VEGAS algorithm - Revision history

Fadesga: /* References */

2022-07-20T02:59:48Z

References

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	[[Category:Statistical algorithms]]		[[Category:Statistical algorithms]]
	[[Category:Variance reduction]]		[[Category:Variance reduction]]


			{{compu-physics-stub}}

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2021-12-29T17:07:04Z

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← Previous revision		Revision as of 17:07, 29 December 2021
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	The '''VEGAS algorithm''', due to [[G. Peter Lepage]],<ref name=Lepage1978>{{cite journal\|last=Lepage\|first=G.P.\|title=A New Algorithm for Adaptive Multidimensional Integration\|journal=Journal of Computational Physics\|date=May 1978\|volume=27\|issue=2\|pages=192–203\|doi=10.1016/0021-9991(78)90004-9\|bibcode=1978JCoPh..27..192L}}</ref><ref name=Lepage1980>{{cite journal\|last=Lepage\|first=G.P.\|title=VEGAS: An Adaptive Multi-dimensional Integration Program\|journal=Cornell Preprint\|volume=CLNS 80-447\|date=March 1980}}</ref><ref name=Ohl1999>{{cite journal\|last=Ohl\|first=T.\|title=Vegas revisited: Adaptive Monte Carlo integration beyond factorization\|journal=Computer Physics Communications\|date=July 1999\|volume=120\|issue=1\|pages=13–19\|doi=10.1016/S0010-4655(99)00209-X\|arxiv=hep-ph/9806432\|bibcode=1999CoPhC.120...13O\|s2cid=18194240}}</ref> is a method for [[variance reduction\|reducing error]] in [[Monte Carlo simulation]]s by using a known or approximate [[probability distribution]] function to concentrate the search in those areas of the [[integrand]] that make the greatest contribution to the final [[integral]].		The '''VEGAS algorithm''', due to [[G. Peter Lepage]],<ref name=Lepage1978>{{cite journal\|last=Lepage\|first=G.P.\|title=A New Algorithm for Adaptive Multidimensional Integration\|journal=Journal of Computational Physics\|date=May 1978\|volume=27\|issue=2\|pages=192–203\|doi=10.1016/0021-9991(78)90004-9\|bibcode=1978JCoPh..27..192L}}</ref><ref name=Lepage1980>{{cite journal\|last=Lepage\|first=G.P.\|title=VEGAS: An Adaptive Multi-dimensional Integration Program\|journal=Cornell Preprint\|volume=CLNS 80-447\|date=March 1980}}</ref><ref name=Ohl1999>{{cite journal\|last=Ohl\|first=T.\|title=Vegas revisited: Adaptive Monte Carlo integration beyond factorization\|journal=Computer Physics Communications\|date=July 1999\|volume=120\|issue=1\|pages=13–19\|doi=10.1016/S0010-4655(99)00209-X\|arxiv=hep-ph/9806432\|bibcode=1999CoPhC.120...13O\|s2cid=18194240}}</ref> is a method for [[variance reduction\|reducing error]] in [[Monte Carlo simulation]]s by using a known or approximate [[probability distribution]] function to concentrate the search in those areas of the [[integrand]] that make the greatest contribution to the final [[integral]].

	The VEGAS algorithm is based on [[importance sampling]]. It samples points from the probability distribution described by the function <math>\|f\|,</math> so that the points are concentrated in the regions that make the largest contribution to the integral. The [[GNU Scientific Library]] (GSL) provides a ~~[https://www.gnu.org/software/gsl/doc/html/montecarlo.html?highlight=vegas~~ VEGAS routine].		The VEGAS algorithm is based on [[importance sampling]]. It samples points from the probability distribution described by the function <math>\|f\|,</math> so that the points are concentrated in the regions that make the largest contribution to the integral. The [[GNU Scientific Library]] (GSL) provides a VEGAS routine.

