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Monte Carlo Methods
Published in Dirk P. Kroese, Zdravko I. Botev, Thomas Taimre, Radislav Vaisman, Data Science and Machine Learning, 2019
Dirk P. Kroese, Zdravko I. Botev, Thomas Taimre, Radislav Vaisman
One of the most important variance reduction techniques is importance sampling. This technique is especially useful for the estimation of very small probabilities. The standard setting is the estimation of a quantity () μ=EfH(X)=∫H(x)f(x)dx,importance sampling
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Published in Harald Paganetti, Proton Therapy Physics, 2018
Variance reduction techniques aim at improving computational efficiency by giving more emphasis to certain physical quantities of interest. For example, the splitting of primary particles in regions of high interest is often used. By splitting at specific locations, one predominantly considers particles that have a high likelihood of contributing in regions of interest. If primary particles are split, their simulated effects are subsequently treated with a weighting factor <1 (which will also be inherited by subsequent secondary particles). A related technique is Russian roulette, which, instead of splitting, combines several particles into one particle. Another method is the importance sampling, i.e., a method to oversample certain regions of interest while reducing the weight of events to maintain the statistical balance. Some of these techniques have been implemented specifically for proton therapy applications [13–15].
Neutronics
Published in Kenneth D. Kok, Nuclear Engineering Handbook, 2016
This simplified description has assumed that the exact physical probabilities are utilized to determine the outcome of every decision; when this is done, the resulting simulation is termed an analog simulation. More sophisticated statistical treatments are included in modern computer codes that utilize nonphysical distributions with corrections (in a defined particle weight) to keep the results of the simulation unbiased; these can be shown to improve the efficiency of the simulation. These methods are called “variance reduction” methods, although this is somewhat of a misnomer because many of these methods increase efficiency by saving computer time, not by reducing variance. The exact theory and technique for doing this is beyond the scope of this handbook but is well described in Monte Carlo descriptions such as in Lewis and Miller (1993).
Neutron Importance Estimation via New Recursive Monte Carlo Method for Deep Penetration Neutron Transport
Published in Nuclear Science and Engineering, 2023
Delgersaikhan Tuya, Yasunobu Nagaya
The Monte Carlo method is used in a wide range of radiation transport applications. Although it is accurate and offers a powerful capability for handling complex physics processes and complicated geometry, the method is not computationally efficient in its basic form. To increase the efficiency of the Monte Carlo method in practice, various variance reduction techniques have been developed. In most Monte Carlo neutron transport codes, for instance, Russian roulette/splitting and implicit capture variance reduction techniques are employed to increase computational efficiency. For deep-penetration or shielding calculations, however, the basic variance reduction techniques are still insufficient. Typically, importance sampling variance reduction techniques are needed to make deep-penetration or shielding calculations efficient or even feasible.
Probabilistic modeling and prediction of out-of-plane unidirectional composite lamina properties
Published in Mechanics of Advanced Materials and Structures, 2021
Jiaxin Zhang, Michael Shields, Stephanie TerMaath
Importance sampling is a variance reduction technique applied to estimate a statistical expectation with respect to a target probability distribution using samples drawn from an alternative distribution Specifically, the expected value with respect to is formulated by where denotes expectation with respect to Defining importance weights the importance sampling estimator of is
Reliability approximation of k-out-of-n pairs: G balanced systems with spatially distributed units
Published in IISE Transactions, 2018
Dingguo Hua, Elsayed A. Elsayed
Importance sampling is a variance reduction method that is widely used to improve simulation efficiency. The basic idea is to draw samples from an importance distribution that overweighs the target region instead of the original distribution and then adjust the estimation with a likelihood ratio, namely, the ratio of the probability density function (PDF) of the original distribution over the PDF of the importance distribution. Due to oversampling in the region of concern, the importance sampling method converges in significantly fewer simulation runs (Geist and Smotherman, 1989). This method is especially used in the case of highly reliable systems and, hence, a small failure region. Research and surveys in this area are found in Kumamoto et al. (1977), Heidelberger (1993), and Juneja and Shahabuddin (2001).