergodic satisfies the detailed balance condition = convergence to the unique fixed point distribution given by the density Π(U) M. Günther and M. Wandelt, HMC for Lattice QCD 4/30

6 Hybrid Monte Carlo Method of choice in Lattice QCD uses augmented Markov chain constructs samples of pairs of links U Ω and momenta P ˆΩ = (su(3)) N samples generated according to the probability density ν(u, P ) given by ν(u, P ) = (2π)N/2 1 exp( S(U)) exp( P, P /2) Z } H (2π) N/2 {{}}{{} :=Π(U) :=ϕ(p ) 1 = exp( H(U, P )) with Z H Z H = ν(u, P )d(u, P ), H(U, P ) = P, P /2 + S(U) Ω ˆΩ M. Günther and M. Wandelt, HMC for Lattice QCD 5/30

7 Transition (U 0, P 0 ) (U, P ) Proposal step (U 0, P 0 ) g(u 0, P 0 ) for the moment: arbitrary mapping g : Ω ˆΩ Ω ˆΩ Acceptance step (U, P ) := g(u 0, P 0 ) g(u 0, P 0 ) is accepted with probability ( α((u 0, P 0 )g(u 0, P 0 )) = min 1, ν(g(u ) 0, P 0 )) ν(u 0, P 0 ) = min ( 1, e ) (H(g(U 0,P 0 )) H(U 0,P 0 )) otherwise the old configuration is kept: (U, P ) := (U 0, P 0 ) M. Günther and M. Wandelt, HMC for Lattice QCD 6/30

8 Demand on g Detailed balance condition holds if g is time-reversible: with S = S g(s g(u 0, P 0 ) ) = (U 0, P 0 ) ( ) I 0 flipping the momenta 0 I g is volume-preserving: ( ) det g(u0, P 0 ) = 1. (U 0, P 0 ) Choice of g? M. Günther The and M. Hamiltonian Wandelt, HMC for Lattice flow! QCD 7/30

9 Detailed Balance condition I Transition kernel in HMC K((U 0, P 0 ), (Ω ˆΩ )) = P ((U, P ) (Ω ˆΩ ) (U 0, P 0 )) = α((u 0, P 0 ), g(u 0, P 0 ))δ g(u0,p 0)(Ω ˆΩ ) + (1 α((u 0, P 0 ), g(u 0, P 0 )))δ (U0,P 0)(Ω ˆΩ ). Restricted Transition kernel K L (U 0, Ω ) = P (U 0 Ω U 0 ) = K((U 0, P ), (Ω ˆΩ))ϕ(P )dp ˆΩ Detailed Balance condition for all A, B σ(ω) K L (U, B)Π(U)dU = K L (U, A)Π(U)dU A B M. Günther and M. Wandelt, HMC for Lattice QCD 8/30

10 Detailed Balance condition II Detailed Balance condition for all A, B σ(ω) K L (U, B)Π(U)dU = K L (U, A)Π(U)dU A Left-hand side ( ) ν(g(u, P )) min 1, δ g(u,p ) (B A ˆΩ ν(u, P ) ˆΩ)ϕ(P )Π(U)dP du + ( ( )) ν(g(u, P )) 1 min 1, ϕ(p )Π(U)dP du. ν(u, P ) A B ˆΩ Right-hand side ( ) ν(g(u, P )) min 1, δ g(u,p ) (A B ˆΩ ν(u, P ) ˆΩ)ϕ(P )Π(U)dP du + ( ( )) ν(g(u, P )) 1 min 1, ϕ(p )Π(U)dP du. ν(u, P ) B A ˆΩ M. Günther and M. Wandelt, HMC for Lattice QCD 9/30 B

11 Detailed Balance condition III To be shown min (ν(u, P ), ν(g(u, P ))) δ g(u,p ) (B ˆΩ)d(P, U) = A ˆΩ min (ν(u, P ), ν(g(u, P ))) δ g(u,p ) (A ˆΩ)d(P, U). B ˆΩ With δ g(u,p ) (B ˆΩ) = δ (U,P ) (g 1 )(B ˆΩ)) min (ν(u, P ), ν(g(u, P ))) d(p, U) = (A ˆΩ) g 1 (B ˆΩ) }{{} (B ˆΩ) g 1 (A ˆΩ) F 1= min (ν(u, P ), ν(g(u, P ))) d(p, U) } {{ } F 2= M. Günther and M. Wandelt, HMC for Lattice QCD 10/30

