Chasing Shadows A D Kennedy University of Edinburgh Tuesday, 06 October 2015 QCD & NA, Regensburg

2 Outline Functional Integrals, Markov Chains, and HMC Symplectic Integrators and Shadow Hamiltonians Symplectic 2-form and Hamiltonian vector fields Poisson brackets BCH formula Shadow Hamiltonians Shadows for gauge theories Improved integrators Hessian (force-gradient) integrators Instabilities Free field theory Pseudofermions Multiple pseudofermions Hasenbusch, RHMC, DD-HMC Multiple shifts and multiple sources Tuesday, 06 October 2015QCD & NA, Regensburg

3 Functional Integrals Tuesday, 06 October 2015 In lattice quantum field theory the goal is to compute the expectation value of some interesting operator Integral is over all field configurations (At least) one integral for each lattice site This become an infinite dimensional integral in the continuum limit QCD & NA, Regensburg

4 Fermions Tuesday, 06 October 2015 Fermion fields are Grassmann-valued Grassmann integration of quadratic form gives a determinant We replace these by bosonic integrals over pseudofermion fields with inverse kernel QCD & NA, Regensburg

5 Fermionic Operators Tuesday, 06 October 2015 Operators involving Fermion fields can be expressed as non-local bosonic operators Add Fermion sourcesComplete the squareShift Fermion fieldsEvaluate Grassmann integral Evaluating the inverses of the Fermion kernel is usually the expensive part of measuring operators on dynamical gauge configurations QCD & NA, Regensburg

6 Markov Chains Tuesday, 06 October 2015 Evaluate functional integrals by Monte Carlo Generate configurations from Markov chain Iterating ergodic Markov step(s) converges to fixed point distribution Detailed balance  fixed point Metropolis accept/reject  detailed balance QCD & NA, Regensburg

7 Hybrid Monte Carlo (HMC) Tuesday, 06 October 2015 Introduce fictitious momenta p for each q and generate phase space configurations with distribution with the Hamiltonian Solution of Hamilton’s equations would then generate a new phase space configuration which satisfies detailed balance Reversible (after a momentum flip) Measure preserving (Liouville’s theorem) Equiprobable (energy conservation) QCD & NA, Regensburg Momentum & pseudofermion heatbath  ergodicity?

8 HMC and Symplectic Integrators Tuesday, 06 October 2015 Symmetric symplectic integrators provide discrete approximations to the solution of Hamilton’s equations Symmetric  r eversible Symplectic  area preserving Such integrator only approximately conserve energy, but this may be corrected by Metropolis step with acceptance probability QCD & NA, Regensburg

9 Hamiltonian Dynamics Tuesday, 06 October 2015 The basic idea of a Hamiltonian system is that its state at time t is represented by a point in phase space (the cotangent bundle over configuration space ) Hamilton’s equations give the time evolution in terms of derivatives of a Hamiltonian which is a function (0-form) on phase space QCD & NA, Regensburg

10 Hamiltonian Dynamics Tuesday, 06 October 2015 More generally the time evolution of the system follows the integral curves of the Hamiltonian vector field, which is related to H through the symplectic 2-form through QCD & NA, Regensburg

11 Hamiltonian Dynamics Tuesday, 06 October 2015 Once can define a Hamiltonian vector field for any 0-form A on phase space The commutator of two such Hamiltonian vector fields is itself a Hamiltonian vector field where is the Poisson bracket QCD & NA, Regensburg

12 Proofs: I Tuesday, 06 October 2015 Definition: Exterior derivative dF of a 0- form F applied to a vector X is Hence for a Hamiltonian vector field Corollary QCD & NA, Regensburg

13 Proofs: II Tuesday, 06 October 2015 Observe that Definition: Exterior derivative of a 2-form field is Hence for Hamiltonian vector fields QCD & NA, Regensburg

14 Proofs: III Tuesday, 06 October 2015 This means that 0-forms on a symplectic manifold form a Lie algebra with a Lie bracket that is not a commutator Thus the closure of the symplectic 2-form implies the Jacobi identity for Poisson brackets Using the Jacobi identity we finally get for any F QCD & NA, Regensburg

15 Symplectic Integrators We are interested in finding the classical trajectory in phase space of a system described by the Hamiltonian The basic idea of a symplectic integrator is to write the time evolution operator as Tuesday, 06 October 2015QCD & NA, Regensburg

