Monte Carlo Algorithms

Transcription

1 Last Modified Wednesday, 15 June 011 Monte Carlo Algorithms Tait Institute, School of Physics & Astronomy, The University of Edinburgh Tuesday, 14 June 011 STRONGnet Summer School 011, Bielefeld

2 Outline Monte Carlo integration Markov Chain Monte Carlo Autocorrelations Hybrid Monte Carlo Symplectic Integrators Shadow Hamiltonians Gauge Fields Fermions Tuesday, 14 June 011

3 Monte Carlo Integration Tuesday, 14 June 011 3

4 Monte Carlo Integration Monte Carlo integration is based on the identification of probabilities with measures There are much better methods of carrying out low dimensional quadrature All other methods become hopelessly expensive for large dimensions In lattice QFT there is one integration per degree of freedom We are approximating an infinite dimensional functional integral Tuesday, 14 June 011 4

5 Monte Carlo Integration Generate a sequence of random field configurations ( φ, φ,, φ,, φ ) chosen from the probability distribution 1 S ( φt ) P( φt) dφt = e dφt Z Measure the value of Ω on each configuration and compute the average Ω 1 t N 1 N N t = 1 Ω( φ ) t Tuesday, 14 June 011 5

6 Central Limit Theorem Law of Large Numbers Central Limit Theorem Ω = lim N Ω Ω Ω+O ( ) N C where the variance of the distribution of Ω is ( ) C Ω Ω The Laplace DeMoivre Central Limit theorem is an asymptotic expansion for the probability distribution of Ω Distribution of values for a single sample ω = Ω(φ) ( ) ( ) ( ) = δ ω Ω( φ) PΩ( ω) dφ P φ δ ω Ω φ ( ) Tuesday, 14 June 011 6

7 Central Limit Theorem Generating function for connected moments The first few cumulants are C 0 3 ( ) ( ) ( ) 4 1 = Ω 4 = Ω Ω 3 = 0 C = Ω Ω C C C C ik W ( k) ln dω P ( ω) e ω Ω = Ω Ω Ω ( φ) ik Ω ( φ ) = ln dφ P e = ln e 3 ik Ω ( ik ) n = 0 n! n C n Note that this is an asymptotic expansion Tuesday, 14 June 011 7

8 Central Limit Theorem Distribution of the average of N samples Connected generating function ik W ( k) ln dω P ( ω) e ω Ω ( ω ) φ1 φn ( φ1) ( φn ) δ ω Ω( φt ) P d d P P Ω Ω N ik = ln dφ1 dφn P φ1 P φn Ω φt N t = 1 ( ) ( ) exp ( ) 1 N N t = 1 = ln ( φ) dφ P e k = NWΩ N ik Ω ( ik ) n = 1 n! ( φ ) n N C N N n n 1 ik = N ln e Ω N Tuesday, 14 June 011 8

9 Central Limit Theorem Take inverse Fourier transform to obtain distribution P Ω ( ) 1 Ω ( ) P ω = dk e e Ω π W k ik ω d C 4 d N N dk ik Ω+ ( ik ) C N ik C + 3! dω 4! dω e e e π ω = e ( ω Ω) 3 4 C3 d C 4 d + C N 3! N 3 4! N 3 4 e d ω d ω π C N Tuesday, 14 June 011 9

10 Central Limit Theorem Re-scale to show convergence to Gaussian distribution P ( ω) F ( ξ) d Ω = ( ) N where ξ ω Ω and ξ d ω F ( ξ ) ( 3 ξ ) C ξ ξ C e = C N πc C 3 3 Tuesday, 14 June

11 Asymptotic Expansions ( ) ( ) f x f x0 ε If dx e exists for ε ε, where x is the absolute minimum of f, show that ( ) ( ) f x f x0 1 1 f ( x )( ) 0 x x 0 + ε ε dx e = dx e ( x ) f 0 ξ / ( ) ( k ε ) = ε dξe 1 + ε + e 1 = ε πf 1 + ε + ( ) ( ) ( k / ε x ) 0 e x x ξ 0 ε Wednesday, 15 June

12 Laplace s Method ( ) ( ) x +Δ ( ) ( ) ( ) ( ) f x f x f x f x f x f x ε ε ε dx e = dx e + dx e x Δ x x >Δ 0 0 ( ) ( ) x 0 +Δ f x f x0 K Δ = ε ε dx e + e x 0 Δ x 0 +Δ 1 f ( x0 ) ξ j j d e + c j x Δ j = 0 ε ξ ε ξ 0 x 0 +Δ j j + c j d e j = 0 x Δ = ε ε ξξ 0 1 f ( x ) 0 ξ j j + c j d e j 0 = ε ε ξξ ( x ) ξ 1 f 0 Wednesday, 15 June 011 1

13 Proof for Laplace s Method ( ) ( ) f x f x K x x ( ) ( + ) f x f x δ 0 0 ( ) ( ) Δ f x f x K x x ( + ) f x δ 0 f ( x 0 ) ( ) ( ) K Δ ε Δ ( ) ( ) f x f x0 ε Iε = dx e +Δ f x f x dx e e ε ε x + δ x 0 0 x 0 0 Δ = + δ min ( δ, δ ) K ( x x ) K Δ 0 I e ε ε ε f ( x ) Wednesday, 15 June

14 Markov Chain Monte Carlo Wednesday, 15 June

15 Markov Chains State space Ω (Ergodic) stochastic transitions P : Ω Ω Deterministic evolution of probability distribution P: Q Q Distribution converges to unique fixed point Q Tuesday, 14 June

16 Convergence of Markov Chains ( ) ( ) ( ) Define a metric d Q1, Q dx Q1 x Q x on the space of (equivalence classes of) probability distributions ( ) ( ) ( ) Prove that d PQ with 1, PQ 1 α d Q1, Q α > 0, so the Markov process P is a contraction mapping The sequence Q, PQ, P²Q, P³Q, is Cauchy The space of probability distributions is complete, so the sequence converges to a unique fixed point Q = lim P n Q n Tuesday, 14 June

