Simulation and the Monte Carlo Method
Abstract
Simulation and the Monte Carlo Method, Second Edition reflects the latest developments in the field and presents a fully updated and comprehensive account of the major topics that have emerged in Monte Carlo simulation since the publication of the classic First Edition over twenty-five years ago. While maintaining its accessible and intuitive approach, this revised edition features a wealth of up-to-date information that facilitates a deeper understanding of problem solving across a wide array of subject areas, such as engineering, statistics, computer science, mathematics, and the physical and life sciences." "Requiring only a basic, introductory knowledge of probability and statistics, Simulation and the Monte Carlo Method, Second Edition is an excellent text for upper-undergraduate and beginning graduate courses in simulation and Monte Carlo techniques. The book also serves as a valuable reference for professionals who would like to achieve a more formal understanding of the Monte Carlo method.
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Convergence to low variability in terms of mean and standard deviations is reached at M=10^3.
“Convergence to low variability in terms of mean and standard deviations is reached at M=103, where both metrics stabilized with minimal variation beyond that point and correspond to the plateau of the coefficient of variation [40].”
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“Monte Carlo systems [87] are general stochastic simulation systems, that is, techniques performing repeated sampling experiments on the model of the system under consideration by making use of a stochastic component in the state sampling and/or transition rules.”
BackgroundLatin hypercube sampling and the propagation of uncertainty in analyses of complex systems
Helton, J. C., & Davis, F. J. (2003) · Reliability Engineering & System Safety · doi:10.1016/s0951-8320(03)00058-9 · Cited by 2,719 · see citing works
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Estimation of small failure probabilities in high dimensions by subset simulation
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“9 ( 7 ) % 7 % 80>9 # ' ) % ( ; ' $ ) % ( ; ' + % ( ; ' ( ; ' ) % , # ) % % ' & ' $ ) % ) % ( ( ' ( 4 ' * 0 (* * 4 ' 0 ) # ' % ) ' % # $ ) 8/ / < + 4 % * 0 ) * !”
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“The AntNet routing system can be seen as a collection of mobile agents collecting data about the network status by concurrently performing on-line Monte Carlo simulations (Rubistein, 1981; Streltsov & Vakili, 1996).”
Methodsdsm — a decision support tool for the assessment of regional climate change impacts
Wilby, R. L., Dawson, C. W., & Barrow, E. (2002) · Environmental Modelling & Software · doi:10.1016/s1364-8152(01)00060-3 · Cited by 1,440 · see citing works
“Following the method of Rubinstein (1981), a pseudo random number generator reproduces values from a normal distribution with standard deviation equal to the calibration standard error.”
MethodRandom waypoint considered harmful
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“It is a common practice in the scientific simulation community to warm-up a simulation (also known as initial data deletion [11]) aimed at eliminating the effect of the transient part by discarding the data from the initial period of a certain length.”
BackgroundA New Study on Reliability-Based Design Optimization
Tu, J., Choi, K. K., & Park, Y. H. (1999) · Journal of Mechanical Design · doi:10.1115/1.2829499 · Cited by 991 · see citing works
“The Monte Carlo simulation (MCS) (Rubinstein, 1981) provides a convenient approximation for both reliability analysis and inverse reliability analysis because it directly approximates the βG~g relationship.”
BackgroundMethodInfluentialGlobal Optimization Methods in Geophysical Inversion
Sen, M. K., & Stoffa, P. L. (2013) · Cambridge University Press eBooks · doi:10.1017/cbo9780511997570 · Cited by 994 · see citing works
Measuring and modeling anisotropic reflection
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Bayesian Updating of Structural Models and Reliability using Markov Chain Monte Carlo Simulation
Beck, J. L., & Au, S. (2002) · Journal of Engineering Mechanics · doi:10.1061/(asce)0733-9399(2002)128:4(380 · Cited by 859 · see citing works
Uncertainties in risk analysis: Six levels of treatment
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The design of simulation studies in medical statistics
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“SE(̂)/ √ B, but this can be computational expensive and therefore variance reduction techniques could be employed [24].”