16 PQP Integrator Define and so that Since the kinetic energy T is a function only of p and the potential energy S is a function only of q, it follows that the action of and may be evaluated trivially We can thus construct an approximate PQP leapfrog integrator Tuesday, 06 October 2015QCD & NA, Regensburg

17 Baker-Campbell-Hausdorff formula More precisely, where and The B n are Bernoulli numbers If and belong to any (non-commutative) algebra then, where is constructed from commutators of and Such commutators are in the Free Lie algebra Tuesday, 06 October 2015QCD & NA, Regensburg

18 BCH formula Explicitly, the first few terms are We only include commutators that are not related by antisymmetry or the Jacobi relation These are chosen from a Hall basis Tuesday, 06 October 2015QCD & NA, Regensburg

19 Symmetric Symplectic Integrators In order to construct reversible integrators we use symmetric symplectic integrators From the BCH formula the PQP integrator corresponds to the vector field Tuesday, 06 October 2015QCD & NA, Regensburg

20 Shadow Hamiltonian: I A symplectic integrator exactly follows the vector field where is constructed from commutators of Hamiltonian vector fields Commutators of Hamiltonian vector fields are themselves Hamiltonian vector fields, Therefore is itself the Hamiltonian vector field corresponding to the shadow Hamiltonian obtained by replacing the commutators in with Poisson brackets This only defines the shadow Hamiltonian as a series expansion in ; we will later consider what happens when this series fails to converge Tuesday, 06 October 2015QCD & NA, Regensburg

21 Shadow Hamiltonian: II The PQP integrator follows a trajectory that is the exact solution of Hamilton’s equations for the (exactly conserved) shadow Hamiltonian Tuesday, 06 October 2015QCD & NA, Regensburg

22 Scalar Theory Evaluating the Poisson brackets gives 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 Tuesday, 06 October 2015QCD & NA, Regensburg

23 How to Tune Integrators For any (symmetric) symplectic integrator the conserved Hamiltonian is constructed from the same Poisson brackets The procedure is therefore Measure average Poisson brackets during a preliminary run Optimize the integrator (number of pseudofermions, step- sizes, order of integration scheme, etc.) offline using these measured values This can be done because the acceptance rate (and instabilities) are completely determined by See talk by Paulo Silva for details Tuesday, 06 October 2015QCD & NA, Regensburg

24 On a Lie group there are left invariant forms dual to the generators that satisfy the Maurer-Cartan equations Classical Mechanics on Group Manifolds Tuesday, 06 October 2015 We first formulate classical mechanics on a Lie group manifold, and then rewrite it in terms of the usual constrained variables (U and P matrices) Haar measure is QCD & NA, Regensburg

25 Fundamental 2-form We may therefore define a closed symplectic fundamental 2-form This specifies the Poisson bracket Tuesday, 06 October 2015QCD & NA, Regensburg

26 Hamiltonian Vector Field Tuesday, 06 October 2015 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 such that QCD & NA, Regensburg

27 Poisson Brackets Tuesday, 06 October 2015 For we have vector fields QCD & NA, Regensburg

28 and the Hamiltonian vector corresponding to it More Poisson Brackets Tuesday, 06 October 2015 We thus compute the lowest-order Poisson bracket QCD & NA, Regensburg

29 Even More Poisson Brackets Tuesday, 06 October 2015QCD & NA, Regensburg

30 Integrators Tuesday, 06 October 2015QCD & NA, Regensburg

31 Hessian Integrators Tuesday, 06 October 2015 We may therefore evaluate the integrator explicitly An interesting observation is that the Poisson bracket depends only of q The force for this integrator involves second derivatives of the action Using this type of step we can construct efficient Hessian (Force-Gradient) integrators QCD & NA, Regensburg

32 Higher-Order Integrators We can eliminate all the leading order Poisson brackets in the shadow Hamiltonian leaving errors of The coefficients of the higher-order Poisson brackets are much smaller than those from the Campostrini integrator Tuesday, 06 October 2015QCD & NA, Regensburg