17 Simple but Inadequate Proof (, ) ( ) ( ) d PQ1 PQ = dx PQ1 x PQ x ( ) ( ) ( ) ( ) = dx dy P x y Q y dy P x y Q y 1 dx dy P ( x y ) Q ( y ) ΔQ ( y ) Q1( y ) Q( y ) = Δ = dy dx P ( x y ) ΔQ ( y ) dy Q ( y ) = d ( Q 1, Q) dx P ( x y ) = 1 = Δ ( ) ( ) dx dy P x y ΔQ y dx f ( x ) dx f ( x ) Tuesday, 14 June

18 ( 1, ) = 1( ) ( ) = dx dy P ( x y ) Q1( y ) dy P ( x y ) Q( y ) ΔQ ( y ) Q1( y ) Q( y ) = dx dy P ( x y ) ΔQ ( y ) θ ( y ) + θ ( y ) = 1 = ( ) Δ ( ) θ Δ ( ) + θ Δ d PQ PQ dx PQ x PQ x Proof ( ) ( ( )) ( ) dx dy P x y Q y Q y Q y a b = a + b min a, b = dx dy P ( x y ) ΔQ ( y ) ( ) ( ) θ ( ) Δ ±Δ ( ) dx min dy P x y Q y Q y ± Tuesday, 14 June

19 ( 1, PQ) dx P ( x y ) = 1 dy Q ( y ) dx min dy P ( x y ) Q ( y ) θ Q ( y ) d PQ Proof = Δ Δ ±Δ ± ( ) ( ) inf ( ) min ( ) θ ( ) ( ) dy ΔQ y dx P x y dy Δ Q y ±ΔQ y y ± dy ΔQ ( y ) θ ( Δ Q ( y )) + dy ΔQ ( y ) θ ( ΔQ ( y )) dy Q ( y ) dy Q ( y ) dy Q ( y ) 1 ( ) θ ( ) ( ) inf ( ) ( ) y 0 α dx infp ( x y ) = Δ = 1 = 1 1 = 0 dy Δ Q y ( ±Δ Q y ) = dy ΔQ ( y ) (1 α) d ( Q1, Q) dy ΔQ y dx P x y dy ΔQ y < = y Tuesday, 14 June

20 Banach Fixed-Point Theorem We show that the sequence of distributions is Cauchy n m 1 m n m+ j m+ j + 1 d( P Q, P Q) d( P Q, P Q) n m 1 j = 0 j m m+ 1 m m+ ( 1 α ) d( P Q, P Q) d( P Q, P 1 Q) ( 1 α ) j = 0 j = 0 m m+ 1 m (, Q) ( 1 α ) d P Q P = = α α d( Q, PQ) < ε for any ε > 0, provided m is large enough and α > 0. Hence the sequence converges to the unique fixed point probability distribution Q = lim P n Q n j Tuesday, 14 June 011 0

21 Markov Chains Use Markov chains to sample from Q Suppose we can construct an ergodic Markov process P which has distribution Q as its fixed point Start with an arbitrary state ( field configuration ) Iterate the Markov process until it has converged ( thermalized ) Thereafter, successive configurations will be distributed according to Q But in general they will be correlated To construct P we only need relative probabilities of states Do not know the normalisation of Q Cannot use Markov chains to compute integrals directly We can compute ratios of integrals Tuesday, 14 June 011 1

22 Markov Chains How do we construct a Markov process with a specified fixed point Q ( x ) = dy P ( x y ) Q ( y )? Detailed balance P y x Q x = P x y Q y Integrate w.r.t. y to obtain fixed point condition Sufficient but not necessary for fixed point Metropolis algorithm P ( x y ) ( ) ( ) ( ) ( ) ( ) ( ) min 1, Q x = Q y Consider cases Q ( x) > Q( y) and Q ( x) < Q( y) separately to obtain detailed balance condition Sufficient but not necessary for detailed balance ( ) Other choices are possible, e.g., P x y = Q ( x ) ( ) + Q ( y ) Q x Wednesday, 15 June 011

23 Markov Chains Composition of Markov steps Let P 1 and P be two Markov steps which have the desired fixed point distribution They need not be ergodic Then the composition of the two steps P P 1 will also have the desired fixed point And it may be ergodic This trivially generalises to any (fixed) number of steps For the case where P 1 is not ergodic but (P 1 ) n is the terminology weakly and strongly ergodic are sometimes used Wednesday, 15 June 011 3

24 Markov Chains This result justifies sweeping through a lattice performing single site updates Each individual single site update has the desired fixed point because it satisfies detailed balance The entire sweep therefore has the desired fixed point, and is ergodic But the entire sweep does not satisfy detailed balance Of course it would satisfy detailed balance if the sites were updated in a random order But this is not necessary And it is undesirable because it puts too much randomness into the system Wednesday, 15 June 011 4

25 Markov Chains Coupling from the Past Propp and Wilson (1996) Use fixed set of random numbers Flypaper principle: If states coalesce they stay together forever Eventually, all states coalesce to some state α with probability one Any state from t = - will coalesce to α α is a sample from the fixed point distribution α Wednesday, 15 June 011 5

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27 Exponential Autocorrelations The unique fixed point of an ergodic Markov process corresponds to a unique eigenvector with eigenvalue 1 All its other eigenvalues must lie within the unit circle In particular, the largest subleading eigenvalue is λ max < 1 The eigenvectors dy P ( x y ) u y = λu x satisfy ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) λ dx u x = dx dy P x y u y = dy dx P x y u y = dy u y so either λ = 1 or dx u ( x ) = 0 Hence we may expand any probability density as Q = Q + c ju j N N N N / N P Q Q = c jλj u j λmax c j u j = Ke λ < 1 λ < 1 j with the exponential autocorrelation time j N exp max exp λ < 1 1 > 0 ln λ j Wednesday, 15 June 011 7

28 Integrated Autocorrelations Consider the autocorrelation of operator Ω Without loss of generality we assume Ω = 0 t t 1 N N t = 1 t = 1 ( φ ) ( φ ) = Ω ( φt ) + Ω( φt ) Ω( φt ) Ω = Ω Ω N 1 N N N 1 N t = 1 t = 1 t = t + 1 Define autocorrelation function ( ) 1 N Ω = Ω + N C Ω N C 1 ( ) Ω ( ) N = 1 Ω Ω ( φt + ) Ω( φt ) Ω ( φ ) Wednesday, 15 June 011 8