MethodA survey on metaheuristics for stochastic combinatorial optimization
Bianchi, L., Dorigo, M., Gambardella, L. M., & Gutjahr, W. J. (2008) · Natural Computing · doi:10.1007/s11047-008-9098-4 · Cited by 738 · see citing works
“for speeding up the sample average computations, it could be useful the application of variance-reduction techniques, such as, for example, those belonging to the field of Rare Event Simulation (Rubinstein 1981).”
MethodChance constrained programming with fuzzy parameters
Liu, B., & Iwamura, K. (1998) · Fuzzy Sets and Systems · doi:10.1016/s0165-0114(96)00236-9 · Cited by 666 · see citing works
Robust Monte Carlo methods for light transport simulation
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The Collapsed Gibbs Sampler in Bayesian Computations with Applications to a Gene Regulation Problem
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Reinforcement learning with replacing eligibility traces
Singh, S. P., & Sutton, R. S. (1996) · Machine Learning · doi:10.1007/bf00114726 · Cited by 581 · see citing works
“This suggests that one might estimate a state's value simply by averaging all the returns that follow it. This is what is classically done in Monte Carlo (MC) prediction methods ( Rubinstein, 1981; Curtiss, 1954; Wasow, 1952; Barto & Duff, 1994).”
BackgroundMethodOptimization of stochastic systems
Glynn, P. W. (1986) · Winter Simulation Conference · doi:10.1145/318242.318260 · Cited by 578 · see citing works
“(See Rubinstein (1981).)”
Mathematics for Machine Learning
Deisenroth, M. P., Faisal, A. A., & Ong, C. S. (2020) · Cambridge University Press eBooks · doi:10.1017/9781108679930 · Cited by 583 · see citing works
A new adaptive importance sampling scheme for reliability calculations
Au, S., & Beck, J. L. (1999) · Structural Safety · doi:10.1016/s0167-4730(99)00014-4 · Cited by 565 · see citing works
“The general situation of almost disconnected important regions, including the case of disconnected multiple failure surfaces, could be handled by augmenting the current methodology with the technique of strati®ed sampling [2,7,11], in which case a Markov chain would be initiated within each failure region and the impo…”
MethodA univariate dimension-reduction method for multi-dimensional integration in stochastic mechanics
Rahman, S., & Xu, H. J. (2004) · Probabilistic Engineering Mechanics · doi:10.1016/j.probengmech.2004.04.003 · Cited by 568 · see citing works
“Direct Monte Carlo simulation [21] is the most widely used simulation method and involves the generation of independent samples of all input random variables, repeated deterministic trials (analyses) to obtain corresponding simulated samples of response variables, and standard statistical analysis to estimate probabil…”
MethodEfficient Monte Carlo Procedures for Generating Points Uniformly Distributed over Bounded Regions
Smith, R. L. (1984) · Operations Research · doi:10.1287/opre.32.6.1296 · Cited by 548 · see citing works
Information Theoretic Learning
Deniz Erdogmus, B., & Príncipe, J. C. (2009) · IGI Global eBooks · doi:10.4018/978-1-59904-849-9.ch133 · Cited by 536 · see citing works
Monte Carlo and Quasi-Monte Carlo Sampling
Lemieux, C. (2009) · Springer series in statistics · doi:10.1007/978-0-387-78165-5 · Cited by 502 · see citing works
“This topic has been widely studied and is surveyed, for example, in [42, 152, 221, 226, 295, 355], which also give several other references.”