33 Campostrini Integrator Tuesday, 06 October 2015QCD & NA, Regensburg

34 Hessian Integrators Tuesday, 06 October 2015QCD & NA, Regensburg

35 Multiple Timescale Integrators Use different integration step sizes for different contributions to the action (Sexton— Weingarten) Evaluate cheap forces that give a large contribution to the shadow Hamiltonian more frequently Original application failed because largest contribution to shadow Hamiltonian was also the most expensive Tuesday, 06 October 2015QCD & NA, Regensburg

36 Instabilities in Free Field Theory Tuesday, 06 October 2015QCD & NA, Regensburg For free field theory the Hamiltonian may be diagonalized to for some set of frequencies The shadow Hamiltonian is exactly This has a singularity at

37 Integrator Instabilities When the step size exceeds some critical value the BCH expansion for the shadow Hamiltonian diverges No real conserved shadow Hamiltonian Symplectic integrators become exponentially unstable in trajectory length Tuesday, 06 October 2015QCD & NA, Regensburg

38 Symptoms Tuesday, 06 October 2015QCD & NA, Regensburg Acceptance rate falls (exponentially) with trajectory length for Floating point rounding errors can also cause this For light fermions maximum frequency corresponds to inverse of smallest eigenvalue of fermion kernel This can grow as gauge fields equilibrate Often erroneously ascribed to exceptional configurations Real problem is exponential amplification due to instabilities

39 Higher Order Integrator Instabilities Tuesday, 06 October 2015QCD & NA, Regensburg The values of at which instabilities occur depends on the choice of integrator IntegratorCriticalvalue PQP, QPQ2 P2MN22.22 P24FG2.61 P5MN43.14 Q5MN4FG3.10 Campostrini1.57 Mike Clark Omelyan, Mryglod, Folk Computer Physics Communcations 151 (2003) Joo et al Phys Rev D62 (2000) (hep-lat/ )

40 Pseudofermions Tuesday, 06 October 2015 We must distinguish the intrinsic fermion part from that due to pseudofermionic noise The pseudofermion contribution to the shadow Hamiltonian has one more power of the inverse fermion kernel than the intrinsic fermion contribution The pseudofermions are fixed during a trajectory, so probably do not suppress this extra inverse power QCD & NA, Regensburg

41 This does not change the intrinsic fermion part at all Instability is driven by most singular kernel Must balance cost of applying kernels with reduction in contribution to shadow Hamiltonian No improvement with more than two or three pseudofermions so far Acceptance rate no longer limited by instability but by bulk contributions? Opportunity to use improved/higher-order integrators Rôle of finite density of fermion spectrum near zero? Eventually limited by intrinsic fermion part Multiple Pseudofermions Tuesday, 06 October 2015 The effect of the extra inverse power can be reduced by introducing multiple pseudofermions with less singular kernels QCD & NA, Regensburg

42 Multiple Pseudofermion Techniques Tuesday, 06 October 2015 Hasenbusch: Heavy pseudofermion cheap to invert Particularly cost-effective if we make use of existing pseudofermion (e.g., s quark): RHMC No parameter tuning necessary Can implement fractional flavours (e.g., for above) Lüscher DD-HMC Dominated by Schur complement part towards chiral limit? Evidence of this from PAC-CS data? Other benefits of DD? QCD & NA, Regensburg

43 Multi-Everything Solvers For RHMC we evaluate fractional powers of the fermion kernel using optimal (Chebyshev) partial fraction rational approximations We thus need to find solutions for this kernel (the Dirac operator or its square) for Multiple shifts: shifts are poles of rational approximation Multiple right hand sides: for multiple pseudofermions We already use a multishift solver, but How to do so for multiple right hand sides? Can we use an initial guess extrapolated from the past? Can we restart the solver to avoid stagnation? Tuesday, 06 October 2015QCD & NA, Regensburg

44 RHMC 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 Use a coarser timescale for expensive smaller shifts Invert small shifts less accurately Cannot use chronological inverter with multishift solver anyhow Tuesday, 06 October 2015QCD & NA, Regensburg

45 Outlook Large-scale computations with light dynamical fermions require Avoiding integrator instabilities as far as possible Use multiple pseudofermion fields Reducing volume contributions to H Shadow Tune integrators by measuring Poisson brackets Use good higher order integrators, such as Hessian/Force Gradient Using longer trajectories Reduces autocorrelations as physical correlation lengths grow Amortizes cost of pseudofermion heatbath and Metropolis energy calculation Improving solver technology Tuesday, 06 October 2015QCD & NA, Regensburg