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Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be deterministic in principle. They are often used in physical and mathematical problems and are most useful when it is difficult or impossible to use other approaches. Monte Carlo methods are mainly used in three problem classes: optimization, numerical integration, and generating draws from a probability distribution.
In statistics, Markov chain Monte Carlo (MCMC) methods comprise a class of algorithms for sampling from a probability distribution. By constructing a Markov chain that has the desired distribution as its equilibrium distribution, one can obtain a sample of the desired distribution by recording states from the chain. The more steps that are included, the more closely the distribution of the sample matches the actual desired distribution. Various algorithms exist for constructing chains, including the Metropolis–Hastings algorithm.
In numerical analysis, the quasi-Monte Carlo method is a method for numerical integration and solving some other problems using low-discrepancy sequences. This is in contrast to the regular Monte Carlo method or Monte Carlo integration, which are based on sequences of pseudorandom numbers.
In mathematics, Monte Carlo integration is a technique for numerical integration using random numbers. It is a particular Monte Carlo method that numerically computes a definite integral. While other algorithms usually evaluate the integrand at a regular grid, Monte Carlo randomly chooses points at which the integrand is evaluated. This method is particularly useful for higher-dimensional integrals.
John Henry Michell, FRS was an Australian mathematician, Professor of Mathematics at the University of Melbourne.
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Joseph Frederick Traub was an American computer scientist. He was the Edwin Howard Armstrong Professor of Computer Science at Columbia University and External Professor at the Santa Fe Institute. He held positions at Bell Laboratories, University of Washington, Carnegie Mellon, and Columbia, as well as sabbatical positions at Stanford, Berkeley, Princeton, California Institute of Technology, and Technical University, Munich.
Information-based complexity (IBC) studies optimal algorithms and computational complexity for the continuous problems that arise in physical science, economics, engineering, and mathematical finance. IBC has studied such continuous problems as path integration, partial differential equations, systems of ordinary differential equations, nonlinear equations, integral equations, fixed points, and very-high-dimensional integration. All these problems involve functions of a real or complex variable. Since one can never obtain a closed-form solution to the problems of interest one has to settle for a numerical solution. Since a function of a real or complex variable cannot be entered into a digital computer, the solution of continuous problems involves partial information. To give a simple illustration, in the numerical approximation of an integral, only samples of the integrand at a finite number of points are available. In the numerical solution of partial differential equations the functions specifying the boundary conditions and the coefficients of the differential operator can only be sampled. Furthermore, this partial information can be expensive to obtain. Finally the information is often contaminated by noise.
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High-dimensional integrals in hundreds or thousands of variables occur commonly in finance. These integrals have to be computed numerically to within a threshold . If the integral is of dimension then in the worst case, where one has a guarantee of error at most , the computational complexity is typically of order . That is, the problem suffers the curse of dimensionality. In 1977 P. Boyle, University of Waterloo, proposed using Monte Carlo (MC) to evaluate options. Starting in early 1992, J. F. Traub, Columbia University, and a graduate student at the time, S. Paskov, used quasi-Monte Carlo (QMC) to price a Collateralized mortgage obligation with parameters specified by Goldman Sachs. Even though it was believed by the world's leading experts that QMC should not be used for high-dimensional integration, Paskov and Traub found that QMC beat MC by one to three orders of magnitude and also enjoyed other desirable attributes. Their results were first published in 1995. Today QMC is widely used in the financial sector to value financial derivatives; see list of books below.
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Harald G. Niederreiter is an Austrian mathematician known for his work in discrepancy theory, algebraic geometry, quasi-Monte Carlo methods, and cryptography.
Lisa J. Fauci is an American mathematician who applies computational fluid dynamics to biological processes such as sperm motility and phytoplankton dynamics. More generally, her research interests include numerical analysis, scientific computing, and mathematical biology. She is the Pendergraft Nola Lee Haynes Professor of Mathematics at Tulane University, and president of the Society for Industrial and Applied Mathematics.
Ilya Meyerovich Sobol is a Russian mathematician of Jewish Lithuanian origin, known for his work on Monte Carlo methods. His research spans several applications, from nuclear studies to astrophysics, and has contributed significantly to the field of sensitivity analysis.
Antoinette A. Tordesillas is a professor of applied mathematics at the University of Melbourne, Australia. She heads the Micromechanics of Granular Media Group in the School of Mathematics and Statistics at Melbourne.
Ngamta "Natalie" Thamwattana is a Thai mathematician who works in Australia as a Professor of Applied Mathematics at the University of Newcastle (Australia). In 2014 she won the J. H. Michell Medal of ANZIAM for her "pioneering contributions in the areas of granular materials and nanotechnology".
Stanisław Krystyn Zaremba was a Polish climber, mountaineer and mathematician. He was the son of Stanisław Zaremba (1863–1942), also a mathematician.
Yvonne Marie Stokes is an Australian mathematician whose research involves fluid mechanics, mathematical biology, and industrial applications of mathematics. She is a professor and Australian Research Council Future Fellow at the University of Adelaide.
Jennifer A. Flegg is an Australian mathematician and is an Associate Professor of applied mathematics in the School of Mathematics and Statistics at the University of Melbourne.
Gerhard Larcher is an Austrian mathematician and professor of financial mathematics at the Johannes Kepler University (JKU) in Linz, Austria. He is the head of the Institute of Financial Mathematics.
References
- ↑ Kuo, Frances Y. (2002), Constructive approaches to quasi-Monte Carlo methods for multiple integration (Doctoral thesis), Hamilton: Waikato Research Commons, hdl:10289/14052, Wikidata Q111966326
- 1 2 Kuo, Frances, Liberating the dimension: The story of my PhD days, Australian Mathematical Society, retrieved 2020-01-07
- ↑ Frances Kuo at the Mathematics Genealogy Project
- ↑ "Professor Frances Kuo", Find a researcher, University of New South Wales, retrieved 2020-01-07
- ↑ The 2011 JH Michell Medal, ANZIAM, 27 March 2011, retrieved 2020-01-07
- ↑ "Joseph F. Traub – Prize", Journal of Complexity, retrieved 2020-01-07