BackgroundMethodInfluentialTranslating Collocations for Bilingual Lexicons: A Statistical Approach
McKeown, K. R., Smadja, F., & Hatzivassiloglou, V. (1996) · doi:10.7916/d8c82m3r · Cited by 489 · see citing works
A nonparametric model of random uncertainties for reduced matrix models in structural dynamics
Soize, C. (2000) · Probabilistic Engineering Mechanics · doi:10.1016/s0266-8920(99)00028-4 · Cited by 475 · see citing works
Hajivassiliou, V. A., McFadden, D. L., & Ruud, P. A. (1996) · Journal of Econometrics · doi:10.1016/0304-4076(94)01716-6 · Cited by 462 · see citing works
Incorporating modeling uncertainties in the assessment of seismic collapse risk of buildings
Liel, A. B., Haselton, C. B., Deierlein, G. G., et al. (2008) · Structural Safety · doi:10.1016/j.strusafe.2008.06.002 · Cited by 471 · see citing works
“An alternative approach uses Monte Carlo methods to determine the effect of modeling uncertainties on the structural response predictions [17,18].”
MethodStochastic global optimization methods part I: Clustering methods
Rinnooy Kan, A. H. G., & Timmer, G. T. (1987) · Mathematical Programming · doi:10.1007/bf02592070 · Cited by 463 · see citing works
“Proposition 1 (Rubinstein, 1981; cf. Devroye, 1978).”
Adaptive sampling — an iterative fast Monte Carlo procedure
Bucher, C. G. (1988) · Structural Safety · doi:10.1016/0167-4730(88)90020-3 · Cited by 463 · see citing works
A generalized dimension‐reduction method for multidimensional integration in stochastic mechanics
Xu, H. J., & Rahman, S. (2004) · International Journal for Numerical Methods in Engineering · doi:10.1002/nme.1135 · Cited by 444 · see citing works
“Simulation and sampling methods, such as direct Monte Carlo simulation [21], quasi-Monte Carlo simulation [22–24], importance sampling [25, 26], directional simulation [27, 28], and others [29–32], are well known in the statistics and reliability literature.”
Dynamics of offshore floating wind turbines—model development and verification
Jonkman, J. M. (2009) · Wind Energy · doi:10.1002/we.347 · Cited by 439 · see citing works
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A critical appraisal of methods to determine failure probabilities
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A Quick Algorithm for Pricing European Average Options
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Eligibility Traces for Off-Policy Policy Evaluation
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Constructed wetlands for pollution control: Processes, performance, design and operation
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“For this, Monte Carlo (MC) methods of stochastic discrete event simulation have been generally accepted as a gold standard [88,106,122,166,228].”
MethodIncremental dynamic analysis for estimating seismic performance sensitivity and uncertainty
Vamvatsikos, D., & Fragiadakis, M. (2009) · Earthquake Engineering & Structural Dynamics · doi:10.1002/eqe.935 · Cited by 399 · see citing works
“However, in our opinion, two of the most important contributions in this field have come from parallel research efforts that proposed the use of Monte Carlo simulation [8] within the framework of IDA [9] to incorporate parameter uncertainty.”
BackgroundSubset Simulation and its Application to Seismic Risk Based on Dynamic Analysis
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Important sampling in high dimensions
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A ray tracing solution for diffuse interreflection
Ward, G. J., Rubinstein, F., & Clear, R. D. (2007) · ACM SIGGRAPH Courses · doi:10.1145/1281500.1281626 · Cited by 374 · see citing works
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Chapter 63 Econometric Tools for Analyzing Market Outcomes
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Monte Carlo Experiments: Design and Implementation
Paxton, P. M., Curran, P. J., Bollen, K. A., Kirby, J., & Chen, F. (2001) · Structural Equation Modeling A Multidisciplinary Journal · doi:10.1207/s15328007sem0802_7 · Cited by 347 · see citing works
“Although there are a few books on the technique (e.g., Mooney, 1997; Rubinstein, 1981; Smith, 1973), none directly relates the method to structural equation models.”
