A Retrospective and Prospective Survey of the Monte Carlo Method

Halton, John H.

doi:10.1137/1012001

Cited by 295 publications

(116 citation statements)

References 161 publications

(187 reference statements)

Supporting

Mentioning

114

Contrasting

Unclassified

Order By: Relevance

“…We then form the estimators t 1 , tz, ... ,tn Cn ~ k) as the linear combination i=1, ... ,n CS. 24) where the Cx1;l are known.…”

Section: Regression Methodsmentioning

confidence: 99%

Introduction to the Monte-Carlo method

2000

Choice Reviews Online

View full text Add to dashboard Cite

“…We then form the estimators t 1 , tz, ... ,tn Cn ~ k) as the linear combination i=1, ... ,n CS. 24) where the Cx1;l are known.…”

Section: Regression Methodsmentioning

confidence: 99%

Introduction to the Monte-Carlo method

2000

Choice Reviews Online

View full text Add to dashboard Cite

“…By a total degree truncation set τ = τ(e, d), we mean a truncation set obtained by choosing e of the most important variates and then collecting the monomials α in those variates with total degree |α| ≡ Σ ω a ω ≤ d. Such a truncation set has size (30) and is related to a standard orthogonal decomposition of a Gaussian Hilbert space [17,37]. By a rectangular truncation set τ = τ(μ), we mean a truncation set consisting of all α ≺ μ for some fixed monomial μ = ∏ ω ω m ω .…”

Section: Truncationmentioning

confidence: 99%

“…Here we instead use such ideas to evaluate the accuracy of PC expansions, rather than to improve MC. A PC solution provides a polynomial "easy function" [30] that can be used to accelerate MC by variance reduction. This acceleration provides a natural, easily estimated measure of the accuracy of the approximate PC coefficients.…”

Section: Monte Carlo Simulationmentioning

confidence: 99%

“…Certain pitfalls must be avoided when using MC methods: for example, the pseudorandom variates used for simulation can fail in important ways [53]. These methods are not easily conditionalized [61], and their slow convergence has led to interest in acceleration techniques [13,30]. Transformation of the underlying probability space using importance sampling was suggested in [49], using a spatially constant random conductivity field with one-dimensional flow, but to the best of our knowledge it has not been considered for nontrivial stochastic flow problems.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

An analysis of polynomial chaos approximations for modeling single-fluid-phase flow in porous medium systems

Rupert

Miller

2007

Journal of Computational Physics

View full text Add to dashboard Cite

We examine a variety of polynomial-chaos-motivated approximations to a stochastic form of a steady state groundwater flow model. We consider approaches for truncating the infinite dimensional problem and producing decoupled systems. We discuss conditions under which such decoupling is possible and show that to generalize the known decoupling by numerical cubature, it would be necessary to find new multivariate cubature rules. Finally, we use the acceleration of Monte Carlo to compare the quality of polynomial models obtained for all approaches and find that in general the methods considered are more efficient than Monte Carlo for the relatively small domains considered in this work. A curse of dimensionality in the series expansion of the log-normal stochastic random field used to represent hydraulic conductivity provides a significant impediment to efficient approximations for large domains for all methods considered in this work, other than the Monte Carlo method.

show abstract

“…This is consistent with the Monte Carlo simulation and with patient data, where 4 of 17 (23.5%) data points had pvr > 1.62. 4 The second density bound is a 1. joint Gaussian density with the parameters listed in Table 1 2. in which the maximum density value is limited to that of a jointly uniform density with the same means and standard deviations.…”

Section: Other Techniquesmentioning

confidence: 99%

Finding the likely behaviors of static continuous nonlinear systems

Yeh¹

1990

Ann Math Artif Intell

View full text Add to dashboard Cite

This paper describes two methods for predicting the likely behaviors of static continuous nonlinear systems with varying input values. The methods use a parameterized equation model and upper or lower bounds on the joint input density to bound the likelihood of a behavior, such as a state variable being inside a numeric range. Using a bound on the density instead of the density itself is desirable because the density's parameters and shape are not exactly known. The first method is limited to using lower density bounds. It finds rough bounds at first, and then refines them as more iterations of the method are allowed. The second method is a hit-or-miss version of sample-mean Monte Carlo. Unlike the first method, the second method can also handle upper density bounds, which are more useful than lower density bounds, but the generated probability bounds are only approximate.However, standard deviations on the bounds are given and become small as the sample size increases. In contrast to other researchers' methods, the two methods described here (1) find all the possible system behaviors, and tell how likely they 1 are, (2) do not just approximate the distribution of possible outcomes without some measure of the error magnitude, (3) do not use discretized variable values, which limit the events one can find probability bounds for, (4) can handle density bounds, and (5) can handle such criteria as two state variables both being inside a numeric range.

show abstract

A Retrospective and Prospective Survey of the Monte Carlo Method

Cited by 295 publications

References 161 publications

Introduction to the Monte-Carlo method

Introduction to the Monte-Carlo method

An analysis of polynomial chaos approximations for modeling single-fluid-phase flow in porous medium systems

Finding the likely behaviors of static continuous nonlinear systems

Contact Info

Product

Resources

About