12 Detailed Balance condition IV ϕ(p ) = ϕ( P ), ν(u, P ) = ν(u, P ) & g time reversible ν(g(u, P )) = ν(g 1 (U, P )) & g(u, P ) = g 1 (U, P ) Rewrite F 1 as F 1 = min (ν(u, P ), ν(g(u, P ))) δ (U,P ) (A ˆΩ)d(P, U) = g 1 (B ˆΩ) ( min ν(g 1 ) (U, P )), ν(u, P ) δ g 1 (U,P ) (A ˆΩ) B ˆΩ det dg 1 (U, P ) d(p, U) = d(u, P ) min (ν(g(u, P )), ν(u, P )) δ g 1 (U,P ) (A ˆΩ) B ˆΩ det dg 1 (U, P ) d(p, U) = d(u, P ) min (ν(g(u, P )), ν(u, P )) δ g(u, P ) (A ˆΩ) B ˆΩ det dg 1 (U, P ) d(p, U) = d(u, P ) min (ν(g(u, P )), ν(u, P )) (B ˆΩ) g 1 (A ˆΩ) det dg 1 (U, P ) d(p, U) = F d(u, P ) 2 } {{ } = 1 M. Günther and M. Wandelt, HMC for Lattice QCD 11/30

13 Choice of g Demand on g: time-reversible and volume-preserving mapping Natural choice: use Hamiltonian flow H(U, P ) = P, P /2 + S(U) Problem: analytical flow not available Way-out: use numerical approximation using geometric integration accuracy of less importance (only to get high acceptance rate) Demand on numerical approximation of g time-reversible & volume preserving numerical flow compatible with non-abelian structure of U (SU(3)) N and P (su(3)) N M. Günther and M. Wandelt, HMC for Lattice QCD 12/30

14 Derivation of Equation of Motion Time Derivative of Links U µ (x) SU(3) derived via an infinitesimal rotation on the group manifold U µ (x) = P µ (x)u µ (x) Time Derivative of Momenta P µ (x) su(3) usually: solve Ḣ(U, P ) = 0 for P µ (x) via the link differential operator P µ (x) = x,µ H(U, P ) = x,µ S(U) x,µf(u) = T i i x,µf(u) with i x,µf(u) = df(us) ds s=0 { (U s) ν(y) = e st i U µ(x) if (y, ν) = (x, µ) U ν(y) otherwise. M. Günther and M. Wandelt, HMC for Lattice QCD 13/30

15 Example: Wilson action S g (U) = β Re tr(u N µ(x)σ µ(x)) + terms indep. of U µ (x) Staples Σ µ (x) defined by [ Uν(x)U µ(x + aˆν)u ν(x + aˆµ) 1 + U ν(x aˆν) 1 U µ(x aˆν)u ν(x aˆν + aˆµ) ] ν µ i x,µ Sg(U) = β N Re tr { T i U µ(x)σ µ (x) } x,µs g(u) = β { } N T i Re tr T i U µ(x)σ µ(x) = β 2N {Uµ(x)Σ µ(x)} T A Time Derivative of Momenta P µ (x) su(3) P µ (x) = x,µ S g (U) = β 2N {U µ(x)σ µ(x)} T A M. Günther and M. Wandelt, HMC for Lattice QCD 14/30

16 Numerical integration on Lie groups Structure of ODE-IVP U = diag(p µ (x)u µ (x)), U(t 0 ) = U 0 P = x,µ S(U), P (t 0 ) = P 0 with U (SU(3)) N, P (su(3)) N. Problem for Numerical integration schemes multiplicative structure of Lie group does not fit the additive structure of integrations schemes (RK and others) way-out: bypass via the Lie algebra with its additive structure (Magnus, 1954) M. Günther and M. Wandelt, HMC for Lattice QCD 15/30

17 Bypass via Lie algebra Lie algebra parameterization of U U(t) = Ψ(Ω(t))U 0 with Ψ(t 0 ) = 0 Transformation of ODE ( ) d U = dω Ψ(Ω(t)) Ω(t)U 0 ( )) = (dψ Ω(t) Ω(t) Ψ(Ω(t))U 0 ( )) = (dψ Ω(t) Ω(t) U(t) }{{} =P (t)! ( ) P (t) = dψ Ω(t) Ω(t) Ω(t) = dψ 1 Ω(t) (P (t)). M. Günther and M. Wandelt, HMC for Lattice QCD 16/30