MethodProbability Integration by Directional Simulation
Bjerager, P. (1988) · Journal of Engineering Mechanics · doi:10.1061/(asce)0733-9399(1988)114:8(1285 · Cited by 338 · see citing works
Uncertainty Analysis of Transport-Transformation Models
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Reliability-based design sensitivity by efficient simulation
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“, it may be fair to say that Monte Carlo-based simulation procedures [18,19] are the only ones that are generally applicable.”
BackgroundMethod for Probabilistic Evaluation of Seismic Structural Damage
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A mixture likelihood approach for generalized linear models
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Reliability‐Based Service‐Life Assessment of Aging Concrete Structures
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The Shapley Value in Machine Learning
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Uncertainty in the biomass of Amazonian forests: An example from Rondônia, Brazil
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Probabilistic assessment of power systems
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“The simulation approach selects states using the concept of random numbers [32].”
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Applications of simulation within the healthcare context
Katsaliaki, K., & Mustafee, N. (2010) · Journal of the Operational Research Society · doi:10.1057/jors.2010.20 · Cited by 276 · see citing works
“Those who wish to have an introduction to the aforementioned techniques can refer to Rubinstein (1981) for MCS, Robinson (1994) for DES, and Sterman (2001) for SD. ABS is the most recent of the four simulation methods used since the mid-1990s.”
MethodSampling power-law distributions
Pickering, G., Bull, J. M., & Sanderson, D. J. (1995) · Tectonophysics · doi:10.1016/0040-1951(95)00030-q · Cited by 286 · see citing works
“The samples are produced using the Inverse Transform Method (Rubinstein, 1981).”
MethodFunctional clustering: identifying strongly interactive brain regions in neuroimaging data
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“This is a special case of the general procedure for generating multivariate Gaussian deviates of any given correlation matrix (Rubinstein, 1981).”
BackgroundGoidts, E., van Wesemael, B., & Crucifix, M. C. (2009) · European Journal of Soil Science · doi:10.1111/j.1365-2389.2009.01157.x · Cited by 282 · see citing works
First excursion probabilities for linear systems by very efficient importance sampling
Au, S., & Beck, J. L. (2001) · Probabilistic Engineering Mechanics · doi:10.1016/s0266-8920(01)00002-9 · Cited by 282 · see citing works
Sempau, J., Badal, A., & Brualla, L. (2011) · Medical Physics · doi:10.1118/1.3643029 · Cited by 279 · see citing works
The Shapley Value in Machine Learning
Rózemberczki, B., Watson, L., Bayer, P., Yang, H.-T., Kiss, O., Nilsson, S., & Sarkar, R. (2022) · International Joint Conference on Artificial Intelligence · doi:10.24963/ijcai.2022/778 · Cited by 342 · see citing works
“For example, Mitchell et al. [ 2021 ] uses the trapezoid rule to sample q at fixed intervals while Okhrati et al. [ 2021 ] proposes incorporating antithetic sampling as a variance reduction technique.”
MethodMore on a Stochastic Asset Model for Actuarial Use
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An introduction to numerical methods for stochastic differential equations
Platen, E. (1999) · Acta Numerica · doi:10.1017/s0962492900002920 · Cited by 273 · see citing works
“Books that include chapters on random number generation include Ermakov (1975), Yakowitz (1977), Rubinstein (1981), Ripley (19836), Morgan (1984), Ross (1991), Mikhailov (1992), Fishman (1992) and Gentle (1998).”
BackgroundPapadrakakis, M., Papadopoulos, V., & Lagaros, N. D. (1996) · Computer Methods in Applied Mechanics and Engineering · doi:10.1016/0045-7825(96)01011-0 · Cited by 273 · see citing works
Oil Spill Modeling: A Critical Review on Current Trends, Perspectives, and Challenges
Κεραμέα, Π., Spanoudaki, K., Zodiatis, G., et al. (2021) · Journal of Marine Science and Engineering · doi:10.3390/jmse9020181 · Cited by 273 · see citing works
“Since the simulated behavior is stochastic, the number of particles is limited [248], however it is the only stochastic model in the full Delft3D modeling suite.”