	==Sampling method==		==Sampling method==

Citation bot: Alter: journal. Add: s2cid, issue. | Use this bot. Report bugs. | Suggested by Abductive | #UCB_toolbar

2021-11-15T04:52:58Z

Alter: journal. Add: s2cid, issue. | Use this bot. Report bugs. | Suggested by Abductive | #UCB_toolbar

← Previous revision		Revision as of 04:52, 15 November 2021
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	{{short description\|Algorithm}}		{{short description\|Algorithm}}
	The '''VEGAS algorithm''', due to [[G. Peter Lepage]],<ref name=Lepage1978>{{cite journal\|last=Lepage\|first=G.P.\|title=A New Algorithm for Adaptive Multidimensional Integration\|journal=Journal of Computational Physics\|date=May 1978\|volume=27\|pages=192–203\|doi=10.1016/0021-9991(78)90004-9\|bibcode=1978JCoPh..27..192L}}</ref><ref name=Lepage1980>{{cite journal\|last=Lepage\|first=G.P.\|title=VEGAS: An Adaptive Multi-dimensional Integration Program\|journal=Cornell ~~preprint~~\|volume=CLNS 80-447\|date=March 1980}}</ref><ref name=Ohl1999>{{cite journal\|last=Ohl\|first=T.\|title=Vegas revisited: Adaptive Monte Carlo integration beyond factorization\|journal=Computer Physics Communications\|date=July 1999\|volume=120\|issue=1\|pages=13–19\|doi=10.1016/S0010-4655(99)00209-X\|arxiv=hep-ph/9806432\|bibcode=1999CoPhC.120...13O}}</ref> is a method for [[variance reduction\|reducing error]] in [[Monte Carlo simulation]]s by using a known or approximate [[probability distribution]] function to concentrate the search in those areas of the [[integrand]] that make the greatest contribution to the final [[integral]].		The '''VEGAS algorithm''', due to [[G. Peter Lepage]],<ref name=Lepage1978>{{cite journal\|last=Lepage\|first=G.P.\|title=A New Algorithm for Adaptive Multidimensional Integration\|journal=Journal of Computational Physics\|date=May 1978\|volume=27\|issue=2\|pages=192–203\|doi=10.1016/0021-9991(78)90004-9\|bibcode=1978JCoPh..27..192L}}</ref><ref name=Lepage1980>{{cite journal\|last=Lepage\|first=G.P.\|title=VEGAS: An Adaptive Multi-dimensional Integration Program\|journal=Cornell Preprint\|volume=CLNS 80-447\|date=March 1980}}</ref><ref name=Ohl1999>{{cite journal\|last=Ohl\|first=T.\|title=Vegas revisited: Adaptive Monte Carlo integration beyond factorization\|journal=Computer Physics Communications\|date=July 1999\|volume=120\|issue=1\|pages=13–19\|doi=10.1016/S0010-4655(99)00209-X\|arxiv=hep-ph/9806432\|bibcode=1999CoPhC.120...13O\|s2cid=18194240}}</ref> is a method for [[variance reduction\|reducing error]] in [[Monte Carlo simulation]]s by using a known or approximate [[probability distribution]] function to concentrate the search in those areas of the [[integrand]] that make the greatest contribution to the final [[integral]].

	The VEGAS algorithm is based on [[importance sampling]]. It samples points from the probability distribution described by the function <math>\|f\|,</math> so that the points are concentrated in the regions that make the largest contribution to the integral. The [[GNU Scientific Library]] (GSL) provides a [https://www.gnu.org/software/gsl/doc/html/montecarlo.html?highlight=vegas VEGAS routine].		The VEGAS algorithm is based on [[importance sampling]]. It samples points from the probability distribution described by the function <math>\|f\|,</math> so that the points are concentrated in the regions that make the largest contribution to the integral. The [[GNU Scientific Library]] (GSL) provides a [https://www.gnu.org/software/gsl/doc/html/montecarlo.html?highlight=vegas VEGAS routine].

