Cost of Generalised HMC Algorithms for Free Field Theory

We study analytically the computational cost of the Generalised Hybrid Monte Carlo (GHMC) algorithm for free field theory. We calculate the autocorrelation functions of operators quadratic in the fields, and optimise the GHMC momentum mixing angle, the trajectory length, and the integration stepsize. We show that long trajectories are optimal for GHMC, and that standard HMC is much more efficient than algorithms based on the Second Order Langevin (L2MC) or Kramers Equation. We show that contrary to naive expectations HMC and L2MC have the same volume dependence, but their dynamical critical exponents are z=1 and z=3/2 respectively.Comment: LATTICE99(Algorithms and Machines) - 3 pages, 1 PostScript figur

Instabilities and Non-Reversibility of Molecular Dynamics Trajectories

The theoretical justification of the Hybrid Monte Carlo algorithm depends upon the molecular dynamics trajectories within it being exactly reversible. If computations were carried out with exact arithmetic then it would be easy to ensure such reversibility, but the use of approximate floating point arithmetic inevitably introduces violations of reversibility. In the absence of evidence to the contrary, we are usually prepared to accept that such rounding errors can be made small enough to be innocuous, but in certain circumstances they are exponentially amplified and lead to blatantly erroneous results. We show that there are two types of instability of the molecular dynamics trajectories which lead to this behavior, instabilities due to insufficiently accurate numerical integration of Hamilton's equations, and intrinsic chaos in the underlying continuous fictitious time equations of motion themselves. We analyze the former for free field theory, and show that it is essentially a finite volume effect. For the latter we propose a hypothesis as to how the Liapunov exponent describing the chaotic behavior of the fictitious time equations of motion for an asymptotically free quantum field theory behaves as the system is taken to its continuum limit, and explain why this means that instabilities in molecular dynamics trajectories are not a significant problem for Hybrid Monte Carlo computations. We present data for pure $SU(3)$ gauge theory and for QCD with dynamical fermions on small lattices to illustrate and confirm some of our results.Comment: 28 pages latex with 19 color postscript figures included by eps

Tuning the generalized Hybrid Monte Carlo algorithm

We discuss the analytic computation of autocorrelation functions for the generalized Hybrid Monte Carlo algorithm applied to free field theory and compare the results with numerical results for the $O(4)$ spin model in two dimensions. We explain how the dynamical critical exponent $z$ for some operators may be reduced from two to one by tuning the amount of randomness introduced by the updating procedure, and why critical slowing down is not a problem for other operators.Comment: 4 pages, to be published in the Proceedings of Lattice 95, uuencoded PostScript fil

Non-Reversibility of Molecular Dynamics Trajectories

We study the non-reversibility of molecular dynamics trajectories arising from the amplification of rounding errors. We analyse the causes of such behaviour and give arguments, indicating that this does not pose a significant problem for Hybrid Monte Carlo computations. We present data for pure SU(3) gauge theory and for QCD with dynamical fermions on small lattices to illustrate and to support some of our ideas.Comment: 3 pages LATEX, 4 color figures included using epsf. Talk presented at LATTICE96(algorithms

The RHMC Algorithm for 2 Flavours of Dynamical Staggered Fermions

We describe an implementation of the Rational Hybrid Monte Carlo (RHMC) algorithm for dynamical computations with two flavours of staggered quarks. We discuss several variants of the method, the performance and possible sources of error for each of them, and we compare the performance and results to the inexact R algorithm.Comment: Lattice2003(machine) 3 pages, 1 figure. Added referenc

Comparing the R algorithm and RHMC for staggered fermions

The R algorithm is widely used for simulating two flavours of dynamical staggered fermions. We give a simple proof that the algorithm converges to the desired probability distribution to within O(dt^2) errors, but show that the relevant expansion parameter is (dt/m)^2, m being the quark mass. The Rational Hybrid Monte Carlo (RHMC) algorithm provides an exact (i.e., has no step size errors) alternative for simulating the square root of the staggered Dirac operator. We propose using it to test the validity of the R algorithm for simulations carried out with dt m.Comment: 3 pages, proceedings from Lattice 2002 poster presentatio

