1. Shadow Hamiltonians and Force-Gradient Symplectic Integrators
A D Kennedy
University of Edinburgh
(Mike Clark and Paulo Silva)
Sunday, 24 February 2019
Perspectives on Light Quark Simulations

2. Object
To review why we need symmetric symplectic integrators, how they sometimes fail, and what properties we will require in the near future
To give a pedagogical introduction to shadow Hamiltonians and how they can help to understand and tune symplectic integrators
Not to present new results: you will have to wait until the summer (perhaps)!
Sunday, 24 February 2019
A D Kennedy

3. Caveat
A recent (published) paper had near the beginning the passage “The object of this paper is to prove (something very important).” It transpired with great difficulty, and not till near the end, that the “object” was an unachieved one. Littlewood, “A Mathematician’s Miscellany”
Sunday, 24 February 2019
A D Kennedy

4. Why Integrators?
HMC algorithm requires reversible area-preserving integrators
We need small integration step sizes for small and thus good acceptance rates
Naïve expectation is for order n integrator
Long history...
Campostrini—Rossi
Sexton—Weingarten
Takaishi—deForcrand
... and a whole subfield of numerical analysts
Sunday, 24 February 2019
A D Kennedy

5. Meyer, Simma, Sommer, Della Morte, Witzel, and Wolff
Long Trajectories
We need long trajectories to generate independent gauge field configurations
where is the relevant correlation length, probably an inverse hadron size in lattice units
It is easy to fool oneself into the false economy of choosing trajectories that are too short!
Meyer, Simma, Sommer, Della Morte, Witzel, and Wolff
Sunday, 24 February 2019
A D Kennedy

6. Mike Clark, Lattice 2006 (Tucson)
Towards Light Quarks
Step size limited by integrator instabilities
Not by exceptional configurations!
Instabilities caused by excessively noisy estimates of fermionic force
Not (yet) by fermionic forces themselves
“Cured” by using more pseudofermion estimators
Hasenbusch, DDHMC, RHMC
Mike Clark, Lattice 2006 (Tucson)
Sunday, 24 February 2019
A D Kennedy

7. Approaching the Physical m
We now need larger and finer lattices
Bulk effects become more important
Higher-order integrators will become increasingly important
Mike Clark, Lattice 2006 (Tucson)
Sunday, 24 February 2019
A D Kennedy

8. Problems to be Solved
Multiple pseudofermions reduce the fermionic force
But what exactly is this “fermion force” that is to be minimized?
Painful tuning problem
Higher-order integrators have lots of free parameters
They are often built of longer sub-steps
Which make instabilities worse
These are painful to tune too
Sunday, 24 February 2019
A D Kennedy

9. 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 in the BCH expansion of with the Poisson bracket
Sunday, 24 February 2019
A D Kennedy

10. Why and How?
Classical mechanics is not specified just by a Hamiltonian H but also by a closed fundamental 2-form
For every function (0-form) A this defines a Hamiltonian vector field
Which just means that for all X
Sunday, 24 February 2019
A D Kennedy

11. Concrete Shadows
To be a little less abstract consider the familiar case where
and
we have so
Sunday, 24 February 2019
A D Kennedy

12. Classical Trajectories
Classical trajectories are then integral curves of the Hamiltonian vector field of the Hamiltonian H
In other words, this vector field is always tangent to the classical trajectory
Sunday, 24 February 2019
A D Kennedy

13. Poisson Brackets
Consider the action of a Hamiltonian vector field of a function (0-form)
Where we have introduced the Poisson bracket of two functions
These obey the Jacobi identity
This follows from the closure of the fundamental 2-form
It is not trivial: Poisson brackets are not commutators
Functions form a Lie algebra with PBs as the Lie product
Sunday, 24 February 2019
A D Kennedy

14. Concrete Poisson Brackets
To make this more familiar when the Poisson bracket becomes
Sunday, 24 February 2019
A D Kennedy

15. 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
Sunday, 24 February 2019
A D Kennedy

16. Commutators
So far this is just a fancy (and complicated) way of rewriting Hamilton’s equations, but now we derive a surprising new result
The commutator of Hamiltonian vector fields is itself a Hamiltonian vector field
To see why this is useful we next consider...
Sunday, 24 February 2019
A D Kennedy

17. Symplectic Integrators
We are interested in finding the classical trajectory in phase space of a system described by the Hamiltonian
Define the corresponding Hamiltonian vector fields (with ) and so that
Sunday, 24 February 2019
A D Kennedy

18. Symplectic Integrators
Formally the solution of Hamilton’s equations with trajectory length is the exponential of the Hamiltonian Hamiltonian vector field,
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 (Taylor’s theorem!)
Sunday, 24 February 2019
A D Kennedy

