579,387 research outputs found
Divergence of the multilevel Monte Carlo Euler method for nonlinear stochastic differential equations
The Euler-Maruyama scheme is known to diverge strongly and numerically weakly
when applied to nonlinear stochastic differential equations (SDEs) with
superlinearly growing and globally one-sided Lipschitz continuous drift
coefficients. Classical Monte Carlo simulations do, however, not suffer from
this divergence behavior of Euler's method because this divergence behavior
happens on rare events. Indeed, for such nonlinear SDEs the classical Monte
Carlo Euler method has been shown to converge by exploiting that the Euler
approximations diverge only on events whose probabilities decay to zero very
rapidly. Significantly more efficient than the classical Monte Carlo Euler
method is the recently introduced multilevel Monte Carlo Euler method. The main
observation of this article is that this multilevel Monte Carlo Euler method
does - in contrast to classical Monte Carlo methods - not converge in general
in the case of such nonlinear SDEs. More precisely, we establish divergence of
the multilevel Monte Carlo Euler method for a family of SDEs with superlinearly
growing and globally one-sided Lipschitz continuous drift coefficients. In
particular, the multilevel Monte Carlo Euler method diverges for these
nonlinear SDEs on an event that is not at all rare but has probability one. As
a consequence for applications, we recommend not to use the multilevel Monte
Carlo Euler method for SDEs with superlinearly growing nonlinearities. Instead
we propose to combine the multilevel Monte Carlo method with a slightly
modified Euler method. More precisely, we show that the multilevel Monte Carlo
method combined with a tamed Euler method converges for nonlinear SDEs with
globally one-sided Lipschitz continuous drift coefficients and preserves its
strikingly higher order convergence rate from the Lipschitz case.Comment: Published in at http://dx.doi.org/10.1214/12-AAP890 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Off-diagonal Wave Function Monte Carlo Studies of Hubbard Model I
We propose a Monte Carlo method, which is a hybrid method of the quantum
Monte Carlo method and variational Monte Carlo theory, to study the Hubbard
model. The theory is based on the off-diagonal and the Gutzwiller type
correlation factors which are taken into account by a Monte Carlo algorithm. In
the 4x4 system our method is able to reproduce the exact results obtained by
the diagonalization. An application is given to investigate the half-filled
band case of two-dimensional square lattice. The energy is favorably compared
with quantum Monte Carlo data.Comment: 9 pages, 11 figure
Metropolis Methods for Quantum Monte Carlo Simulations
Since its first description fifty years ago, the Metropolis Monte Carlo
method has been used in a variety of different ways for the simulation of
continuum quantum many-body systems. This paper will consider some of the
generalizations of the Metropolis algorithm employed in quantum Monte Carlo:
Variational Monte Carlo, dynamical methods for projector monte carlo ({\it
i.e.} diffusion Monte Carlo with rejection), multilevel sampling in path
integral Monte Carlo, the sampling of permutations, cluster methods for lattice
models, the penalty method for coupled electron-ionic systems and the Bayesian
analysis of imaginary time correlation functions.Comment: Proceedings of "Monte Carlo Methods in the Physical Sciences"
Celebrating the 50th Anniversary of the Metropolis Algorith
Transition Matrix Monte Carlo Method
We analyze a new Monte Carlo method which uses transition matrix in the space
of energy. This method gives an efficient reweighting technique. The associated
artificial dynamics is a constrained random walk in energy, producing the
result that correlation time is proportional to the specific heat.Comment: LaTeX, 8 pages, 1 figur
Self-Learning Monte Carlo Method
Monte Carlo simulation is an unbiased numerical tool for studying classical
and quantum many-body systems. One of its bottlenecks is the lack of general
and efficient update algorithm for large size systems close to phase transition
or with strong frustrations, for which local updates perform badly. In this
work, we propose a new general-purpose Monte Carlo method, dubbed self-learning
Monte Carlo (SLMC), in which an efficient update algorithm is first learned
from the training data generated in trial simulations and then used to speed up
the actual simulation. We demonstrate the efficiency of SLMC in a spin model at
the phase transition point, achieving a 10-20 times speedup.Comment: add more refs and correct some typo
Transition matrix Monte Carlo method for quantum systems
We propose an efficient method for Monte Carlo simulation of quantum lattice
models. Unlike most other quantum Monte Carlo methods, a single run of the
proposed method yields the free energy and the entropy with high precision for
the whole range of temperature. The method is based on several recent findings
in Monte Carlo techniques, such as the loop algorithm and the transition matrix
Monte Carlo method. In particular, we derive an exact relation between the DOS
and the expectation value of the transition probability for quantum systems,
which turns out to be useful in reducing the statistical errors in various
estimates.Comment: 6 pages, 4 figure
Numerical approximation of statistical solutions of scalar conservation laws
We propose efficient numerical algorithms for approximating statistical
solutions of scalar conservation laws. The proposed algorithms combine finite
volume spatio-temporal approximations with Monte Carlo and multi-level Monte
Carlo discretizations of the probability space. Both sets of methods are proved
to converge to the entropy statistical solution. We also prove that there is a
considerable gain in efficiency resulting from the multi-level Monte Carlo
method over the standard Monte Carlo method. Numerical experiments illustrating
the ability of both methods to accurately compute multi-point statistical
quantities of interest are also presented
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