29 Integrated Autocorrelations The autocorrelation function must fall faster that the Ω λmax = N exp exponential autocorrelation ( ) For a sufficiently large number of samples { A } C Ω N exp 1 C Ω ( ) 1 = 1 N N Ω = + + Define integrated autocorrelation function e Ω N exp Ω = 1+ Ω 1+ N N A Ω C = 1 Ω ( ) Wednesday, 15 June 011 9

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31 Hybrid Monte Carlo In order to carry out Monte Carlo computations including the effects of dynamical fermions we would like to find an algorithm which Updates the fields globally Because single link updates are not cheap if the action is not local Take large steps through configuration space Because small-step methods carry out a random walk which leads to critical slowing down with a dynamical critical exponent z= z relates the autocorrelation to the correlation length of the system, A Ω Does not introduce any systematic errors ξ z Wednesday, 15 June

32 Hybrid Monte Carlo A useful class of algorithms with these properties is the (Generalised) Hybrid Monte Carlo (HMC) method Introduce a fictitious momentum p corresponding to each dynamical degree of freedom q Find a Markov chain with fixed point exp[-h(q,p) ] where H is the fictitious Hamiltonian ½p + S(q) The action S of the underlying QFT plays the rôle of the potential in the fictitious classical mechanical system This gives the evolution of the system in a fifth dimension, fictitious or computer time This generates the desired distribution exp[-s(q) ] if we ignore the momenta p (i.e., the marginal distribution) Wednesday, 15 June 011 3

33 Hybrid Monte Carlo The HMC Markov chain alternates two Markov steps Molecular Dynamics Monte Carlo (MDMC) (Partial) Momentum Refreshment Both have the desired fixed point Together they are ergodic Wednesday, 15 June

34 MDMC If we could integrate Hamilton s equations exactly we could follow a trajectory of constant fictitious energy This corresponds to a set of equiprobable fictitious phase space configurations Liouville s theorem tells us that this also preserves the functional integral measure dp dq as required Any approximate integration scheme which is reversible and area preserving may be used to suggest configurations to a Metropolis accept/reject test With acceptance probability min[1,exp(-δh)] Wednesday, 15 June

35 MDMC We build the MDMC step out of three parts Molecular Dynamics (MD), an approximate integrator U τ : q, p q, p which is exactly ( ) ( ) ( ) q, p Area preserving, detu * = det = 1 ( q, p) Reversible, F U ( τ ) F U ( τ ) = 1 A momentum flip F : p p A Metropolis accept/reject step ( ) The composition of these gives q ( ) ( δh ) 1 ( δh q F U τ ϑ e y ϑ y e ) = + p p with y being a uniformly distributed random number in [0,1] Wednesday, 15 June

36 Partial Momentum Refreshment This mixes the Gaussian distributed momenta p with Gaussian noise ξ p cosθ sinθ p F ξ = sinθ cosθ ξ The Gaussian distribution of p is invariant under F The extra momentum flip F ensures that for small θ the momenta are reversed after a rejection rather than after an acceptance For θ = π / all momentum flips are irrelevant Wednesday, 15 June

37 Acceptance Rates The normalization of the equilibrium distribution is 1 H( q, p) 1 1 = H( q, p ) dqdp e = dq dp e Z Z 1 H q p δh (, ) = dq dp e Z = dqdp e Z since δ H H q, p H q, p ( ) ( ) 1 H q p δh (, ) = e δh and dq dp = dq dp δh For small we have 1 δh e = 1 δh + δh +, hence δh 1 δh n Thus if δh δτ we have = ( ) ( n δh = δτ ) Wednesday, 15 June

38 1 Thermodynamic Limit If box size L ξ (correlation length) cluster decomposition and central limit theorem ( ) = ( π ) so we must have V = δh ( δ ) = = ( ) h h e dh Pδ H h e = P h V e dh e πv h H H hv and thus the average acceptance rate P acc is 0 1 = dh e + dh e π δh 0 h = dh e ( ) 1 = erfc δh π min( 1, e h ) 1 δh δh V = e 1 ( V δh ) ( h δh ) ( h δh ) ( h δh ) 1 h 4 δh 4 δh V Wednesday, 15 June

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40 Hamiltonian Vector Fields Classical mechanics is not specified just by a Hamiltonian H but also by a closed fundamental - formω For every function (0- form) A this defines a Hamiltonian vector field ˆ A such that da = i ω ( ) = ω ( ˆ, A X ) ˆ A Which just means that for all X da X Wednesday, 15 June

41 Hamiltonian Vector Fields To be a little less abstract consider the familiar case where ω = dq dp and A A da dq + dp; X X q + X p q p q p so A A da( X ) = X q + X p = ω q p ˆ ˆ A A ˆ q + A q p ( ˆ, A X ) = ˆ AX ˆ AX A A = p p q q p q p p q Wednesday, 15 June

42 Classical Trajectories Classical trajectories are then integral curves of the Hamiltonian vector field Ĥ of the Hamiltonian H f f f q + p = ˆ Hf q p H f H f = p q q p H H q = ; p = p q In other words, this vector field is always tangent to the classical trajectory Wednesday, 15 June 011 4

43 Symplectic Integrators We are interested in finding the classical trajectory in phase space of a system described by the Hamiltonian ( ) ( ) ( ) 1, = + = + ( ) H q p T p S q p S q Define the corresponding Hamiltonian vector fields (with ) ω = dq dp T T ( p) and q S S ( q) so that H = T + S p Wednesday, 15 June

44 Symplectic Integrators Formally the solution of Hamilton s equations with Since the kinetic energy T is a function only of p and the potential energy S is a function only of q, it follows Ŝ that the action of and may be evaluated trivially e τ (Taylor s theorem!) τ T τ S τ trajectory length is the exponential of the Hamiltonian Hamiltonian vector field, e τĥ ˆ e τt ( ) + τ ( ) ( ) e : f q, p f q T p, p ( ) τ ( ) ( ) e : f q, p f q, p S q Wednesday, 15 June

