94,379 research outputs found
Error in Monte Carlo, quasi-error in Quasi-Monte Carlo
While the Quasi-Monte Carlo method of numerical integration achieves smaller
integration error than standard Monte Carlo, its use in particle physics
phenomenology has been hindered by the abscence of a reliable way to estimate
that error. The standard Monte Carlo error estimator relies on the assumption
that the points are generated independently of each other and, therefore, fails
to account for the error improvement advertised by the Quasi-Monte Carlo
method. We advocate the construction of an estimator of stochastic nature,
based on the ensemble of pointsets with a particular discrepancy value. We
investigate the consequences of this choice and give some first empirical
results on the suggested estimators.Comment: 41 pages, 19 figure
Discrepancy-based error estimates for Quasi-Monte Carlo. III: Error distributions and central limits
In Quasi-Monte Carlo integration, the integration error is believed to be
generally smaller than in classical Monte Carlo with the same number of
integration points. Using an appropriate definition of an ensemble of
quasi-randompoint sets, we derive various results on the probability
distribution of the integration error, which can be compared to the standard
Central Limit theorem for normal stochastic sampling. In many cases, a Gaussian
error distribution is obtained.Comment: 15 page
Applicability of Quasi-Monte Carlo for lattice systems
This project investigates the applicability of quasi-Monte Carlo methods to
Euclidean lattice systems in order to improve the asymptotic error scaling of
observables for such theories. The error of an observable calculated by
averaging over random observations generated from ordinary Monte Carlo
simulations scales like , where is the number of observations. By
means of quasi-Monte Carlo methods it is possible to improve this scaling for
certain problems to , or even further if the problems are regular
enough. We adapted and applied this approach to simple systems like the quantum
harmonic and anharmonic oscillator and verified an improved error scaling of
all investigated observables in both cases.Comment: on occasion of the 31st International Symposium on Lattice Field
Theory - LATTICE 2013, July 29 - August 3, 2013, Mainz, Germany, 7 Pages, 4
figure
A first look at quasi-Monte Carlo for lattice field theory problems
In this project we initiate an investigation of the applicability of Quasi-Monte Carlo methods to lattice field theories in order to improve the asymptotic error behavior of observables for such theories. In most cases the error of an observable calculated by averaging over random observations generated from an ordinary Monte Carlo simulation behaves like Nâ1/2, where N is the number of observations. By means of Quasi-Monte Carlo methods it is possible to improve this behavior for certain problems to up to Nâ1. We adapted and applied this approach to simple systems like the quantum harmonic and anharmonic oscillator and verified an improved error scaling.Peer Reviewe
On Bounding and Approximating Functions of Multiple Expectations using Quasi-Monte Carlo
Monte Carlo and Quasi-Monte Carlo methods present a convenient approach for
approximating the expected value of a random variable. Algorithms exist to
adaptively sample the random variable until a user defined absolute error
tolerance is satisfied with high probability. This work describes an extension
of such methods which supports adaptive sampling to satisfy general error
criteria for functions of a common array of expectations. Although several
functions involving multiple expectations are being evaluated, only one random
sequence is required, albeit sometimes of larger dimension than the underlying
randomness. These enhanced Monte Carlo and Quasi-Monte Carlo algorithms are
implemented in the QMCPy Python package with support for economic and parallel
function evaluation. We exemplify these capabilities on problems from machine
learning and global sensitivity analysis
Multilevel and quasi-Monte Carlo methods for uncertainty quantification in particle travel times through random heterogeneous porous media
In this study, we apply four Monte Carlo simulation methods, namely, Monte Carlo, quasi-Monte Carlo, multilevel Monte Carlo and multilevel quasi-Monte Carlo to the problem of uncertainty quantification in the estimation of the average travel time during the transport of particles through random heterogeneous porous media. We apply the four methodologies to a model problem where the only input parameter, the hydraulic conductivity, is modelled as a log-Gaussian random field by using direct KarhunenâLoĂ©ve decompositions. The random terms in such expansions represent the coefficients in the equations. Numerical calculations demonstrating the effectiveness of each of the methods are presented. A comparison of the computational cost incurred by each of the methods for three different tolerances is provided. The accuracy of the approaches is quantified via the mean square error
Combining Normalizing Flows and Quasi-Monte Carlo
Recent advances in machine learning have led to the development of new
methods for enhancing Monte Carlo methods such as Markov chain Monte Carlo
(MCMC) and importance sampling (IS). One such method is normalizing flows,
which use a neural network to approximate a distribution by evaluating it
pointwise. Normalizing flows have been shown to improve the performance of MCMC
and IS. On the other side, (randomized) quasi-Monte Carlo methods are used to
perform numerical integration. They replace the random sampling of Monte Carlo
by a sequence which cover the hypercube more uniformly, resulting in better
convergence rates for the error that plain Monte Carlo. In this work, we
combine these two methods by using quasi-Monte Carlo to sample the initial
distribution that is transported by the flow. We demonstrate through numerical
experiments that this combination can lead to an estimator with significantly
lower variance than if the flow was sampled with a classic Monte Carlo
Adaptive Multidimensional Integration Based on Rank-1 Lattices
Quasi-Monte Carlo methods are used for numerically integrating multivariate
functions. However, the error bounds for these methods typically rely on a
priori knowledge of some semi-norm of the integrand, not on the sampled
function values. In this article, we propose an error bound based on the
discrete Fourier coefficients of the integrand. If these Fourier coefficients
decay more quickly, the integrand has less fine scale structure, and the
accuracy is higher. We focus on rank-1 lattices because they are a commonly
used quasi-Monte Carlo design and because their algebraic structure facilitates
an error analysis based on a Fourier decomposition of the integrand. This leads
to a guaranteed adaptive cubature algorithm with computational cost ,
where is some fixed prime number and is the number of data points
Computational investigations of low-discrepancy point sets
The quasi-Monte Carlo method of integration offers an attractive solution to the problem of evaluating integrals in a large number of dimensions; however, the associated error bounds are difficult to obtain theoretically. Since these bounds are associated with the L2 discrepancy of the set of points used in the integration. Numerical calculations of the L2 discrepancy for several types of quasi-Monte Carlo formulae are presented
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