scholarly journals Primary-space Adaptive Control Variates Using Piecewise-polynomial Approximations

2021 ◽  
Vol 40 (3) ◽  
pp. 1-15
Author(s):  
Miguel Crespo ◽  
Adrian Jarabo ◽  
Adolfo Muñoz
Keyword(s):  
Monte Carlo ◽  
Importance Sampling ◽  
High Frequency ◽  
Low Frequency ◽  
Monte Carlo Integration ◽  
Piecewise Polynomial ◽  
Integration Algorithm ◽  
Participating Media ◽  
Control Variate ◽  
Primary Space

We present an unbiased numerical integration algorithm that handles both low-frequency regions and high-frequency details of multidimensional integrals. It combines quadrature and Monte Carlo integration by using a quadrature-based approximation as a control variate of the signal. We adaptively build the control variate constructed as a piecewise polynomial, which can be analytically integrated, and accurately reconstructs the low-frequency regions of the integrand. We then recover the high-frequency details missed by the control variate by using Monte Carlo integration of the residual. Our work leverages importance sampling techniques by working in primary space, allowing the combination of multiple mappings; this enables multiple importance sampling in quadrature-based integration. Our algorithm is generic and can be applied to any complex multidimensional integral. We demonstrate its effectiveness with four applications with low dimensionality: transmittance estimation in heterogeneous participating media, low-order scattering in homogeneous media, direct illumination computation, and rendering of distribution effects. Finally, we show how our technique is extensible to integrands of higher dimensionality by computing the control variate on Monte Carlo estimates of the high-dimensional signal, and accounting for such additional dimensionality on the residual as well. In all cases, we show accurate results and faster convergence compared to previous approaches.

Effects of measurement error on Monte Carlo integration estimators of tree volume: sample diameters measured at random upper-stem heights

2015 ◽  
Vol 45 (4) ◽  
pp. 471-479 ◽  
Author(s):  
Thomas B. Lynch
Keyword(s):  
Monte Carlo ◽  
Measurement Error ◽  
Importance Sampling ◽  
Sampling Error ◽  
Monte Carlo Integration ◽  
Stem Diameter ◽  
Tree Level ◽  
Stem Volume ◽  
Control Variate ◽  
And Control

A study of the effects of measurement error was conducted on importance sampling and control variate sampling estimators of tree stem volume in which sample diameters are measured at randomly selected upper-stem heights. It was found that these estimators were unbiased in the presence of additive mean zero and multiplicative mean one measurement error applied to random samples of upper-stem diameter squared. However, biases due to measurement error are present if additive or multiplicative error is applied to upper-stem diameter rather than to upper-stem diameter squared. This is significant, as it appears that most of the previous studies on the magnitude of upper-stem diameter measurement error implicitly assume that the mean error is centered around the diameter rather than about the square of the diameter. Application of typical upper-stem measurement error obtained from previous studies to bias formulae derived here indicates that the bias could be a concern for small trees and with additive measurement error within ranges found in previous studies. Formulae for the variances of importance sampling and control variate sampling are derived, which include the contribution of both measurement error and sampling error. Results from previous studies of Monte Carlo integration estimator sampling error are combined with results from studies of upper-stem measurement error to obtain estimates of the typical magnitude of the contribution of measurement error to total estimator variability. Increases in upper-stem sample size may be warranted due to the impact of measurement error if precise estimates of stem volume at the individual-tree level are desired.

scholarly journals Control variate selection for Monte Carlo integration

2021 ◽  
Vol 31 (4) ◽  
Author(s):  
Rémi Leluc ◽  
François Portier ◽  
Johan Segers
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Integration ◽  
Control Variate ◽  
Selection For

Effects of measurement error on Monte Carlo integration estimators of tree volume: critical height sampling and vertical Monte Carlo methods