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Quick Computation of Spatial Autoregressive Estimators
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Simulating Probability Distributions in the Coalescent
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Impacts of irrigated agriculture on food–energy–water–CO2 nexus across metacoupled systems
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Disentangling Label Distribution for Long-tailed Visual Recognition
Hong, Y., Han, S., Choi, K., Seo, S., Kim, B., & Chang, B. (2021) · IEEE Computer Society Conference on Computer Vision and Pattern Recognition/Proceedings - IEEE Computer Society Conference on Computer Vision and Pattern Recognition/Proceedings · doi:10.1109/cvpr46437.2021.00656 · Cited by 265 · see citing works
A comparison of production-line control mechanisms
Bonvik, A. M., Couch, C. E., & Gershwin, S. B. (1997) · International Journal of Production Research · doi:10.1080/002075497195713 · Cited by 252 · see citing works
Asymptotic Expansions for Reliability and Moments of Uncertain Systems
Papadimitriou, C., Beck, J. L., & Katafygiotis, L. S. (1997) · Journal of Engineering Mechanics · doi:10.1061/(asce)0733-9399(1997)123:12(1219 · Cited by 249 · see citing works
Optimization — the basis of code-making and reliability verification
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The Old and the New: Can Physics-Informed Deep-Learning Replace Traditional Linear Solvers?
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A Brief Introduction to Monte Carlo Simulation
Bonate, P. L. (2001) · Clinical Pharmacokinetics · doi:10.2165/00003088-200140010-00002 · Cited by 243 · see citing works
“The term ‘Monte Carlo’ was first coined by Ulam and von Neumann [2] during their work on development of the atomic bomb during World War II and was a reference to the gaming city in Monaco.”
BackgroundGeneralized information potential criterion for adaptive system training
Deniz Erdogmus, B., & Príncipe, J. C. (2002) · IEEE Transactions on Neural Networks · doi:10.1109/tnn.2002.1031936 · Cited by 237 · see citing works
“2 shows that the TDNN trained using the entropy criterion achieved a better pdf fit to the distribution of the desired signal data compared to the TDNN trained using MSE.”
MethodThe geometry of random vibrations and solutions by FORM and SORM
Kiureghian, A. D. (2000) · Probabilistic Engineering Mechanics · doi:10.1016/s0266-8920(99)00011-9 · Cited by 239 · see citing works
Transfer learning based multi-fidelity physics informed deep neural network
Chakraborty, S. (2020) · Journal of Computational Physics · doi:10.1016/j.jcp.2020.109942 · Cited by 241 · see citing works
“results using Monte Carlo simulation (MCS) [62] are generated.”
MethodSimulated annealing for constrained global optimization
Romeijn, H. E., & Smith, R. L. (1994) · Journal of Global Optimization · doi:10.1007/bf01100688 · Cited by 235 · see citing works
“where S is a compact body in R a whose boundary has Lebesgue measure zero and where f is continuous on S. Following Pincus (1968), Rubinstein (1981) suggests for approximating the global optimum that we generate points from the distribution ~'r with density”
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“A random variate T with an underlying exponential distribution has a Probability density function (p.d.0 [ 13 ]:”
BackgroundMethodFuzzification of continuous-value spatial evidence for mineral prospectivity mapping
Yousefi, M., & Carranza, E. J. M. (2014) · Computers & Geosciences · doi:10.1016/j.cageo.2014.10.014 · Cited by 233 · see citing works
“…expected value function because uncertain variables of fuzzy sets, where there is vague knowledge about evidential scores (Xu, 2007a,b), can be integrated using expected value functions (e.g., Heilpern, 1992, 1997; Rubinstein, 1981; Wang and Chin, 2011; Ye, 2011; Liu, 2013; Gupta et al., 2013).”