I dream of horses: AWB cleanup patrol

2020-09-24T02:19:19Z

AWB cleanup patrol

← Previous revision		Revision as of 02:19, 24 September 2020
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	{{short description\|Algorithm}}		{{short description\|Algorithm}}
	The '''VEGAS algorithm''', due to [[G. Peter Lepage]],<ref name=Lepage1978>{{cite journal\|last=Lepage\|first=G.P.\|title=A New Algorithm for Adaptive Multidimensional Integration\|journal=Journal of Computational Physics\|date=May 1978\|volume=27\|pages=~~192-203~~\|doi=10.1016/0021-9991(78)90004-9\|bibcode=1978JCoPh..27..192L}}</ref><ref name=Lepage1980>{{cite journal\|last=Lepage\|first=G.P.\|title=VEGAS: An Adaptive Multi-dimensional Integration Program\|journal=Cornell preprint\|volume=CLNS 80-447\|date=March 1980}}</ref><ref name=Ohl1999>{{cite journal\|last=Ohl\|first=T.\|title=Vegas revisited: Adaptive Monte Carlo integration beyond factorization\|journal=Computer Physics Communications\|date=July 1999\|volume=120\|issue=1\|pages=13–19\|doi=10.1016/S0010-4655(99)00209-X\|arxiv=hep-ph/9806432\|bibcode=1999CoPhC.120...13O}}</ref> is a method for [[variance reduction\|reducing error]] in [[Monte Carlo simulation]]s by using a known or approximate [[probability distribution]] function to concentrate the search in those areas of the [[integrand]] that make the greatest contribution to the final [[integral]].		The '''VEGAS algorithm''', due to [[G. Peter Lepage]],<ref name=Lepage1978>{{cite journal\|last=Lepage\|first=G.P.\|title=A New Algorithm for Adaptive Multidimensional Integration\|journal=Journal of Computational Physics\|date=May 1978\|volume=27\|pages=192–203\|doi=10.1016/0021-9991(78)90004-9\|bibcode=1978JCoPh..27..192L}}</ref><ref name=Lepage1980>{{cite journal\|last=Lepage\|first=G.P.\|title=VEGAS: An Adaptive Multi-dimensional Integration Program\|journal=Cornell preprint\|volume=CLNS 80-447\|date=March 1980}}</ref><ref name=Ohl1999>{{cite journal\|last=Ohl\|first=T.\|title=Vegas revisited: Adaptive Monte Carlo integration beyond factorization\|journal=Computer Physics Communications\|date=July 1999\|volume=120\|issue=1\|pages=13–19\|doi=10.1016/S0010-4655(99)00209-X\|arxiv=hep-ph/9806432\|bibcode=1999CoPhC.120...13O}}</ref> is a method for [[variance reduction\|reducing error]] in [[Monte Carlo simulation]]s by using a known or approximate [[probability distribution]] function to concentrate the search in those areas of the [[integrand]] that make the greatest contribution to the final [[integral]].

	The VEGAS algorithm is based on [[importance sampling]]. It samples points from the probability distribution described by the function <math>\|f\|,</math> so that the points are concentrated in the regions that make the largest contribution to the integral. The [[GNU Scientific Library]] (GSL) provides a [https://www.gnu.org/software/gsl/doc/html/montecarlo.html?highlight=vegas VEGAS routine].		The VEGAS algorithm is based on [[importance sampling]]. It samples points from the probability distribution described by the function <math>\|f\|,</math> so that the points are concentrated in the regions that make the largest contribution to the integral. The [[GNU Scientific Library]] (GSL) provides a [https://www.gnu.org/software/gsl/doc/html/montecarlo.html?highlight=vegas VEGAS routine].
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	==References==		==References==
	{{reflist}}<br />		{{reflist}}<br />

	[[Category:Monte Carlo methods]]		[[Category:Monte Carlo methods]]
	[[Category:Computational physics]]		[[Category:Computational physics]]

Histohob: Adding short description: "Algorithm" (Shortdesc helper)

2020-08-01T11:54:06Z

Adding short description: "Algorithm" (Shortdesc helper)