19. Symplectic Integrators
Baker-Campbell-Hausdorff (BCH) formula
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 )
More precisely, where and
Sunday, 24 February 2019
A D Kennedy

20. 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
Sunday, 24 February 2019
A D Kennedy

21. 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
Sunday, 24 February 2019
A D Kennedy

22. Shadow Hamiltonian
But more significantly the PQP integrator follows the integral curves of exactly
And is constructed from commutators of the Hamiltonian vector fields and
Therefore it is the Hamiltonian vector field of the corresponding combination of Poisson brackets
This is called the shadow Hamiltonian
Sunday, 24 February 2019
A D Kennedy

23. Shadow Hamiltonian For the PQP integrator we have
Sunday, 24 February 2019
A D Kennedy

24. What use are Shadows?
Find a Hamiltonian H’ whose shadow is the Hamiltonian H the we want?
No luck! The shadow is not the sum of a kinetic term and a potential term in general
Sunday, 24 February 2019
A D Kennedy

25. What use are Shadows? Use the shadow to tune an integrator
A precise definition of a “large” fermionic force is a large contribution to the shadow Hamiltonian
An integrator becomes unstable when the BCH expansion for its shadow fails to converge
In which case there is no (real) conserved shadow Hamiltonian
Optimize the integrator by minimizing ?
Not quite, as we shall see later
Sunday, 24 February 2019
A D Kennedy

26. 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?
Sunday, 24 February 2019
A D Kennedy

27. All manifolds are locally flat
Review
Symplectic 2-form
Hamiltonian vector field
Equations of motion
Poisson bracket
Flat Manifold
General
Darboux theorem:
All manifolds are locally flat
Sunday, 24 February 2019
A D Kennedy

28. Maurer—Cartan Equations
The generators of a Lie algebra satisfy the commutation relations
These may be extended to a frame of “left invariant” vector fields over the Lie group
The dual left invariant forms with satisfy the Maurer—Cartan equations
Sunday, 24 February 2019
A D Kennedy

29. Fundamental 2-form We can invent any Classical Mechanics we want…
So we may therefore define the closed fundamental 2-form which globally defines an invariant Poisson bracket by
Sunday, 24 February 2019
A D Kennedy

30. Hamiltonian Vector Field
We may now follow the usual procedure to find the equations of motion
Introduce a Hamiltonian function (0-form) on the cotangent bundle (phase space) over the group manifold
Define a vector field such that
Sunday, 24 February 2019
A D Kennedy

31. Poisson Brackets For any Hamiltonian vector field
So for we have vector fields
Sunday, 24 February 2019
A D Kennedy

32. More Poisson Brackets We thus compute the lowest-order Poisson bracket
and the Hamiltonian vector corresponding to it
Sunday, 24 February 2019
A D Kennedy

33. Even More Poisson Brackets
Sunday, 24 February 2019
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34. 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 Lie-algebra-valued fields living on links into the loops in the action using a “loop walker” algorithm
Sunday, 24 February 2019
A D Kennedy

35. Fermion Poisson Brackets
Fermions are easy to include in the formalism
We only need a few extra linear equation solves
Sunday, 24 February 2019
A D Kennedy

36. 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
Sunday, 24 February 2019
A D Kennedy

37. Clark, Kennedy, and Silva Lattice 2008 (JLab)
What to Tune
As I said a while ago, minimizing is not a good choice
It is much better to minimize the variance of
This is a function of two sets of quantities
The ensemble-averaged Poisson brackets
The integrator parameters
Clark, Kennedy, and Silva Lattice 2008 (JLab)
Sunday, 24 February 2019
A D Kennedy

38. Why Minimize the Variance?
As the system wanders through phase space is constant, so
We hypothesize that the distribution of 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 2008 (JLab)
Sunday, 24 February 2019
A D Kennedy

39. Simplest Integrators
Sunday, 24 February 2019
A D Kennedy

40. Campostrini Integrator
Sunday, 24 February 2019
A D Kennedy

41. Force-Gradient Integrators
An interesting observation is that the Poisson bracket depends only of q
We may therefore evaluate the integrator 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
Sunday, 24 February 2019
A D Kennedy

42. Force-Gradient Integrators
Sunday, 24 February 2019
A D Kennedy

43. Problems? Difficulties?
Cancellations between Poisson Brackets require higher accuracy
Double precision
Accurate linear equation solutions
Approximates solves do not correspond to Hamiltonian vector fields
Perhaps one can still define a pseudofermion action using an approximate ?
Sunday, 24 February 2019
A D Kennedy

44. 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 soon…
Sunday, 24 February 2019
A D Kennedy