45 Leapfrog The simplest example is the Leapfrog PQP 1 1 δτs S δτt δτ U δτ e e e integrator ( ) 0 It consists of three steps 1δτ S f ( q1, p1) = e f q0, p0 = f q0, p0 S q0 ( ) ( ) ( ) ( ) ( ) ( ) δτ T, = 1, 1 = δτ, 1 f q p e f q p f q T p p 1δτ S f ( q3, p3) = e f q, p = f q, p S q ( ) ( ) δτ δτ Wednesday, 15 June

46 Langevin Algorithm The leapfrog update is If we ignore the Metropolis acceptance step (e.g., if we take small enough steps) 1 Rescale time step ε δτ and initial Gaussian noise 1 (momenta) ps ( 0) η ( s ) δτ We obtain the Langevin equation dq δs = + η ds δq δτ δτ p ( 0) ( ( 0) ) = p S q δτ q ( δτ ) = q ( 0) + p δτ δτ δτ p ( δτ ) = p S ( q ( δτ )) ( δτ ) ( ) 1 1 ( ) q q 0 0 = S ( q ( 0) ) + p δτ δτ ps ( 0) = 0; ps ( 0) ps ( 0) = δs, s δs, s ( s ) = 0; ( s ) ( s ) = ( s s ) η η η δ ε Wednesday, 15 June

47 Shadow Hamiltonians Monday, 13 June

48 Into the Shadow World For each symplectic integrator there exists a Hamiltonian H which is exactly conserved This may be obtained by replacing the commutators ˆ ST, ˆ in the BCH expansion ˆ S Tˆ of ln e e with the Poisson bracket ( ) { ST, } Monday, 13 June

49 Poisson Brackets Consider the action of a Hamiltonian vector field on a function (0-form) ˆ AF = df ( ˆ A) = i ω( ˆ A) = ω( ˆ F, ˆ A) { A, F} ˆ F We have introduced the Poisson bracket of two functions These obey the Jacobi identity A, B, C + B, C, A + C, A, B = 0 { { }} { } { } { { }} This follows from the closure of the fundamental -form d ω = 0 It is not trivial: Poisson brackets are not commutators Functions form a Lie algebra with PBs as the Lie product Wednesday, 15 June

50 Jacobi Identity Invariant definition of exterior derivative dθ ( X, Y ) = Xθ ( Y ) Yθ ( X ) θ ( X, Y ) (,, ) = (, ) + (, ) + (, ) dω X Y Z Xω Y Z Yω Z X Zω X Y ( XY,, Z) ( Y, Z, X) ( Z, X, Y) ω ω ω For Hamiltonian vector fields we have ( ) { } { } Xω Y, Z = X Y, Z = X,{ Y, Z} ω( X, Y, Z ) = ω( Z, X, Y ) = dz ( X, Y ) = X, Y Z = ( XY Y X ) Z = XY Z + Y XZ X Y, Z Y X, Z = X, Y, Z + Y, X, Z = { } + { } { } The condition d ω = 0 gives the Jacobi identity { } { { }} { { }} dω ( X, Y, Z ) = 0 = X,{ Y, Z} + Y, Z, X + Z, X, Y df ( X ) = XF { } { { }} Wednesday, 15 June

51 ω Concrete Poisson Brackets To make this more familiar when ω = dq dp the Poisson bracket A, B ω ˆ A, ˆ B becomes ( ˆ ABˆ) ( dq dp) { } ( ) ˆ A A A = p q q p A A B B, =, p q q p p q q p { AB, } A B A B = p q q p Wednesday, 15 June

52 Hamilton s Equations (again) To make this really concrete consider the action of the Hamiltonian Hamiltonian vector field on an arbitrary function f that we saw earlier f f f q + p = ˆ Hf = { H, f } q p { H, f } H f H f = p q q p Monday, 13 June 011 5

53 Commutators So far this is just a fancy (and complicated) way of rewriting Hamilton s equations, but now we derive a surprising new result To derive it consider ( ) ˆ A, ˆ B F = ˆˆ AB ˆˆ BA F = ˆ A B F ˆ B A F A, B, F B, A, F = A, B, F {, } {, } A, BF = { { }} { { }} {{ } } = { } The commutator of Hamiltonian vector fields is itself a Hamiltonian vector field Monday, 13 June

54 Baker Campbell Hausdorff (BCH) Formula If A and B belong to any (non-commutative) A B A+ B +δ algebra then e e = e, where δ is constructed from commutators of A and B i.e., is in the Free Lie Algebra generated by A and B δ A B More precisely, ln e e = c where c and n 1 = A + B 1 n + 1 c n+ 1 = c, n A B ( ) n B m + c,,, k c 1 k A B + m! m = 0 ( ) ( m ) k,, k 1 1 m k + + k = n 1 m n 1 Monday, 13 June

55 Symplectic Integrators Explicitly, the first few terms are A B ( ) { } { } ln ee = A+ B + AB, + A, AB, B, AB, B, A, AB, A, A, A, A, B 4 B, A, A, A, B 6 A, B, A, A, B B, B, A, A, B A, B, B, A, B + B, B, B, A, B In order to construct reversible integrators we use symmetric symplectic integrators The following identity follows directly from the BCH formula A B A 1 ( ) 4 { } { } ln e e e = A + B + A, A, B B, A, B 7 A, A, A, A, B 8 B, A, A, A, B 1 A, B, A, A, B B, B, A, A, B 16 A, B, B, A, B + 8 B, B, B, A, B Monday, 13 June

56 Symplectic Integrators From the BCH formula we find that the PQP symmetric symplectic integrator is given by ( ) τ / δτ 1 δτ S 1 δτt δτs e e e (( ) ) ( ( ) ) (,, T, S, T 3 5 ( ) 1 = + exp T + S δτ S, S, T + T, S, T δτ + 3 δτ 5 4 δτ + δτ ( ) 1 4 = exp τ T S S, S, T T, S, T δτ + ( δτ ) τh ( δτ) τh + ( δτ ) τ H = e = e = e + δτ ( ) In addition to conserving energy to O (δτ² ) such symmetric symplectic integrators are manifestly area preserving and reversible τ δτ Monday, 13 June

