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Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis. It is the study of numerical methods that attempt at finding approximate solutions of problems rather than the exact ones. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences, medicine, business and even the arts. Current growth in computing power has enabled the use of more complex numerical analysis, providing detailed and realistic mathematical models in science and engineering. Examples of numerical analysis include: ordinary differential equations as found in celestial mechanics, numerical linear algebra in data analysis, and stochastic differential equations and Markov chains for simulating living cells in medicine and biology.
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.
Computational physics is the study and implementation of numerical analysis to solve problems in physics. Historically, computational physics was the first application of modern computers in science, and is now a subset of computational science. It is sometimes regarded as a subdiscipline of theoretical physics, but others consider it an intermediate branch between theoretical and experimental physics — an area of study which supplements both theory and experiment.
In analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral. The term numerical quadrature is more or less a synonym for "numerical integration", especially as applied to one-dimensional integrals. Some authors refer to numerical integration over more than one dimension as cubature; others take "quadrature" to include higher-dimensional integration.
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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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References
- ↑ Newsletter, Asia Pacific Mathematics (26 August 2013). "An Interview with Ian Sloan". Gonit Sora. Retrieved 13 October 2022.
- 1 2 3 Ian Sloan's web page, maths.unsw.edu.au, retrieved 2012-05-11.
- ↑ "List of Presidents – The Royal Society of NSW". www.royalsoc.org.au. Retrieved 16 June 2018.
- ↑ Citation for 1997 ANZIAM Medal, www.anziam.org.au
- ↑ Thomas Ranken Lyle Medal Archived 2010-11-28 at the Wayback Machine , Australian Academy of Science, retrieved 2010-06-06.
- ↑ Centenary Medal, 1 January 2001, www.itsanhonour.gov.au
For service to Australian society and science mathematics - ↑ The George Szekeres Medal, www.austms.org.au
- ↑ Officer of the Order of Australia (AO), 9 June 2008, www.itsanhonour.gov.au
For service to education through the study of mathematics, particularly in the field of computational mathematics, as an academic, researcher and mentor, and to a range of national and international professional associations. - ↑ List of Fellows of the American Mathematical Society, retrieved 2013-07-20.
- ↑ Reviews:
- Cools, Ronald (1996). "Book Review: Lattice methods for multiple integration". Bulletin of the American Mathematical Society. 33 (3): 363–366. doi: 10.1090/S0273-0979-96-00660-X . ISSN 1088-9485. S2CID 120864895.
- Keast, Patrick (1997). "Review of Lattice Methods for Multiple Integration". Mathematics of Computation. 66 (217): 459–460. ISSN 0025-5718. JSTOR 2153669.