← Previous revision		Revision as of 11:54, 1 August 2020
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			{{short description\|Algorithm}}
	The '''VEGAS algorithm''', due to [[G. Peter Lepage]],<ref name=Lepage1978>{{cite journal\|last=Lepage\|first=G.P.\|title=A New Algorithm for Adaptive Multidimensional Integration\|journal=Journal of Computational Physics\|date=May 1978\|volume=27\|pages=192-203\|doi=10.1016/0021-9991(78)90004-9\|bibcode=1978JCoPh..27..192L}}</ref><ref name=Lepage1980>{{cite journal\|last=Lepage\|first=G.P.\|title=VEGAS: An Adaptive Multi-dimensional Integration Program\|journal=Cornell preprint\|volume=CLNS 80-447\|date=March 1980}}</ref><ref name=Ohl1999>{{cite journal\|last=Ohl\|first=T.\|title=Vegas revisited: Adaptive Monte Carlo integration beyond factorization\|journal=Computer Physics Communications\|date=July 1999\|volume=120\|issue=1\|pages=13–19\|doi=10.1016/S0010-4655(99)00209-X\|arxiv=hep-ph/9806432\|bibcode=1999CoPhC.120...13O}}</ref> is a method for [[variance reduction\|reducing error]] in [[Monte Carlo simulation]]s by using a known or approximate [[probability distribution]] function to concentrate the search in those areas of the [[integrand]] that make the greatest contribution to the final [[integral]].		The '''VEGAS algorithm''', due to [[G. Peter Lepage]],<ref name=Lepage1978>{{cite journal\|last=Lepage\|first=G.P.\|title=A New Algorithm for Adaptive Multidimensional Integration\|journal=Journal of Computational Physics\|date=May 1978\|volume=27\|pages=192-203\|doi=10.1016/0021-9991(78)90004-9\|bibcode=1978JCoPh..27..192L}}</ref><ref name=Lepage1980>{{cite journal\|last=Lepage\|first=G.P.\|title=VEGAS: An Adaptive Multi-dimensional Integration Program\|journal=Cornell preprint\|volume=CLNS 80-447\|date=March 1980}}</ref><ref name=Ohl1999>{{cite journal\|last=Ohl\|first=T.\|title=Vegas revisited: Adaptive Monte Carlo integration beyond factorization\|journal=Computer Physics Communications\|date=July 1999\|volume=120\|issue=1\|pages=13–19\|doi=10.1016/S0010-4655(99)00209-X\|arxiv=hep-ph/9806432\|bibcode=1999CoPhC.120...13O}}</ref> is a method for [[variance reduction\|reducing error]] in [[Monte Carlo simulation]]s by using a known or approximate [[probability distribution]] function to concentrate the search in those areas of the [[integrand]] that make the greatest contribution to the final [[integral]].