2015 ◽  
Vol 45 (4) ◽  
pp. 463-470 ◽  
Author(s):  
Thomas B. Lynch
Keyword(s):  
Monte Carlo ◽  
Measurement Error ◽  
Sampling Error ◽  
Monte Carlo Integration ◽  
Critical Height ◽  
Volume Estimate ◽  
Cross Sectional ◽  
Individual Tree ◽  
Control Variate ◽  
Multiplicative Measurement Error

The effects of measurement error on Monte Carlo (MC) integration estimators of individual-tree volume that sample upper-stem heights at randomly selected cross-sectional areas (termed vertical methods) were studied. These methods included critical height sampling (on an individual-tree basis), vertical importance sampling (VIS), and vertical control variate sampling (VCS). These estimators were unbiased in the presence of two error models: additive measurement error with mean zero and multiplicative measurement error with mean one. Exact mathematical expressions were derived for the variances of VIS and VCS that include additive components for sampling error and measurement error, which together comprise total variance. Previous studies of sampling error for MC integration estimators of tree volume were combined with estimates of upper-stem measurement error obtained from the mensurational literature to compute typical estimates of total standard errors for VIS and VCS. Through examples, it is shown that measurement error can substantially increase the total root mean square error of the volume estimate, especially for small trees.

Reliability Prediction of RV Reducer Based on Fault Tree and Monte-Carlo Simulation

2013 ◽  
Vol 274 ◽  
pp. 663-666 ◽  
Author(s):  
Y.G. Sun ◽  
G.B. Yu ◽  
J. Zhang ◽  
F. Wang ◽  
Y.Q. Sun ◽  
...  
Keyword(s):  
Monte Carlo ◽  
Frequency Analysis ◽  
Time Resolution ◽  
High Frequency ◽  
Fault Tree ◽  
Low Frequency ◽  
Frequency Resolution ◽  
Potential System ◽  
Rv Reducer ◽  
System Life

Wavelet de-nosing method for complex signal is put forward in this paper. The time resolution and frequency resolution are changed as wavelet transform for signal analysis. It uses high-frequency resolution and low time resolution in low frequency analysis, and uses low-frequency resolution and high time resolution in high frequency analysis. So it fit the uncertainty principle and realize the signal time domain and in frequency domain at the same time. In this paper, the configuration of RV reducer is briefly introduced, and its fault tree is constructed taking the fault of “Output shaft can not transfer torque” as top event through illuminating potential system unit failures and analyze the effect on whole system. And then system reliability qualitative and quantitative analysis are conducted. The fault tree qualitative analysis is operated based on the minimal cut set. Followed establishing simulation model of RV reducer, system life time is obtained using Monte-Carlo random sampling method. Furthermore, system life distribution is deduced, and point and confidence interval of distribution parameters and reliability characters are given by Maximum Likelihood Estimate. Finally, simulation experiment and results analysis are given to show the effectiveness of this method.

scholarly journals Corrigenda

1965 ◽  
Vol 61 (3) ◽  
pp. 613-613
Keyword(s):  
Monte Carlo ◽  
Importance Sampling ◽  
Monte Carlo Integration ◽  
Cambridge Philos

The author wishes to make the following corrections to his paper, entitled ‘On the relative merits of correlated and importance sampling for Monte Carlo integration’, which appeared in Proc. Cambridge Philos. Soc. 61 (1965), 497–498

Distributed Monte Carlo Integration Using Corba and Java

1998 ◽  
Vol 09 (07) ◽  
pp. 903-915 ◽  
Author(s):  
Fritz Solms ◽  
Willi-Hans Steeb
Keyword(s):  
Monte Carlo ◽  
Distributed Computing ◽  
Scientific Computing ◽  
Object Oriented ◽  
Monte Carlo Integration ◽  
The Internet ◽  
Integration Algorithm

The object-oriented middleware standard, CORBA, is a very useful platform for distributed computing, and in particular for sharing a workload among a collection of possibly polymorphic computers. CORBA has, however, received relatively little attention from the scientific computing community. In this article we demonstrate how CORBA and Java can be used to implement a distributed multi-dimensional Monte Carlo integration algorithm which runs on the internet.

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