Cost of the Generalised Hybrid Monte Carlo Algorithm for Free Field Theory

We study analytically the computational cost of the Generalised Hybrid Monte Carlo (GHMC) algorithm for free field theory. We calculate the Metropolis acceptance probability for leapfrog and higher-order discretisations of the Molecular Dynamics (MD) equations of motion. We show how to calculate autocorrelation functions of arbitrary polynomial operators, and use these to optimise the GHMC momentum mixing angle, the trajectory length, and the integration stepsize for the special cases of linear and quadratic operators. We show that long trajectories are optimal for GHMC, and that standard HMC is more efficient than algorithms based on Second Order Langevin Monte Carlo (L2MC), sometimes known as Kramers Equation. We show that contrary to naive expectations HMC and L2MC have the same volume dependence, but their dynamical critical exponents are z = 1 and z = 3/2 respectively.Comment: 54 pages, 3 figure

On the Dynamics of Light Quarks in QCD

We describe recent results concerning the behavior of lattice QCD with light dynamical Wilson and Staggered quarks. We show that it is possible to reach regions of parameter space with light pions $m_\pi\approx 0.2/a$ using Wilson fermions. If the Hybrid Molecular Dynamics (HMD) algorithm is used with the same parameters it gives incorrect results. We also present preliminary results using a higher-order integration scheme.Comment: 4 pages (all in postscript), proceedings of LAT'9

The LHMC Algorithm for Free Field Theory: Reexamining Overrelaxation

We analyze the autocorrelations for the LHMC algorithm in the context of free field theory. In this case this is just Adler's overrelaxation algorithm. We consider the algorithm with even/odd, lexicographic, and random updates, and show that its efficiency depends crucially on this ordering of sites when optimized for a given class of operators. In particular, we show that, contrary to previous expectations, it is possible to eliminate critical slowing down (z[int]=0) for a class of interesting observables, including the magnetic susceptibility: this can be done with lexicographic updates but is not possible with even/odd (z[int]=1) or random (z[int]=2) updates. We are considering the dynamical critical exponent z[int] for integrated autocorrelations rather than for the exponential autocorrelation time; this is reasonable because it is the integrated autocorrelation which determines the cost of a Monte Carlo computation.Comment: LaTeX, 33 pages, 3 postscript figure

Use of Discriminant and Fourth-Derivative Analyses With High-resolution Absorption Spectra for Phytoplankton Research: Limitations at Varied Signal-to-Noise Ratio and Spectral Resolution

Future management efforts aimed at inhibiting harmful algal blooms will require extensive temporal and spatial monitoring of phytoplankton community composition. A cost-effective approach to delineating phytoplankton community composition may be through analysis of absorption spectra, measured in situ with instruments deployed on moorings or by remote sensing. Classification techniques relying on absorption spectra include discriminant and fourth-derivative analysis. We investigated how well these techniques performed theoretically at varied signal-to-noise ratio and spectral resolution representative of a new absorption and attenuation instrument called HiStar. Our findings suggest that discriminant analysis of absorption spectra is a highly useful technique for categorizing green algae, cyanobacteria, noxious bloom-forming dinoflagellates, diatoms, and other chrysophytes. For the purposes of discriminating dinoflagellates from the other algae groups, discriminant analysis worked well with either low- or high-resolution spectral data. The discriminant analysis technique was able to delineate a noxious bloom-forming dinoflagellate species, Prorocentrum minimum, at signal-to-noise ratios as low as ~17. The current noise level in the HiStar, however, is ~28-fold too high to allow correct classification of this dinoflagellate at concentrations where shellfisheries are closed. Improvements to the discriminant analysis (e.g., inclusion of scatter properties) or to the HiStar must be accomplished before this technique becomes useful for harmful algal bloom management applications. Fourth-derivative analysis of absorption spectra, also a useful classification technique and a possible approach to assess physiological state of some algae, required at least 4 nm spectral resolution for assessment of chlorophylls a and b. The spectral resolution of HiStar (3.3 nm) meets this requirement