Definition
Any random process can, in principle, be represented by a computer program using pseudorandom numbers. If near-perfectly independent random numbers between 0 and 1 can be produced, appropriate functions of these numbers can simulate realizations of practically any probability distribution needed in technical and scientific applications. The development of Monte Carlo methods grew out of our ability to find those functions and algorithms.
The literature on Monte Carlo methods is vast, and the methods are many. We cannot cover everything here, but the intention is to provide a brief description of some of the most important techniques and their principles. After having reviewed basic techniques for random simulation, we shall move on to three high-level categories of Monte Carlo methods: testing, inference, and optimization. We shall also present some hybrid methods, as there is often no sharp borderline between the categories.
In the following, unless otherwise stated, we...
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References
Deutsch CV, Journel AG (1994) Application of simulated annealing to stochastic reservoir modeling. SPE Adv Technol Ser 2:222–227
Efron B (1993) An introduction to the bootstrap. Chapman & Hall\CRC
Evensen G (1994) Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error statistics. J Geophys Res 99(C5):10,143–10,162. https://doi.org/10.1029/94JC00572
Evensen G (2003) The ensemble Kalman filter: theoretical formulation and practical implementation. Ocean Dyn 53(4):343–367. https://doi.org/10.1007/s10236-003-0036-9
Geman S, Geman D (1984) Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Trans Pattern Anal Mach Intell 6(6):721–741. https://doi.org/10.1109/TPAMI.1984.4767596
Geyer CJ (1991) Markov chain Monte Carlo maximum likelihood. Interface Foundation of North America
Green PJ (1995) Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. Biometrika 82(4):711–732. https://doi.org/10.1093/biomet/82.4.711
Hastings WK (1970) Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57(1):97–109
Holland JH (1975) Adaptation in natural and artificial systems. University of Michigan Press
Kalman RE (1960) A new approach to linear filtering and prediction problems. J Basic Eng 82(1):35–45. https://doi.org/10.1115/1.3662552
Kirkpatrick S, Gelatt CD, Vecchi MP (1983) Optimization by simulated annealing. Science 220(4598):671–680. https://doi.org/10.1126/science.220.4598.671
Marinari E, Parisi G (1992) Simulated tempering: A new Monte Carlo scheme. Europhys Lett 19(6):451–458. https://doi.org/10.1209/0295-5075/19/6/002
Metropolis N, Rosenbluth A, Rosenbluth M, Teller A, Teller E (1953) Equations of state calculations by fast computing machines. J Chem Phys 21(6):1087–1092
Mosegaard K, Tarantola A (1995) Monte Carlo sampling of solutions to inverse problems. J Geophys Res 100:12431–12447
Mosegaard K, Vestergaard PD (1991) A simulated annealing approach to seismic model optimization with sparse prior information. Geophys Prosp 39(05):599–612
Mrkvička T, Soubeyrand S, Myllymäki M, Grabarnik P, Hahn U (2016) Monte Carlo testing in spatial statistics, with applications to spatial residuals. Spatial Stat 18:40–53, spatial Statistics Avignon: Emerging Patterns
Nakamura G, Potthast R (2015) Inverse modeling. IOP Publishing, Bristol
Reich S, Cotter C (2015) Probabilistic forecasting and Bayesian data assimilation. Cambridge University Press, Cambridge, UK
Salamon P, Sibani P, Frost R (2002) Facts, conjectures, and improvements for simulated annealing. Society of Industrial and Applied Mathematics
Sambridge M (1999a) Geophysical inversion with a neighbourhood algorithm – i. searching a parameter space. Geophys J Int 138(2):479–494. https://doi.org/10.1046/j.1365-246X.1999.00876.x
Sambridge M (1999b) Geophysical inversion with a neighbourhood algorithm – ii. appraising the ensemble. Geophys J Int 138(3):727–746. https://doi.org/10.1046/j.1365-246x.1999.00900.x
Sambridge M (2013) A parallel tempering algorithm for probabilistic sampling and multimodal optimization. Geophys J Int 196(1):357–374. https://doi.org/10.1093/gji/ggt342
Sambridge M, Gallagher K, Jackson A, Rickwood P (2006) Trans-dimensional inverse problems, model comparison and the evidence. Geophys J Int 167(2):528–542. https://doi.org/10.1111/j.1365-246X.2006.03155.x
Strebelle S (2002) Conditional simulation of complex geological structures using multiple-point statistics. Math Geol 34:1–21
van Leeuwen P, Cheng Y, Reich S (2015) Nonlinear data assimilation. Springer, Berlin
von Neumann J (1951) Various techniques used in connection with random digits. Monte Carlo methods. Nat Bureau Standards 12:36–38
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Mosegaard, K. (2022). Monte Carlo Method. In: Daya Sagar, B.S., Cheng, Q., McKinley, J., Agterberg, F. (eds) Encyclopedia of Mathematical Geosciences. Encyclopedia of Earth Sciences Series. Springer, Cham. https://doi.org/10.1007/978-3-030-26050-7_431-2
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Latest
Monte Carlo Method- Published:
- 05 August 2022
DOI: https://doi.org/10.1007/978-3-030-26050-7_431-2
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Monte Carlo Method- Published:
- 28 August 2021
DOI: https://doi.org/10.1007/978-3-030-26050-7_431-1