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

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 A flexible importance sampling method for integrating subgrid processes

2016 ◽  
Vol 9 (1) ◽  
pp. 413-429 ◽  
Author(s):  
E. K. Raut ◽  
V. E. Larson
Keyword(s):  
Monte Carlo ◽  
Importance Sampling ◽  
Sampling Error ◽  
Numerical Models ◽  
Atmospheric Modeling ◽  
Monte Carlo Integration ◽  
Spatial Integration ◽  
Importance Sampling Method ◽  
Subgrid Scales ◽  
Sample Points

Abstract. Numerical models of weather and climate need to compute grid-box-averaged rates of physical processes such as microphysics. These averages are computed by integrating subgrid variability over a grid box. For this reason, an important aspect of atmospheric modeling is spatial integration over subgrid scales. The needed integrals can be estimated by Monte Carlo integration. Monte Carlo integration is simple and general but requires many evaluations of the physical process rate. To reduce the number of function evaluations, this paper describes a new, flexible method of importance sampling. It divides the domain of integration into eight categories, such as the portion that contains both precipitation and cloud, or the portion that contains precipitation but no cloud. It then allows the modeler to prescribe the density of sample points within each of the eight categories. The new method is incorporated into the Subgrid Importance Latin Hypercube Sampler (SILHS). The resulting method is tested on drizzling cumulus and stratocumulus cases. In the cumulus case, the sampling error can be considerably reduced by drawing more sample points from the region of rain evaporation.

scholarly journals A flexible importance sampling method for integrating subgrid processes

2015 ◽  
Vol 8 (10) ◽  
pp. 9147-9191 ◽  
Author(s):  
E. K. Raut ◽  
V. E. Larson
Keyword(s):  
Monte Carlo ◽  
Importance Sampling ◽  
Sampling Error ◽  
Numerical Models ◽  
Atmospheric Modeling ◽  
Monte Carlo Integration ◽  
Importance Sampling Method ◽  
Weather And Climate ◽  
Sample Points ◽  
Is Integration

Abstract. Numerical models of weather and climate need to compute grid-box-averaged rates of physical processes such as microphysics. These averages are computed by integrating subgrid variability over a grid box. For this reason, an important aspect of atmospheric modeling is integration. The needed integrals can be estimated by Monte Carlo integration. Monte Carlo integration is simple and general but requires many evaluations of the physical process rate. To reduce the number of function evaluations, this paper describes a new, flexible method of importance sampling. It divides the domain of integration into eight categories, such as the portion that contains both precipitation and cloud, or the portion that contains precipitation but no cloud. It then allows the modeler to prescribe the density of sample points within each of the eight categories. The new method is incorporated into the Subgrid Importance Latin Hypercube Sampler (SILHS). The resulting method is tested on drizzling cumulus and stratocumulus cases. In the cumulus case, the sampling error can be considerably reduced by drawing more sample points from the region of rain evaporation.

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.

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