Comp.arch at 15:39, 21 June 2020

2020-06-21T15:39:39Z

← Previous revision		Revision as of 15:39, 21 June 2020
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	The '''VEGAS algorithm''', due to [[G. Peter Lepage]],<ref name=Lepage1978>{{cite journal\|last=Lepage\|first=G.P.\|title=A New Algorithm for Adaptive Multidimensional Integration\|journal=Journal of Computational Physics\|date=May 1978\|volume=27\|pages=192-203\|doi=10.1016/0021-9991(78)90004-9\|bibcode=1978JCoPh..27..192L}}</ref><ref name=Lepage1980>{{cite journal\|last=Lepage\|first=G.P.\|title=VEGAS: An Adaptive Multi-dimensional Integration Program\|journal=Cornell preprint\|volume=CLNS 80-447\|date=March 1980}}</ref><ref name=Ohl1999>{{cite journal\|last=Ohl\|first=T.\|title=Vegas revisited: Adaptive Monte Carlo integration beyond factorization\|journal=Computer Physics Communications\|date=July 1999\|volume=120\|issue=1\|pages=13–19\|doi=10.1016/S0010-4655(99)00209-X\|arxiv=hep-ph/9806432\|bibcode=1999CoPhC.120...13O}}</ref> is a method for [[variance reduction\|reducing error]] in [[Monte Carlo simulation]]s by using a known or approximate [[probability distribution]] function to concentrate the search in those areas of the [[integrand]] that make the greatest contribution to the final [[integral]].		The '''VEGAS algorithm''', due to [[G. Peter Lepage]],<ref name=Lepage1978>{{cite journal\|last=Lepage\|first=G.P.\|title=A New Algorithm for Adaptive Multidimensional Integration\|journal=Journal of Computational Physics\|date=May 1978\|volume=27\|pages=192-203\|doi=10.1016/0021-9991(78)90004-9\|bibcode=1978JCoPh..27..192L}}</ref><ref name=Lepage1980>{{cite journal\|last=Lepage\|first=G.P.\|title=VEGAS: An Adaptive Multi-dimensional Integration Program\|journal=Cornell preprint\|volume=CLNS 80-447\|date=March 1980}}</ref><ref name=Ohl1999>{{cite journal\|last=Ohl\|first=T.\|title=Vegas revisited: Adaptive Monte Carlo integration beyond factorization\|journal=Computer Physics Communications\|date=July 1999\|volume=120\|issue=1\|pages=13–19\|doi=10.1016/S0010-4655(99)00209-X\|arxiv=hep-ph/9806432\|bibcode=1999CoPhC.120...13O}}</ref> is a method for [[variance reduction\|reducing error]] in [[Monte Carlo simulation]]s by using a known or approximate [[probability distribution]] function to concentrate the search in those areas of the [[integrand]] that make the greatest contribution to the final [[integral]].

	The VEGAS algorithm is based on [[importance sampling]]. It samples points from the probability distribution described by the function <math>\|f\|,</math> so that the points are concentrated in the regions that make the largest contribution to the integral.		The VEGAS algorithm is based on [[importance sampling]]. It samples points from the probability distribution described by the function <math>\|f\|,</math> so that the points are concentrated in the regions that make the largest contribution to the integral. The [[GNU Scientific Library]] (GSL) provides a [https://www.gnu.org/software/gsl/doc/html/montecarlo.html?highlight=vegas VEGAS routine].
	The [[GNU Scientific Library]] (GSL) provides a [https://www.gnu.org/software/gsl/doc/html/montecarlo.html?highlight=vegas VEGAS routine]

	==Sampling method==		==Sampling method==
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	* [[Importance sampling]]		* [[Importance sampling]]

	== References ==		==References==
	{{reflist}}<br />		{{reflist}}<br />
	[[Category:Monte Carlo methods]]		[[Category:Monte Carlo methods]]