57 Shadow Hamiltonian But more significantly the PQP integrator follows the integral curves of H ( δτ ) exactly And H ( δτ ) is constructed from commutators of the Hamiltonian vector fields S and T Therefore it is the Hamiltonian vector field of the corresponding combination of Poisson brackets H ( δτ ) This is called the shadow Hamiltonian Monday, 13 June

58 Leapfrog Shadow Hamiltonian For the PQP integrator we have {,, } {,{, }} δτ ( ) 4 { } 7 S, { S, { S, { S, T }}} + 8 T, S, { S, { S, T }} H PQP δτ = T + S + S S T T S T { } { } {{ ST} { S{ ST} }} T { T { S { S T }}} { } 4 δτ ,,, + 3,,,, 16,,, + 8,,,, { } {{ ST} } T { ST} { } { { }} { } { T T T S T } Monday, 13 June

59 PQP Leapfrog Shadow Hamiltonian Evaluating the Poisson brackets with ω = dq dp gives δτ ( ) { δτ = + } H H p S S 4 { ( 4) p 4 S 6p ( SS S ) 3S S } O ( δτ 6 ) 4 δτ Note that H PQP cannot be written as the sum of a p-dependent kinetic term and a q-dependent potential term So, sadly, it is not possible to construct an integrator that conserves the Hamiltonian we started with 59

60 What use are Shadows? An integrator becomes unstable when the BCH (asymptotic) expansion for its shadow fails to converge In which case there is no (real) conserved shadow Hamiltonian Use the shadow to tune an integrator Reduce large contributions to the shadow Hamiltonian Optimize the integrator by minimizing ΔH H H? Not quite, as we shall see later Monday, 13 June

61 Gauge Theories But first there are a few details that we shouldn t overlook Can we compute Poisson brackets and shadow Hamiltonians for gauge fields and fermions? Monday, 13 June

62 Inexact Algorithms Let us take a small detour to consider what happens the an HMC-like algorithm if we omit the Metropolis step. If such an algorithm is ergodic it has a unique fixed point e ( ) ( S( q ) +Δ S( q )) S ( q ) +Δ S ( q ) (, ) ΔS( q) H q p 1 p δ ( ) ( ) 1 ( H S ) U δ ( ) ( ) = dq e dp e q q = dq dp e δ q q 1 +Δ = dq dp ( detu * ) e q q where the MD evolution is U :( q, p) q, p and the Jacobian is ( q, p ) det ( ) = det U* =tr ln U q, p * We assume that the momenta a completely refreshed from a Gaussian heatbath before each trajectory Momentum distribution is not Gaussian at the end of a trajectory Monday, 13 June 011 6

63 Inexact Algorithms If we define the quantities 1 δ : Ω Ω U Ω F U F violation of reversibility δ : Ω Ω U F Ω F change in Ω over a trajectory then we obtain ( δ )( H S ) tr lnu* e δ+ +Δ = p This may be expanded as an asymptotic series in the integration step size δτ to obtain an expression for ΔS This shows that ΔS is a power series in δτ up to terms like C / δτ e, where the constant C is not necessarily real The form of ΔS is not known in closed form for long trajectories 1 Monday, 13 June

64 Monday, 13 June

65 Review Symplectic -form Hamiltonian vector field Equations of motion Flat Manifold dq dp General ω : d ω = 0 ˆ H H H = dh = p q q p Ĥ Darboux theorem: H H q =, p = All manifolds are d = ˆ H p q dt σ locally flat i ω Poisson bracket { A, B} A B A B = p q q p { AB} = ω ˆ ˆ, ( AB, ) Monday, 13 June

66 Lie Group Manifolds A Lie Group is a differential manifold with a globally well-defined smooth multiplication h : g hg This induces a smooth map on 0-forms (functions) h : f f h *, i.e., hf * ( g) = f ( hg) A left-invariant 0-form satisfies f = h* f for any h, so it is specified by its value at the origin f 1 = g f 1 = f g a constant, not very interesting ( ) ( ) ( ) * Similarly, we have smooth induced map on vector * fields (linear differential operators) h : v v h* * A left-invariant vector field satisfies v = hv ( h) * hv f = v hf = v f h ( ) ( ) ( ) * Monday, 13 June

67 Lie Group Manifolds An frame of vectors { e j } at the origin can therefore be extended to a frame of left-invariant vector fields * over the whole group e h = h e 0 i ( ) ( ) For a classical matrix group this is quite intuitive: each element g of the group is associated with the frame which is obtained from that at the origin by the action of g While there is a globally well-defined basis of left-invariant frame vector fields, but there are no globally well-defined coordinates k Their commutators satisfy ei, e j = cijek with k coefficients called structure constants i Monday, 13 June

68 Maurer Cartan Equations The dual left invariant forms the Maurer Cartan equations { θ i } ( ) i i with θ = δ satisfy e j j dθ = c θ θ i 1 i j k jk jk i, j k j k k i j, k ( ) ( ) ( ) ( ) dθ e e = e θ e e θ e θ e e ( ) = e δ e δ θ c e i i i j k k j jk (, ) 1 i j k = c j k θ θ e j ek j, k i = c jk The Maurer Cartan forms also provide the groupinvariant Haar measure d Ω θ1 θ θn Monday, 13 June

69 Fundamental -form We can invent any Classical Mechanics we want So we may therefore define the closed fundamental -form ω which globally defines an invariant Poisson bracket by ω d θ i i p i i ( θ i dp i p i d θ i ) = ( i i 1 i i j k θ dp p c ) jkθ θ = + i Monday, 13 June

70 Hamiltonian Vector Field We may now follow the usual procedure to find the equations of motion Introduce a Hamiltonian function (0-form) H on the cotangent bundle (phase space) over the group manifold Define a vector field Ĥ such that dh = i ω H H H e c p e ( H ) ˆ k k = i i + ji j i i i p jk p p Ĥ Monday, 13 June

