Timeline for Real world example of use of Monte Carlo method for high dimensional integrals
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
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| Oct 21, 2016 at 16:33 | comment | added | arsmath | I would say almost any area that involves high dimensional integrals that can't be solved in closed form, that Monte Carlo is the preferred method. So just pick an area, and there will be examples. | |
| Oct 21, 2016 at 13:01 | comment | added | Piyush Grover | Boltzmann equations are routinely solved with Monte-Carlo. They are 6D. | |
| Oct 21, 2016 at 1:58 | answer | added | Dirk | timeline score: 5 | |
| Oct 20, 2016 at 22:26 | answer | added | Gerry Myerson | timeline score: 2 | |
| Oct 20, 2016 at 22:10 | comment | added | Gerry Myerson | Have a look at the work of Stephen Joe and Ian Sloan. Their paper, Imbedded lattice rules for multidimensional integration, SIAM J. Numer. Anal. 29 (1992), no. 4, 1119–1135, gives examples with up to 4 000 317 440 points in 20 dimensions. Their book, Lattice methods for multiple integration, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1994. xii+239 pp. ISBN: 0-19-853472-8, is aimed "not only at graduate students ..., but also at practical scientists and other who have high-dimensional integrals which they need to approximate''. | |
| Oct 20, 2016 at 21:29 | comment | added | user36212 | Bollobas, Balister, Sarkar and Walters have some (recent!) results of the form: Suppose that (integral) is at least (threshold). Then (random geometric model) exhibits percolation.Here (integral) is a high-dimensional, complicated integral which seems hopeless to bound rigorously (they tried, quite hard); but by use of these Monte Carlo methods one becomes confident that it indeed exceeds (threshold) and hence the geometric percolation statement is true. | |
| Oct 20, 2016 at 21:22 | comment | added | Nawaf Bou-Rabee | I would be amiss if I didn't mention the 1953 paper of Metropolis et al. en.wikipedia.org/wiki/… In that paper, the authors also relate their work to that of Mayer and Ulam. | |
| Oct 20, 2016 at 20:39 | comment | added | Nawaf Bou-Rabee | Please consult a book on molecular dynamics or data science where high-dimensional integrals of this sort are routinely computed. For example, store.elsevier.com/Understanding-Molecular-Simulation/… or springer.com/us/book/9780387763699 | |
| Oct 20, 2016 at 20:28 | comment | added | john mangual | what would you consider realistic? is other academic research (perhaps in Computer Science, Engineering, Physics) considered realistic? Are you looking for financial applications? | |
| Oct 20, 2016 at 20:17 | comment | added | Neal | This may be useful to you- permalink.lanl.gov/object/tr?what=info:lanl-repo/lareport/… In particular, I believe the first couple of paragraphs on p 133 (second full page), suitably interpreted, describe the Monte Carlo estimate of an integral of a region in a phase space. | |
| Oct 20, 2016 at 20:03 | history | asked | David | CC BY-SA 3.0 |