Rjwilmsi: /* top */Journal cites:, added 1 Bibcode

2020-05-30T12:38:48Z

top: Journal cites:, added 1 Bibcode

← Previous revision		Revision as of 12:38, 30 May 2020
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	The '''VEGAS algorithm''', due to [[G. Peter Lepage]],<ref name=Lepage1978>{{cite journal\|last=Lepage\|first=G.P.\|title=A New Algorithm for Adaptive Multidimensional Integration\|journal=Journal of Computational Physics\|date=May 1978\|volume=27\|pages=192-203\|doi=10.1016/0021-9991(78)90004-9}}</ref><ref name=Lepage1980>{{cite journal\|last=Lepage\|first=G.P.\|title=VEGAS: An Adaptive Multi-dimensional Integration Program\|journal=Cornell preprint\|volume=CLNS 80-447\|date=March 1980}}</ref><ref name=Ohl1999>{{cite journal\|last=Ohl\|first=T.\|title=Vegas revisited: Adaptive Monte Carlo integration beyond factorization\|journal=Computer Physics Communications\|date=July 1999\|volume=120\|issue=1\|pages=13–19\|doi=10.1016/S0010-4655(99)00209-X\|arxiv=hep-ph/9806432\|bibcode=1999CoPhC.120...13O}}</ref> is a method for [[variance reduction\|reducing error]] in [[Monte Carlo simulation]]s by using a known or approximate [[probability distribution]] function to concentrate the search in those areas of the [[integrand]] that make the greatest contribution to the final [[integral]].		The '''VEGAS algorithm''', due to [[G. Peter Lepage]],<ref name=Lepage1978>{{cite journal\|last=Lepage\|first=G.P.\|title=A New Algorithm for Adaptive Multidimensional Integration\|journal=Journal of Computational Physics\|date=May 1978\|volume=27\|pages=192-203\|doi=10.1016/0021-9991(78)90004-9\|bibcode=1978JCoPh..27..192L}}</ref><ref name=Lepage1980>{{cite journal\|last=Lepage\|first=G.P.\|title=VEGAS: An Adaptive Multi-dimensional Integration Program\|journal=Cornell preprint\|volume=CLNS 80-447\|date=March 1980}}</ref><ref name=Ohl1999>{{cite journal\|last=Ohl\|first=T.\|title=Vegas revisited: Adaptive Monte Carlo integration beyond factorization\|journal=Computer Physics Communications\|date=July 1999\|volume=120\|issue=1\|pages=13–19\|doi=10.1016/S0010-4655(99)00209-X\|arxiv=hep-ph/9806432\|bibcode=1999CoPhC.120...13O}}</ref> is a method for [[variance reduction\|reducing error]] in [[Monte Carlo simulation]]s by using a known or approximate [[probability distribution]] function to concentrate the search in those areas of the [[integrand]] that make the greatest contribution to the final [[integral]].

	The VEGAS algorithm is based on [[importance sampling]]. It samples points from the probability distribution described by the function <math>\|f\|,</math> so that the points are concentrated in the regions that make the largest contribution to the integral.		The VEGAS algorithm is based on [[importance sampling]]. It samples points from the probability distribution described by the function <math>\|f\|,</math> so that the points are concentrated in the regions that make the largest contribution to the integral.

Livingthingdan: minor updates to reflect link in the Monte Carlo page linking here.

2020-04-08T21:35:15Z

minor updates to reflect link in the Monte Carlo page linking here.

← Previous revision		Revision as of 21:35, 8 April 2020
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	The VEGAS algorithm is based on [[importance sampling]]. It samples points from the probability distribution described by the function <math>\|f\|,</math> so that the points are concentrated in the regions that make the largest contribution to the integral.		The VEGAS algorithm is based on [[importance sampling]]. It samples points from the probability distribution described by the function <math>\|f\|,</math> so that the points are concentrated in the regions that make the largest contribution to the integral.
		⚫	The [[GNU Scientific Library]] (GSL) provides a [https://www.gnu.org/software/gsl/doc/html/montecarlo.html?highlight=vegas VEGAS routine]

	==Sampling method==		==Sampling method==
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	* [[Las Vegas algorithm]]		* [[Las Vegas algorithm]]
	* [[Monte Carlo integration]]		* [[Monte Carlo integration]]
			* [[Importance sampling]]
⚫	*The [[GNU Scientific Library]] (GSL) provides a [https://www.gnu.org/software/gsl/doc/html/montecarlo.html?highlight=vegas VEGAS routine]

	== References ==		== References ==

17387349L8764: /* Minimal correction and linkage to GSL VEGAS routine URL */

2019-08-17T10:14:12Z

Minimal correction and linkage to GSL VEGAS routine URL

← Previous revision		Revision as of 10:14, 17 August 2019
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	* [[Las Vegas algorithm]]		* [[Las Vegas algorithm]]
	* [[Monte Carlo integration]]		* [[Monte Carlo integration]]
			*The [[GNU Scientific Library]] (GSL) provides a [https://www.gnu.org/software/gsl/doc/html/montecarlo.html?highlight=vegas VEGAS routine]

	== References ==		== References ==
	{{reflist}}		{{reflist}}<br />
	* The [https://www.gnu.org/software/gsl GNU Scientific Library] provides VEGAS routines

