Chasing Shadows

1
Chasing Shadows

• A D Kennedy
• University of Edinburgh

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

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Functional Integrals

• Integral is over all field configurations
• (At least) one integral for each lattice site
• This become an infinite dimensional integral in
  the continuum limit

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Fermions

• 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

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Fermionic Operators

• Operators involving Fermion fields can be
  expressed as non-local bosonic operators

Add Fermion sources
Complete the square
Shift Fermion fields
Evaluate Grassmann integral

• Evaluating the inverses of the Fermion kernel is
  usually the expensive part of measuring operators
  on dynamical gauge configurations

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Markov Chains

• 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

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Hybrid Monte Carlo (HMC)

• Solution of Hamiltons equations would then
  generate a new phase space configuration which
  satisfies detailed balance
• Reversible (after a momentum flip)
• Measure preserving (Liouvilles theorem)
• Equiprobable (energy conservation)

• Momentum pseudofermion heatbath ? ergodicity?

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HMC and Symplectic Integrators

• Symmetric symplectic integrators provide discrete
  approximations to the solution of Hamiltons
  equations
• Symmetric ? reversible
• Symplectic ? area preserving

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Hamiltonian Dynamics
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Hamiltonian Dynamics
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Hamiltonian Dynamics
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Proofs I
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Proofs II
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Proofs III

• This means that 0-forms on a symplectic manifold
  form a Lie algebra with a Lie bracket that is not
  a commutator

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Symplectic Integrators

• The basic idea of a symplectic integrator is to
  write the time evolution operator as

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PQP Integrator
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Baker-Campbell-Hausdorff formula

• Such commutators are in the Free Lie algebra

• The Bn are Bernoulli numbers

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BCH formula

• We only include commutators that are not related
  by antisymmetry or the Jacobi relation
• These are chosen from a Hall basis

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Symmetric Symplectic Integrators

• In order to construct reversible integrators we
  use symmetric symplectic integrators

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Shadow Hamiltonian I
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Shadow Hamiltonian II
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Scalar Theory

• Note that HPQP 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

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How to Tune Integrators
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Classical Mechanicson Group Manifolds

• 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)

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Fundamental 2-form
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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

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Poisson Brackets
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More Poisson Brackets
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Even More Poisson Brackets
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Integrators
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Hessian Integrators

• The force for this integrator involves second
  derivatives of the action
• Using this type of step we can construct
  efficient Hessian (Force-Gradient) integrators

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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

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Campostrini Integrator
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Hessian Integrators
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Multiple Timescale Integrators

• Use different integration step sizes for
  different contributions to the action
  (SextonWeingarten)
• 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

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Instabilities in Free Field Theory
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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

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Symptoms
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Higher Order Integrator Instabilities

• The values of at which instabilities
  occur depends on the choice of integrator

• Mike Clark
• Omelyan, Mryglod, Folk Computer Physics
  Communcations 151 (2003) 272-314
• Joo et al Phys Rev D62 (2000) 114501
  (hep-lat/0005023)

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Pseudofermions

• 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

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Multiple Pseudofermions
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Multiple Pseudofermion Techniques
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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?

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RHMC with Multiple Timescales

• Semiempirical observation The largest force from
  a single pseudofermion does not come from the
  smallest shift

• Use a coarser timescale for expensive smaller
  shifts
• Invert small shifts less accurately
• Cannot use chronological inverter with multishift
  solver anyhow

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Outlook

• Large-scale computations with light dynamical
  fermions require
• Avoiding integrator instabilities as far as
  possible
• Use multiple pseudofermion fields
• Reducing volume contributions to HShadow
• 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