30,806 research outputs found
Simple Monte Carlo and the Metropolis Algorithm
We study the integration of functions with respect to an unknown density. We
compare the simple Monte Carlo method (which is almost optimal for a certain
large class of inputs) and compare it with the Metropolis algorithm (based on a
suitable ball walk).
Using MCMC we prove (for certain classes of inputs) that adaptive methods are
much better than nonadaptive ones. Actually, the curse of dimension (for
nonadaptive methods) can be broken by adaption.Comment: Journal of Complexity, to appea
Simple Monte Carlo and the metropolis algorithm
We study the integration of functions with respect to an unknown
density. Information is available as oracle calls to the integrand and to the
non-normalized density function. We are interested in analyzing the
integration error of optimal algorithms (or the complexity of the problem)
with emphasis on the variability of the weight function. For a corresponding
large class of problem instances we show that the complexity grows linearly
in the variability, and the simple Monte Carlo method provides an almost
optimal algorithm. Under additional geometric restrictions (mainly
log-concavity) for the density functions, we establish that a suitable
adaptive local Metropolis algorithm is almost optimal and outperforms any
non-adaptive algorithm
Universality of the Ising and the S=1 model on Archimedean lattices: A Monte Carlo determination
The Ising model S=1/2 and the S=1 model are studied by efficient Monte Carlo
schemes on the (3,4,6,4) and the (3,3,3,3,6) Archimedean lattices. The
algorithms used, a hybrid Metropolis-Wolff algorithm and a parallel tempering
protocol, are briefly described and compared with the simple Metropolis
algorithm. Accurate Monte Carlo data are produced at the exact critical
temperatures of the Ising model for these lattices. Their finite-size analysis
provide, with high accuracy, all critical exponents which, as expected, are the
same with the well known 2d Ising model exact values. A detailed finite-size
scaling analysis of our Monte Carlo data for the S=1 model on the same lattices
provides very clear evidence that this model obeys, also very well, the 2d
Ising model critical exponents. As a result, we find that recent Monte Carlo
simulations and attempts to define effective dimensionality for the S=1 model
on these lattices are misleading. Accurate estimates are obtained for the
critical amplitudes of the logarithmic expansions of the specific heat for both
models on the two Archimedean lattices.Comment: 9 pages, 11 figure
Comparative Monte Carlo Efficiency by Monte Carlo Analysis
We propose a modified power method for computing the subdominant eigenvalue
of a matrix or continuous operator. Here we focus on defining
simple Monte Carlo methods for its application. The methods presented use
random walkers of mixed signs to represent the subdominant eigenfuction.
Accordingly, the methods must cancel these signs properly in order to sample
this eigenfunction faithfully. We present a simple procedure to solve this sign
problem and then test our Monte Carlo methods by computing the of
various Markov chain transition matrices. We first computed for
several one and two dimensional Ising models, which have a discrete phase
space, and compared the relative efficiencies of the Metropolis and heat-bath
algorithms as a function of temperature and applied magnetic field. Next, we
computed for a model of an interacting gas trapped by a harmonic
potential, which has a mutidimensional continuous phase space, and studied the
efficiency of the Metropolis algorithm as a function of temperature and the
maximum allowable step size . Based on the criterion, we
found for the Ising models that small lattices appear to give an adequate
picture of comparative efficiency and that the heat-bath algorithm is more
efficient than the Metropolis algorithm only at low temperatures where both
algorithms are inefficient. For the harmonic trap problem, we found that the
traditional rule-of-thumb of adjusting so the Metropolis acceptance
rate is around 50% range is often sub-optimal. In general, as a function of
temperature or , for this model displayed trends defining
optimal efficiency that the acceptance ratio does not. The cases studied also
suggested that Monte Carlo simulations for a continuum model are likely more
efficient than those for a discretized version of the model.Comment: 23 pages, 8 figure
Theoretical Analysis of Acceptance Rates in Multigrid Monte Carlo
We analyze the kinematics of multigrid Monte Carlo algorithms by
investigating acceptance rates for nonlocal Metropolis updates. With the help
of a simple criterion we can decide whether or not a multigrid algorithm will
have a chance to overcome critial slowing down for a given model. Our method is
introduced in the context of spin models. A multigrid Monte Carlo procedure for
nonabelian lattice gauge theory is described, and its kinematics is analyzed in
detail.Comment: 7 pages, no figures, (talk at LATTICE 92 in Amsterdam
Grapham: Graphical Models with Adaptive Random Walk Metropolis Algorithms
Recently developed adaptive Markov chain Monte Carlo (MCMC) methods have been
applied successfully to many problems in Bayesian statistics. Grapham is a new
open source implementation covering several such methods, with emphasis on
graphical models for directed acyclic graphs. The implemented algorithms
include the seminal Adaptive Metropolis algorithm adjusting the proposal
covariance according to the history of the chain and a Metropolis algorithm
adjusting the proposal scale based on the observed acceptance probability.
Different variants of the algorithms allow one, for example, to use these two
algorithms together, employ delayed rejection and adjust several parameters of
the algorithms. The implemented Metropolis-within-Gibbs update allows arbitrary
sampling blocks. The software is written in C and uses a simple extension
language Lua in configuration.Comment: 9 pages, 3 figures; added references, revised language, other minor
change
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