71 Poisson Brackets For any Hamiltonian vector field A A A e c p e ( A) ˆ k k = i i + ji j i i i p jk p p ( ) ( ) ( ) So for H q, p = T p + S q we have vector fields  ( ) ˆ S = e S i i p i ˆ T k k T T = e i i + c jip j i i p jk p p i k k j = pei + c jip p if T i ( p) = i jk p p Monday, 13 June

72 More Poisson Brackets We thus compute the lowest-order Poisson bracket { ST, } = ω( ˆ ST, ˆ) ( )( ) i i 1 i i j k ˆ, ˆ i θ dp p c jkθ θ S T p ei ( S ) = + = and the Hamiltonian vector corresponding to it { ST, } { ST, k k } { } ST, = e ({, }) i + c jip ei ST i p jk p p k k j = ei ( S ) ei + c jip e j ( S ) + p eie j ( S ) i p i j i Monday, 13 June 011 7

73 Even More Poisson Brackets { { }} { } S, S, T = e ( S) e ( S) { } { { }} { } i j T, S, T = p p e e ( S) { { }} {{ } { } } i j T, T, S, S, T = p p eie jek( S) ek( S) + eiek( S) e jek( S) i i ST,, T, ST, = c jk ppe j ( S) ee k ( S) + ee k ( S) + { T { S { S { S T }}}} {{ } { { }}} { { }},,,, = 0 i j + pp ek( Seee ) k i j ( S) ee k i ( S) + ee i k( S) ee k j ( S) ST,, S, ST, = e( S) e ( S) e e ( S) { } { { { }}} { } i j k T, T, T, S, T = p p p p e e e e ( S) { S S S S T },,,, = 0 i i i i j j i j i j k Monday, 13 June

74 Computing Poisson Brackets These are quite complicated (some might say disgusting) objects to compute on the lattice Even for the simplest Wilson gauge action They consists sums of complicated lattice loops with momenta inserted in various places Fortunately there is a recursive way of computing them which is tractable even for more complicated gauge actions It involves inserting previously computed Liealgebra-valued fields living on links into the loops in the action using a tower algorithm Monday, 13 June

75 Fermion Poisson Brackets Fermions are easy to include in the formalism: we only need a few extra linear equation solves ( ) 1( ) 1 ( ) SF U = φ M U φ = Tr M U φ φ M U 1 = M M M U 1 1 Monday, 13 June

76 Tuning Your Integrator For any (symmetric) symplectic integrator the conserved Hamiltonian is constructed from the same Poisson brackets A procedure for tuning such integrators is Measure the Poisson brackets during an HMC run Optimize the integrator (number of pseudofermions, step-sizes, order of integration scheme, etc.) offline using these measured values Monday, 13 June

77 What to Tune As I said a while ago, minimizing ΔH H H is not a good choice It is much better to minimize the variance of ΔH This is a function of two sets of quantities The ensemble-averaged Poisson brackets The integrator parameters Clark, Kennedy, and Silva Lattice 008 (JLab) Monday, 13 June

78 Why Minimize the Variance? As the system wanders through phase space is constant, so H δh = H H = δδh f i We hypothesize that the distribution of ΔH is essentially sampled independently and randomly at the start and end of each equilibrium trajectory Therefore we want to minimize the variance of this distribution Clark, Kennedy, and Silva Lattice 008 (JLab) Monday, 13 June

79 Simplest Integrators Integrator Update Steps Shadow Hamiltonian ({,{, }} {,{, }}) S / T S / PQP e e e T + S S S T + T S T T ε T ε S ε ( {,{, }} {,{, }}) ε QPQ e e e T + S + S S T + T S T 4 Sε T ε Sε T ε Sε ε Omelyan SST e e e e e T + S + { S,{ S, T }} 7 ( 3 3) S ε T ε S ε T ( ε 3 ) 3 S ε Omelyan TST e e e e e T + S + ε T, S,T 4 ε 4 { { }} Monday, 13 June

80 Campostrini Integrators Campostrini found an ingenious way of constructing integrators with errors of arbitrarily high order Start with an integrator with errors of order n in the ε integration step size ( ) H ( ε) ε H n n X n ε e = e 1 + ε Δ+ ( ε + ) Construct the wiggle sandwiching a backward step of this integrator between two forward ones ( ) ( ) ( ) ( ) H n n ( ) n X n ε X n σε X n ε = e ε σ 1+ ε σ Δ + ( ε + ) n Eliminate the leading order error by choosing σ = and adjust the step size by setting δτ = ε ( σ) H n to obtain X n ( δτ ) X n ( ε ) X n ( σε ) X n ( ε ) e δτ 1 δτ + + = = + ( ) Monday, 13 June

81 Campostrini Integrators Integrator Update Steps Shadow Hamiltonian e T ε Campostrini e S ε T ε S ε T ε e e e e S ε T ε T + S ( ){ S,{ S,{ S,{ S, T }}}} + ( 0 +3 ){ T,{ T,{ S,{ S, T }}}} ( ){{ ST, },{ T, { ST, }}} + ( ){ T, { S, { S, { ST, }}}} ( ){{ ST} { S{ ST} }} + ( ){ T { T { T { ST} }}} ,,,, 5 +8,,,, e ε 4 Monday, 13 June

82 Force-Gradient Integrators An interesting observation is that the Poisson bracket S, S, T depends only of q We may therefore evaluate the integrator e 3 { S,{ S, T }} δτ { { }} explicitly The force for this integrator involves second derivatives of the action Using this type of step we can construct efficient Force-Gradient (Hessian) integrators Monday, 13 June 011 8

83 Force-Gradient Integrators Integrator Update Steps Shadow Hamiltonian Force-Gradient 1 Force-Gradient e T ε 3Sε T ε e e e { } 3 { { }} 48 Sε S, S, T 19 T ε 3Sε T ε e e e e e S ε T ε 6 ε { { }} T + S { S { S { S S T }}} + T T { S { S T }} { ST} { T { ST} } T S{ S ST } { { }} { } { { { } }} {{ ST} { S{ ST} }} T { T { T { ST} }} 59,,,, 44,,,, + 768,,,, ,,,, + 304,,,, + 896,,,, e 3 { } 48 Sε S, S, T e T ε 7 e S ε 6 ε T + S { } { S { S { S S T }}} T T { S { S T }} { ST} { T { ST} } T S{ S ST } { { }} { } { { { } }} {{ ST} { S{ ST} }} T { T { T { ST} }} 41,,,, +16,,,, +7,,,, +84,,,, +36,,,, +54,,,, { } ε 4 ε 4 Monday, 13 June

