Monte Carlo Variance Reduction Techniques Research Articles

Weighted Multipass Methods for Global Illumination

In multipass rendering, care has to be taken to include all light transport only once in the final solution. Therefore the different methods in current multipass configurations handle a perfectly disjunct part of the light transport. In this paper a Monte Carlo variance reduction technique is presented that probabilistically weights overlapping transport between different methods. A good heuristic for the weights is derived so that strengths of the respective methods are retained. The technique is applied to a combination of radiosity and bidirectional path tracing and significant improvement is obtained over the non‐weighted combination. This method promises to be a very useful extension to other multipass algorithms as well.

Using Monte-Carlo variance reduction in statistical tolerance synthesis

A statistical tolerance synthesis must analyse many sets of tolerances, each of which has a unique probability distribution. The Monte-Carlo technique that is typically used to evaluate the probability distribution must analyse large numbers of individual cases. The result is a huge number of individual analyses, which is computationally expensive. This paper examines two Monte-Carlo variance reduction techniques, importance sampling and correlation, and proposes a method for using them in statistical tolerance synthesis. Correlation is used to reduce the error in the tolerance analyses. Importance sampling is used to estimate the sensitivity of an analysis to the tolerances so that a gradient based optimization algorithm can be used.

A Monte Carlo Variance Reduction Approach for Non-Boltzmann Tallies

Quantities that depend on the collective effects of groups of particles cannot be obtained from the standard Boltzmann transport equation. Monte Carlo estimates of these quantities are called non-Boltzmann tallies and have become increasingly important recently. Standard Monte Carlo variance reduction techniques were designed for tallies based on individual particles rather than groups of particles. Experience with non-Boltzmann tallies and analog Monte Carlo has demonstrated the severe limitations of analog Monte Carlo for many non-Boltzmann tallies. In fact, many calculations absolutely require variance reduction methods to achieve practical computation times. A description is given of how variance reduction techniques can be applied when non-Boltzmann estimates are desired.

Unbiased Combinations of Nonanalog Monte Carlo Techniques and Fair Games

Historically, Monte Carlo variance reduction techniques have been developed one at a time in response to calculational needs. This paper reports that the theoretical basis is provided for obtaining unbiased Monte Carlo estimates from all possible combinations of variance reduction techniques. Hitherto, the techniques have not been proven to be unbiased in arbitrary combinations. The authors are unaware of any Monte Carlo techniques (in any linear process) that are not treated by the theorem herein.

Alternative specifications of the error process in the stochastic simulation of econometric models

SUMMARY This paper analyses the stochastic simulation of econometric models using three different methods for specifying the probability distribution of the structural error terms. The impact of these different assumptions on the simulation bias and model variance is explored empirically. Monte Carlo variance reduction techniques are used to distinguish the effects of the different specifications. A macroeconometric model is a set of behavioural equations which represent endogenous variables as a function of predetermined variables and error terms. The model may be written in the form f(yt, xt) = ut, where yt is a vector of endogenous variables, xt is a vector of predetermined variables, and ut is a vector of stochastic error terms. The parameters of the structural functions f are estimated using econometric methods. When econometric models are used to predict the future, a point prediction is usually obtained with deterministic solution. The deterministic solution is calculated by solving f(yt, xt) = 0 for yt. This is equivalent to assuming that the stochastic terms ut are equal to zero in the future period. It is well known that, if the model is nonlinear, then E(yt I xt) is not equal to the deterministic solution. For example, see, Fisher and Salmon (1986), where large discrepancies between E(yt I xt) arid the deterministic solution are shown in some cases. Alternatively, predictions may be made using stochastic simulation. In stochastic simulation a distribution is assumed for the error vector ut. Then a pseudo-random vector st is drawn from the specified distribution. The model is then solved; the value of Yt which solves f(yt, xt) = st is found. This process is repeated n times and the resulting vectors are averaged. Since this is a Monte Carlo technique, the economist can increase accuracy by performing more replications (increasing n). The results will depend upon the distribution assumed for the error terms ut, and adopted for generating the pseudo-random error terms St. 1 Estimation errors of the structural form parameters could also be considered in simulation studies. However, it was shown in Bianchi and Calzolari (1980) that the impact of errors in estimated coefficients can be treated with analytical or analytical-numerical methods even for nonlinear models, without need of stochastic simulation. Therefore, in this study we shall not deal with errors in coefficients; coefficients will be treated as given.

Importance sampling via a simulacrum

A Monte Carlo variance reduction technique known as “importance sampling” has recently been applied to many problems in data communications. This technique holds the promise of offering vast improvements to traditional Monte Carlo methods. An overview of importance sampling applied to the calculation of tail probabilities is presented, as well as examples for which some popular approaches to importance sampling fail to work. New techniques for the calculation of the resulting variances are introduced, as well as a new approach to importance sampling which offers the promise of substantial variance reduction over previous techniques.

Unbiased combinations of nonanalog Monte Carlo techniques and fair games

Historically, Monte Carlo variance reduction techniques have developed one at a time in response to calculational needs. This paper provides the theoretical basis for obtaining unbiased Monte Carlo estimates from all possible combinations of variance reduction techniques. Hitherto, the techniques have not been proven to be unbiased in arbitrary combinations. The authors are unaware of any Monte Carlo techniques (in any linear process) that are not treated by the theorem herein.

Adaptive estimation procedures for multi-parameter Monte Carlo computations

This paper describes further developments of a Monte Carlo variance reduction technique which was originally devised by Chorin in 1971. The basic idea of expanding the estimator using orthonormal Hermite polynomials as basis functions has now been extended by the authors to include: (i) multi-parameters, (ii) symmetrized estimators of increased efficiency, and (iii) an adaptive series selection algorithm so that only those coefficients determined with sufficient precision are retained in the Hermite polynomial expansion of the estimator. The so-called Tri-Symmetric Chorin (TSC) estimator is described in detail and computational results obtained using the adaptive algorithm are presented for 6- and 12-parameter problems.