1. Saturday, 02 May 2015 Speeding up HMC with better integrators A D Kennedy and M A Clark School of Physics & SUPA, The University of Edinburgh Boston University A D Kennedy and M A Clark School of Physics & SUPA, The University of Edinburgh Boston University

2. 2 2Saturday, 02 May 2015A D Kennedy Abstract We shall discuss how dynamical fermion computations may be made yet cheaper by using symplectic integrators that conserve energy much more accurately without increasing the integration step size. We will first explain why symplectic integrators exactly conserve a “shadow” Hamiltonian close to the desired one, and how this Hamiltonian may be computed in terms of Poisson brackets. We shall then discuss how classical mechanics may be implemented on Lie groups and derive the form of the Poisson brackets and force terms for some interesting integrators such as those making use of second derivatives of the action (Hessian or force gradient integrators). These will be seen to greatly improved energy conservation for only a small additional cost (one extra inversion of the Dirac operator), and we hope to show that their use significantly reduces the cost of dynamical fermion computations.

3. 3 3Saturday, 02 May 2015A D Kennedy Outline Symmetric symplectic integrators in HMC Shadow Hamiltonians and Poisson brackets Tuning integrators using Poisson brackets Hessian or Force-Gradient integrators Symplectic integrators and Poisson brackets on Lie groups Results for single-link updates Symmetric symplectic integrators in HMC Shadow Hamiltonians and Poisson brackets Tuning integrators using Poisson brackets Hessian or Force-Gradient integrators Symplectic integrators and Poisson brackets on Lie groups Results for single-link updates

4. 4 4Saturday, 02 May 2015A D Kennedy Symplectic Integrators We are interested in finding the classical trajectory in phase space of a system described by the Hamiltonian The idea of a symplectic integrator is to write the time evolution operator (Lie derivative) as

5. 5 5Saturday, 02 May 2015A D Kennedy Symplectic Integrators 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

6. 6 6Saturday, 02 May 2015A D Kennedy If A and B belong to any (non-commutative) algebra then, where  constructed from commutators of A and B (i.e., is in the Free Lie Algebra generated by A and B ) Symplectic Integrators More precisely, where and Baker-Campbell-Hausdorff (BCH) formula

7. 7 7Saturday, 02 May 2015A D Kennedy Symplectic Integrators Explicitly, the first few terms are In order to construct reversible integrators we use symmetric symplectic integrators The following identity follows directly from the BCH formula

8. 8 8Saturday, 02 May 2015A D Kennedy Symplectic Integrators From the BCH formula we find that the PQP symmetric symplectic integrator is given by In addition to conserving energy to O (  ² ) such symmetric symplectic integrators are manifestly area preserving and reversible

9. 9 9Saturday, 02 May 2015A D Kennedy Shadow Hamiltonians This may be obtained by replacing the commutators in the BCH expansion of with the Poisson bracket For each symplectic integrator there exists a Hamiltonian H’ which is exactly conserved

10. 10 Saturday, 02 May 2015A D Kennedy Conserved Hamiltonian For the PQP integrator we have

11. 11 Saturday, 02 May 2015A D Kennedy Tuning HMC 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 This can be done because the acceptance rate (and instabilities) are completely determined by δH = H’ - H 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 This can be done because the acceptance rate (and instabilities) are completely determined by δH = H’ - H

12. 12 Saturday, 02 May 2015A D Kennedy Simple Example (Omelyan) Consider the PQPQP integrator The conserved Hamiltonian is thus Measure the “operators” and minimize the cost by adjusting the parameter α

13. 13 Saturday, 02 May 2015A D Kennedy Hessian Integrators 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 very efficient Force-Gradient integrators The force for this integrator involves second derivatives of the action Using this type of step we can construct very efficient Force-Gradient integrators

14. 14 Saturday, 02 May 2015A D Kennedy Higher-Order Integrators We can eliminate all the leading order Poisson brackets in the shadow Hamiltonian leaving errors of O (δτ 2 ) The coefficients of the higher-order Poisson brackets are much smaller than those from the Campostrini integrator We can eliminate all the leading order Poisson brackets in the shadow Hamiltonian leaving errors of O (δτ 2 ) The coefficients of the higher-order Poisson brackets are much smaller than those from the Campostrini integrator

15. 15 Saturday, 02 May 2015A D Kennedy Beyond Scalar Field Theory We need to extend the formalism beyond a scalar field theory Fermions are easy How do we extend all this fancy differential geometry formalism to gauge fields?

16. 16 Saturday, 02 May 2015A D Kennedy Hamiltonian Mechanics Flat ManifoldGeneral Symplectic 2-form Hamiltonian vector field Equations of motion Poisson bracket Darboux theorem: All manifolds are locally flat

17. 17 Saturday, 02 May 2015A D Kennedy Maurer-Cartan Equations The left invariant forms dual to the generators of a Lie algebra satisfy the Maurer- Cartan equations

18. 18 Saturday, 02 May 2015A D Kennedy We can invent any Classical Mechanics we want… So we may therefore define a closed symplectic 2-form which globally defines an invariant Poisson bracket by Fundamental 2-form

19. 19 Saturday, 02 May 2015A D Kennedy 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 Hamiltonian Vector Field Define a vector field such that

20. 20 Saturday, 02 May 2015A D Kennedy Integral Curves The classical trajectories are then the integral curves of h :

21. 21 Saturday, 02 May 2015A D Kennedy Poisson Brackets Recall our Hamiltonian vector field For H(q,p) = T(p) + S(q) we have vector fields

22. 22 Saturday, 02 May 2015A D Kennedy More Poisson Brackets We thus compute the lowest-order Poisson bracket and the Hamiltonian vector corresponding to it

23. 23 Saturday, 02 May 2015A D Kennedy Even More Poisson Brackets

24. 24 Saturday, 02 May 2015A D Kennedy Integrators

25. 25 Saturday, 02 May 2015A D Kennedy Campostrini Integrator

26. 26 Saturday, 02 May 2015A D Kennedy Hessian Integrators

27. 27 Saturday, 02 May 2015A D Kennedy One-Link Results

28. 28 Saturday, 02 May 2015A D Kennedy Scaling Behaviour

29. 29 Saturday, 02 May 2015A D Kennedy Conclusions We hope that very significant performance improvements can be obtained using Force- Gradient integrators For fermions one extra inversion of the Dirac operator is required Pure gauge force terms and Poisson brackets get quite complicated to program Real-life speed-up factors will be measured really soon… We hope that very significant performance improvements can be obtained using Force- Gradient integrators For fermions one extra inversion of the Dirac operator is required Pure gauge force terms and Poisson brackets get quite complicated to program Real-life speed-up factors will be measured really soon…