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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Even More Poisson Brackets Sunday, 24 February 2019 A D Kennedy

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

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

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

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

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

Simplest Integrators Sunday, 24 February 2019 A D Kennedy

Campostrini Integrator Sunday, 24 February 2019 A D Kennedy

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

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

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

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