18 Transformed ODE system Solve ODE-IVP on Lie-algebra Ω µ (x) = dψ 1 Ω (P µ(x) µ(x)), Ω µ (x)(t 0 ) = 0, P = x,µ S(Ψ(Ω)U 0 ), P (t 0 ) = P 0 links can be obtained via the transformation U µ (x)(t) = Ψ(Ω µ (x)(t))u 0 Possible choice of parameterization Exponential map: Ψ(Ω) = exp(ω) = 1 k=0 k! Ωk Cayley map: Ψ(Ω) = cay(ω) = (I Ω) 1 (I + Ω) M. Günther and M. Wandelt, HMC for Lattice QCD 17/30

19 Exponential map ψ(ω) = exp(ω) = k=0 1 k! Ωk Ω(t) = d exp 1 Ω (P (t)) = B k k! adk Ω(P (t)) k 0 = P (t) 1 2 [Ω(t), P (t)] + 1 [Ω(t), [Ω(t), P (t)]] Problem: for numerical integration schemes, infinite sum has to be truncated: k 0 q k=0 Compatability with numerical integration schemes: Order p if a) RK scheme has order p and b) q + 2 p holds (Munthe-Kaas). M. Günther and M. Wandelt, HMC for Lattice QCD 18/30

20 Cayley map ψ(ω) = cay(ω) = (I Ω) 1 (I + Ω) Ω(t) = dcay 1 Ω (P (t)) = 1 (I Ω(t))P (t)(i + Ω(t)) 2 No infinite series, no trunaction needed No compatability condition cheap evaluations! M. Günther and M. Wandelt, HMC for Lattice QCD 19/30

21 The basis scheme: leap frog Exponential map: Update (U n, P n ) (U n+1, P n+1 ) P n+ 1 2 = P n h 2 x,µs(u n ), Ω n+1 = Ω n + h d exp 1 Ω (P n+ 1 ) Ω n + h P 2 n+ 1, 2 U n+1 = exp(ω n+1 )U n, P n+1 = P n+ 1 h 2 2 x,µs(u n+1 ). Elimination of the auxiliary Ω variable U n+1 = exp(h (P n h 2 x,µs(u n )))U n, P n+1 = P n h 2 x,µs(u n ) h 2 x,µs(exp(h (P n h 2 x,µs(u n )))) M. Günther and M. Wandelt, HMC for Lattice QCD 20/30

22 The basis scheme: leap frog Cayley map: Update (U n, P n ) (U n+1, P n+1 ) P n+ 1 2 = P n h 2 x,µs(un), Ω n+1 = Ω n + h dcay 1 Ω (P n+ 1 ) 2 = Ω n + h 2 (I Ωn)P n+ 2 1 (I + Ω n), U n+1 = cay(ω n+1 )U n = (I Ω n+1 ) 1 (I + Ω n+1 )U n, P n+1 = P n+ 1 2 h 2 x,µs(u n+1). Elimination of the auxiliary Ω variable U n+1 = cay(h dcay 1 Ω (Pn h 2 x,µs(un)))un, P n+1 = P n h 2 x,µs(un) h 2 x,µs(cay(h dcay 1 Ω (Pn h 2 x,µs(un)))) M. Günther and M. Wandelt, HMC for Lattice QCD 21/30

23 Higher-order schemes Composition Starting point: basis scheme ϕ h t order p, time-reversible & volume-preserving ( ) ( ) Un+1 = ϕ h Un t P n n+1 P n Composition of m basic schemes Volume-preservation ϕ h t n = ϕ γ 1h t n Condition for time-reversibility: ϕ γ 2h t n ϕ γmh t n γ m+1 k = γ k, k = 1,..., m 1 Order p + 1, if the underlying scheme has order p: m m γ j = 1, γ p j = 0 j=1 j=1 For leap-frog to get p = 4: m = 3, γ 1 = γ 3 = (2 3 2) 1/3, γ 2 = 1 2γ 1. M. Günther and M. Wandelt, HMC for Lattice QCD 22/30