84 Multiple timescales Split Hamiltonian into pieces H( q, p) = T ( p) + S1( q) + S( q) and T T ( p) S ( ), so H = T + S i S i q 1 + S q p Introduce a symmetric symplectic integrator of the form n n n1 1 n ( ) / δτ S δτ S n n1 1 T δτ S n1 1 δτ τ δτ δτ n USW δτ = e e e e e S 1 S If P then the instability in the integrator is n1 n tickled equally by each sub-step This helps if the most expensive force computation does not correspond to the largest force τ / δτ Monday, 13 June

85 Monday, 13 June

86 Pseudofermions Direct simulation of Grassmann fields is not feasible The problem is not that of manipulating anticommuting values in a computer It is that S F M e = e ψ ψ is not positive, and thus we get poor importance sampling We therefore integrate out the fermion fields to M obtain the determinant d d e ψ ψ ψ ψ det ( M ) and always occur quadratically ψ ψ The overall sign of the exponent is unimportant Monday, 13 June

87 Pseudofermions Any operator Ω can be expressed solely in terms of the bosonic fields δ δ Ω ( φ) = Ω φ,, δψ δψ E.g., the fermion propagator is M e ψ 1 ( ) ( ) ( ) φ ψ ψ = ψ = 0 Gψ ( x, y) = ψ x ψ y = M ( x, y) Monday, 13 June

88 Pseudofermions Including the determinant as part of the observable to be det M ( φ) Ω ( φ) SB measured is not feasible Ω = det M φ ( ) The determinant is extensive in the lattice volume, thus again we get poor importance sampling Represent the fermion determinant as a bosonic Gaussian M 1 ( ) integral with a non-local kernel det M ( ) d d e χ φ χ χ φ χ The fermion kernel must be positive definite (all its eigenvalues must have positive real parts) otherwise the bosonic integral will not converge The new bosonic fields are called pseudofermions S B Monday, 13 June

89 Pseudofermions It is usually convenient to introduce two flavours of fermion ( ) and to write ( ) 1 M M det M det M M d d e χ φ = φ φ χ χ χ ( ) ( ( ) ( )) This not only guarantees positivity, but also allows us to generate the pseudofermions from a global heatbath by applying M to a random Gaussian distributed field The equations for motion for the boson (gauge) fields are φ = π ( φ) The evaluation of the pseudofermion action and the corresponding force then requires us to find the solution of a (large) set of linear equations ( M M ) 1 χ = ψ ( φ) S S = ( ) = + ( ) ( ) ( ) φ φ φ φ B 1 B 1 1 π χ M M χ M M χ M M M M χ Monday, 13 June

90 Pseudofermions It is not necessary to carry out the inversions required for the equations of motion exactly There is a trade-off between the cost of computing the force and the acceptance rate of the Metropolis MDMC step The inversions required to compute the pseudofermion action for the accept/reject step does need to be computed exactly, however We usually take exactly to by synonymous with to machine precision Monday, 13 June

91 Reversibility Are HMC trajectories reversible and area preserving in practice? The only fundamental source of irreversibility is the rounding error caused by using finite precision floating point arithmetic For fermionic systems we can also introduce irreversibility by choosing the starting vector for the iterative linear equation solver time-asymmetrically We do this if we to use a Chronological Inverter, which takes (some extrapolation of) the previous solution as the starting vector Floating point arithmetic is not associative It is more natural to store compact variables as scaled integers (fixed point) Saves memory Does not solve the precision problem Monday, 13 June 011 9

92 Reversibility Data for SU(3) gauge theory and QCD with heavy quarks show that rounding errors are amplified exponentially The underlying continuous time equations of motion are chaotic Ляпунов exponents characterise the divergence of nearby trajectories The instability in the integrator occurs when δh» 1 Zero acceptance rate anyhow Monday, 13 June

93 Reversibility In QCD the Ляпунов exponents appear to scale with β as the system approaches the continuum limit β νξ = constant This can be interpreted as saying that the Ляпунов exponent characterises the chaotic nature of the continuum classical equations of motion, and is not a lattice artefact Therefore we should not have to worry about reversibility breaking down as we approach the continuum limit Caveat: data is only for small lattices, and is not conclusive Monday, 13 June

94 Reversibility Data for QCD with lighter dynamical quarks Instability occurs close to region in δτ where acceptance rate is near one May be explained as a few modes becoming unstable because of large fermionic force Integrator goes unstable if too poor an approximation to the fermionic force is used Monday, 13 June

95 Monday, 13 June

96 Polynomial approximation What is the best polynomial approximation p(x) to a continuous function f(x) for x in [0,1]? Best with respect to the appropriate norm where n 1 1 p f = dx p( x) f ( x) n 0 n 1/ n Monday, 13 June

97 Weierstraß theorem Taking n this is the minimax norm p f = minmax p( x) f ( x) p 0 x 1 Weierstraß: Any continuous function can be arbitrarily well approximated by a polynomial Monday, 13 June

98 Бернштейне polynomials The explicit solution is provided by Бернштейне polynomials n k = 0 k n f x (1 x) n k p ( x ) n n n k Monday, 13 June

99 Чебышев s theorem Чебышев: There is always a unique polynomial of any degree d which minimises 0 x 1 ( ) ( ) p f = max p x f x The error p(x) - f(x) reaches its maximum at exactly d+ points on the unit interval Monday, 13 June

100 Чебышев s theorem: Necessity Suppose p-f has less than d+ extrema of equal magnitude Then at most d+1 maxima exceed some magnitude This defines a gap We can construct a polynomial q of degree d which has the opposite sign to p-f at each of these maxima (Lagrange interpolation) And whose magnitude is smaller than the gap The polynomial p+q is then a better approximation than p to f Monday, 13 June