BackgroundFelzer, K. R., Becker, T. W., Abercrombie, R. E., et al. (2002) · Journal of Geophysical Research Atmospheres · doi:10.1029/2001jb000911 · Cited by 233 · see citing works
“[28] To generate aftershock magnitudes and times in our Monte Carlo simulations, we use the inverse transform method [e.g., Rubinstein, 1981] for choosing sample values from an arbitrary probability distribution.”
MethodStyblinski, M. A., & Tang, T. S. (1990) · Neural Networks · doi:10.1016/0893-6080(90)90029-k · Cited by 232 · see citing works
Soize, C. (2005) · Journal of Sound and Vibration · doi:10.1016/j.jsv.2005.07.009 · Cited by 229 · see citing works
Stochastic finite element methods for partial differential equations with random input data
Gunzburger, M., Webster, C., & Zhang, G. (2014) · Acta Numerica · doi:10.1017/s0962492914000075 · Cited by 229 · see citing works
CLASSICAL ESTIMATION METHODS FOR LDV MODELS USING SIMULATION
Hajivassiliou, V. A., & Ruud, P. A. (1993) · London School of Economics and Political Science Research Online (London School of Economics and Political Science) · Cited by 237 · see citing works
Bidding Strategies for Electricity Producers in a Competitive Electricity Marketplace
Gountis, V. P., & Bakirtzis, A. G. (2004) · IEEE Transactions on Power Systems · doi:10.1109/tpwrs.2003.821474 · Cited by 224 · see citing works
“The inverse transform method [23] is used to create random samples that follow a specific pdf.”
Modern Monte Carlo methods for efficient uncertainty quantification and propagation: A survey
Zhang, J. (2020) · Wiley Interdisciplinary Reviews Computational Statistics · doi:10.1002/wics.1539 · Cited by 224 · see citing works
“Typically, methods based on variance reduction (VR) techniques (Rubinstein & Kroese, 2016) are relatively widespread used, such as stratified sampling, Latin hypercube sampling, control variates, importance sampling, etc.”
MethodFixture Failure Diagnosis for Autobody Assembly Using Pattern Recognition
Ceglarek, D. J., & Shi, J. (1996) · Journal of Engineering for Industry · doi:10.1115/1.2803648 · Cited by 224 · see citing works
Effects of noise on modal parameters identified by the Eigensystem Realization Algorithm
Juang, J.-N., & Pappa, R. S. (1986) · Journal of Guidance Control and Dynamics · doi:10.2514/3.20106 · Cited by 223 · see citing works
- Authors
- Rubinstein, R. Y., & Kroese, D. P.
- ORCID
- Kroese, Dirk P.
- Venue
- Wiley Series in Probability and Statistics
- Publisher
- Wiley
- Published
- 2016-11-21
- ISSN
- 1940-6347 (ISSN-L)
- OpenAlex
- W2147632348
- Cited by
- 3,356
- References
- 0
- Citations by year
- 2026: 131 · 2025: 182 · 2024: 168 · 2023: 157 · 2022: 167 · 2021: 162 · 2020: 171 · 2019: 139 · 2018: 118 · 2017: 66 · 2016: 41 · 2015: 47 · 2014: 60 · 2013: 68 · 2012: 126
- FWCI
- 52.82
- Type
- book
- Language
- en
- Open access
- closed
- Locations
- Wiley series in probability and statistics (publishedVersion)
- CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
- International Linear Collider (submittedVersion)
- Topics
- Social Sciences › Decision Sciences › Management Science and Operations Research › Simulation Techniques and Applications
- Physical Sciences › Mathematics › Statistics and Probability › Markov Chains and Monte Carlo Methods
- Physical Sciences › Mathematics › Numerical Analysis › Mathematical Approximation and Integration
- Keywords
- Monte Carlo simulation, stochastic simulation, random number generation, probability distributions, variance reduction, Markov chain Monte Carlo
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