Asymptotics of fixed point distributions for inexact Monte Carlo algorithms

Abstract

We introduce a simple general method for finding the equilibrium distribution for a class of widely used inexact Markov Chain Monte Carlo algorithms. The explicit error due to the noncommutivity of the updating operators when numerically integrating Hamilton's equations can be derived using the Baker-Campbell-Hausdorff formula. This error is manifest in the conservation of a "shadow" Hamiltonian that lies close to the desired Hamiltonian. The fixed point distribution of inexact Hybrid algorithms may then be derived taking into account that the fixed point of the momentum heatbath and that of the molecular dynamics do not coincide exactly. We perform this derivation for various inexact algorithms used for lattice QCD calculations.

Force-gradient integrators

“…As described in [5,6,7] we may define a Hamiltonian system for a gauge field by introducing the symplectic fundamental 2-form ω ≡ −d(p i θ i ) with θ i being the frame of left-invariant MaurerCartan forms; this ensures that the Hamiltonian dynamics is gauge invariant. For every 0-form F on phase space this defines a Hamiltonian vector fieldF satisfying dF = iF ω, and the Hamiltonian evolution for the system corresponds to an integral curve of the Hamiltonian vector fieldĤ for the Hamiltonian function H. We can find a closed-form integral curve ofF, that is evaluate eF τ explicitly, if F depends only on the "positions" (fields) q or momenta p. This is particularly useful when the Hamiltonian is of the form H(q, p) = S(q) + T (p) as then we can integrate the Hamiltonian vector fieldsŜ andT exactly.…”

Investigation of new methods for numerical stochastic perturbation theory in φ4 theory

“…We conclude by pointing out that if one chooses τ = δt, i.e., the trajectory consists of a single step, then the HMD algorithm effectively integrates the Langevin equation (4.1) (q.v., [41] and below). In other words, in this case the algorithm just described can be interpreted as a particular integration scheme for the Langevin equation.…”

Speeding up HMC with better integrators

“…For every symplectic integrator there is a shadow HamiltonianH that is exactly conserved; this may be obtained by replacing the commutators [Ŝ,T ] in the BCH expansion with the Poisson bracket [1]. For example our PQP integrator above exactly conserves the shadow HamiltonianH ≡ T + S − 1 24 {S, {S, T }} + 2{T, {S, T }} δ τ 2 + · · ·.…”