101 Чебышев s theorem: Sufficiency Suppose there is a polynomial p f p f ( ) ( ) ( ) ( ) Then p x f x p x f x at each of the d+ extrema i i i i Therefore p - p must have d+1 zeros on the unit interval Thus p p = 0 as it is a polynomial of degree d Monday, 13 June

102 Чебышев polynomials Convergence is often exponential in d The best approximation of degree d-1 over [-1,1] to x d is d 1 p 1 d 1 x x Td x ( ) ( ) d ( ) Where the Чебышев polynomials are d ( ) 1 ( ) = cos cos ( ) T x d x The notation is an old transliteration of Чебышев! The error is d 1 ( ) ( ) d x p 1 ln d x = Td ( x ) = e d Monday, 13 June

103 Чебышев rational functions Чебышев s theorem is easily extended to rational approximations Rational functions with nearly equal degree numerator and denominator are usually best Convergence is still often exponential Rational functions usually give a much better approximation A simple (but somewhat slow) numerical algorithm for finding the optimal Чебышев rational approximation was given by Ремез Monday, 13 June

104 1 x Чебышев rationals: Example A realistic example of a rational approximation is ( x ) ( x ) ( x ) ( + )( + )( + ) x x x This is accurate to within almost 0.1% over the range [0.003,1] Using a partial fraction expansion of such rational functions allows us to use a multishift linear equation solver, thus reducing the cost significantly. The partial fraction expansion of the rational function above is x + x x x This appears to be numerically stable. Monday, 13 June

105 Polynomials versus rationals Золотарев s formula has L error Optimal L approximation with weight is n j ( ) 4 Tj + 1( x ) ( j + 1) π j = 0 This has L error of O(1/n) n ln Δ e ε 1 1 x Optimal L approximation cannot be too much better (or it would lead to a better L approximation) Monday, 13 June

106 Non-linearity of CG solver Suppose we want to solve A x=b for Hermitian A by CG It is better to solve Ax=y, Ay=b successively Condition number κ(a ) = κ(a) Cost is thus κ(a) < κ(a ) in general Suppose we want to solve Ax=b Why don t we solve A 1/ x=y, A 1/ y=b successively? The square root of A is uniquely defined if A>0 This is the case for fermion kernels All this generalises trivially to n th roots No tuning needed to split condition number evenly How do we apply the square root of a matrix? Monday, 13 June

107 Rational matrix approximation Functions on matrices Defined for a Hermitian matrix by diagonalisation H = U D U -1 f (H) = f (U D U -1 ) = U f (D) U -1 Rational functions do not require diagonalisation α H m + β H n = U (α D m + β D n ) U -1 H -1 = U D -1 U -1 Rational functions have nice properties Cheap (relatively) Accurate Monday, 13 June

108 No Free Lunch Theorem We must apply the rational approximation with each CG iteration M 1/n r(m) The condition number for each term in the partial fraction expansion is approximately κ(m) So the cost of applying M 1/n is proportional to κ(m) Even though the condition number κ(m 1/n )=κ(m) 1/n And even though κ(r(m))=κ(m) 1/n So we don t win this way Monday, 13 June

109 Pseudofermions 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 1 M detm dφ dφe φ φ Monday, 13 June

110 Multipseudofermions 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 M 1/n ] n 1 n 1 n M det M dφ dφe φ φ Monday, 13 June

111 Hasenbusch s method Start with the Wilson fermion kernel M = 1 κh Introduce the quantity M = 1 κ H Use the (associative) identity Introduce separate pseudofermions for each det M = det M det M 1 M determinant ( ) Adjust κ to minimise the cost Easily generalises More than two pseudofermions Wilson-clover action = 1 M MM M Monday, 13 June

112 Violation of NFL Theorem Let s try using our n th root trick to implement multipseudofermions Condition number κ(r(m))=κ(m) 1/n So maximum force is reduced by a factor of nκ(m) (1/n)-1 This is a good approximation if the condition number is dominated by a few isolated tiny eigenvalues This is so in the case of interest Cost reduced by a factor of nκ(m) (1/n)-1 Optimal value n opt ln κ(m) So optimal cost reduction is (e lnκ) /κ This works! Monday, 13 June

113 Rational Hybrid Monte Carlo RHMC algorithm for fermionic kernel ( ) 1 n MM Generate pseudofermion from Gaussian heatbath 1 1 4n χ = ( MM) P( ξ ) e ξ ξ n ( ) 4n P( ) d e ( ) e χ ξξ χ ξ δ χ ξ MM MM χ Use accurate rational approximation r( x) 4 Use less accurate approximation for MD, ξ n x n r( x) r( x) r( x), so there are no double poles Use accurate approximation for Metropolis acceptance step x Monday, 13 June

114 Rational Hybrid Monte Carlo Reminders Apply rational approximations using their partial fraction expansions Denominators are all just shifts of the original fermion kernel All poles of optimal rational approximations are real and positive for cases of interest (Miracle #1) Only simple poles appear (by construction!) Use multishift solver to invert all the partial fractions using a single Krylov space Cost is dominated by Krylov space construction, at least for O(0) shifts Result is numerically stable, even in 3-bit precision All partial fractions have positive coefficients (Miracle #) MD force term is of the usual form for each partial fraction Applicable to any kernel Monday, 13 June

115 Multipseudofermions with multiple timescales Semiempirical observation: The largest force from a single pseudofermion does not come from the smallest shift For example, look at the numerators in the partial fraction expansion we exhibited earlier Make use of this by using a coarser timescale for the more expensive smaller shifts 100% 75% 50% 5% 0% Residue (α) L² Force α/(β+0.15) CG iterations Shift [ln(β)] x + x x x Monday, 13 June

116 L versus L Force Norms β=5.6, Wilson fermion forces (from Urbach et. al.) Monday, 13 June

117 Conclusions (RHMC) Advantages of RHMC Exact No step-size errors; no step-size extrapolations Significantly cheaper than the R algorithm Allows easy implementation of Hasenbusch (multipseudofermion) acceleration Further improvements possible Such as multiple timescales for different terms in the partial fraction expansion Disadvantages of RHMC Costly for FG integrators (numerous right hand sides) Monday, 13 June