24 Higher-order schemes Splitting Starting point: sequence of link and momenta updates Momenta Update U µ (x) = 0 P µ (x) = x,µ S(U) Link Updata U µ (x) = P µ,x U µ,x P µ (x) = 0 in general: order 1 at least order 2, if splitting symmetric important classes: Omelyan and force-gradient schemes M. Günther and M. Wandelt, HMC for Lattice QCD 23/30

25 Higher-order schemes Examples Symmetric five-stage Omelyan scheme P λh U h/2 P (1 2λ)h U h/2 P λh with λ = 1 2 ( ) 1/ ( ) 1/3 Still order 2, but minimal leading error coefficient Five stage force-gradient scheme of order 4 P λh,ξh 3 U h/2 P (1 2λ)h,χh 3 U h/2 P λh,ξh 3 with λ = 1 6, ξ = 0, χ = 1 72 M. Günther and M. Wandelt, HMC for Lattice QCD 24/30

26 Higher-order schemes Discussion Exponential mapping leap-frog scheme used as basic scheme time-reversibility & volume preservation but: truncation order q = 0 Problem: (H(g(U 0, P 0 ) H(U 0, P 0 )) = O(( t) 2 ) only high order only with respect to modified equation of motion Ω µ (x) = P µ (x), U µ,x (t) = exp (Ω µ (x)(t))u µ (x)(0) P µ (x) = x,µ S(U) possible way out: if modified equation of motion based on modified Hamiltonian H(U, P ), use g(u, P ) := H(U, P ) instead of H(U, P ) M. Günther and M. Wandelt, HMC for Lattice QCD 25/30

27 Fermionic fields Additional action: pseudofermionic field Partition function now given by Z = exp ( S g (U))det(DD )du det(dd ) = exp ( S pf (U, φ))dφ, with S pf (U, φ) = φ, (DD ) 1 φ H(U, P ) = 1 2 P, P + S g(u) + S pf (U, φ) M. Günther and M. Wandelt, HMC for Lattice QCD 26/30

28 Computation of pseudofermionic force Additional action: pseudofermionic field x,µ S pf(u,φ) = 2ReT i (DD ) 1 φ, δx,µφ i (δx,µφ)(y) i = 1 δ y,x 2 (1 γ µ)t i U µ (x)d 1 φ(x + ˆµ) +δ y,x+ˆµ 1 2 (1 + γ µ)u 1 µ (x)t i D 1 φ(x) Note: Computation requires two expensive inversions of the Wilson-Dirac operator M. Günther and M. Wandelt, HMC for Lattice QCD 27/30

29 Nested integration - multirate schemes Idea: use different activity levels gauge field S g : fast dynamics, cheap evaluations pseudofermionic field S pf : slow dynamics, expensive evaluations exploit different dynamics by multirate schemes: higher/lower sample rate for fast/slow part introduce additional activity splitting of pseudofermionic field: even-odd preconditioning, domain decomposition and determinant splitting S pf (U, φ ) = M S k [U, φ], m=1 H(U, P ) = P, P /2 + S g (U) + M S k [U, φ]. m=1 M. Günther and M. Wandelt, HMC for Lattice QCD 28/30

30 Sexton and Weingarten First step macro step size h0 micro step size h1 = h 0 /m 1 Leap-frog step ( m1 V H (h 0 ) = V H2 (h 0 /2) V H1 (h 1 )) VH2 (h 0 /2) with M 1 H 1 (U, P ) = P, P /2 + S g[u] + S k [U, φ], H 2 (U, P ) = S M [U, φ] m=1 Nesting next finer step size h2 = h 1 /m 2 further splitting H1 into H 1 (U, P ) = H 11 (U, P ) + H 12 (U, P ) with M 2 H 11 (U, P ) = P, P /2 + S g[u] + S k [U, φ], H 12 (U, P ) = S M 1 [U, φ] m=1 replace VH1 (h 1 ) above by V H1 (h 1 ) = V H12 (h 1 /2) ( V H11 (h 2 )) m2 VH12 (h 1 /2). M. Günther and M. Wandelt, HMC for Lattice QCD 29/30

31 More to read... M. Günther and M. Wandelt, HMC for Lattice QCD 30/30

Polynomial Filtered Hybrid Monte Carlo

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