	[[Category:Monte Carlo methods]]		[[Category:Monte Carlo methods]]
	[[Category:Computational physics]]		[[Category:Computational physics]]

Lihenryhfl: /* Sampling method */ Explicitly wrote out the previously undefined function I(f(x)). See Eq. 3 of A new algorithm for adaptive multidimensional integration (https://www.sciencedirect.com/science/article/pii/0021999178900049)

2019-07-19T03:01:16Z

Sampling method: Explicitly wrote out the previously undefined function I(f(x)). See Eq. 3 of A new algorithm for adaptive multidimensional integration (https://www.sciencedirect.com/science/article/pii/0021999178900049)

← Previous revision		Revision as of 03:01, 19 July 2019
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	==Sampling method==		==Sampling method==
	{{Further\|Importance sampling}}		{{Further\|Importance sampling}}
	In general, if the Monte Carlo integral of <math>f</math> is sampled with points distributed according to a probability distribution described by the function <math>g,</math> we obtain an estimate <math>\mathrm{E}_g(f; N),</math>		In general, if the Monte Carlo integral of <math>f</math> over a volume <math>\Omega</math> is sampled with points distributed according to a probability distribution described by the function <math>g,</math> we obtain an estimate <math>\mathrm{E}_g(f; N),</math>

	:<math>\mathrm{E}_g(f; N) = {1 \over N } \sum_i^N { f(x_i)} / g(x_i) .</math>		:<math>\mathrm{E}_g(f; N) = {1 \over N } \sum_i^N { f(x_i)} / g(x_i) .</math>
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	where <math>\mathrm{Var}(f;N)</math> is the variance of the original estimate, <math>\mathrm{Var}(f; N) = \mathrm{E}(f^2; N) - (\mathrm{E}(f; N))^2.</math>		where <math>\mathrm{Var}(f;N)</math> is the variance of the original estimate, <math>\mathrm{Var}(f; N) = \mathrm{E}(f^2; N) - (\mathrm{E}(f; N))^2.</math>

	If the probability distribution is chosen as <math>g = \|f\|/I(\|f\|)</math> then it can be shown that the variance <math>\mathrm{Var}_g(f; N)</math> vanishes, and the error in the estimate will be zero. In practice it is not possible to sample from the exact distribution g for an arbitrary function, so importance sampling algorithms aim to produce efficient approximations to the desired distribution.		If the probability distribution is chosen as <math>g = \|f\|/\textstyle \int_\Omega \|f(x)\|dx </math> then it can be shown that the variance <math>\mathrm{Var}_g(f; N)</math> vanishes, and the error in the estimate will be zero. In practice it is not possible to sample from the exact distribution g for an arbitrary function, so importance sampling algorithms aim to produce efficient approximations to the desired distribution.

	==Approximation of probability distribution==		==Approximation of probability distribution==

← Previous revision		Revision as of 04:52, 15 November 2021
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	{{short description\|Algorithm}}		{{short description\|Algorithm}}
	The '''VEGAS algorithm''', due to [[G. Peter Lepage]],<ref name=Lepage1978>{{cite journal\|last=Lepage\|first=G.P.\|title=A New Algorithm for Adaptive Multidimensional Integration\|journal=Journal of Computational Physics\|date=May 1978\|volume=27\|pages=192–203\|doi=10.1016/0021-9991(78)90004-9\|bibcode=1978JCoPh..27..192L}}</ref><ref name=Lepage1980>{{cite journal\|last=Lepage\|first=G.P.\|title=VEGAS: An Adaptive Multi-dimensional Integration Program\|journal=Cornell ~~preprint~~\|volume=CLNS 80-447\|date=March 1980}}</ref><ref name=Ohl1999>{{cite journal\|last=Ohl\|first=T.\|title=Vegas revisited: Adaptive Monte Carlo integration beyond factorization\|journal=Computer Physics Communications\|date=July 1999\|volume=120\|issue=1\|pages=13–19\|doi=10.1016/S0010-4655(99)00209-X\|arxiv=hep-ph/9806432\|bibcode=1999CoPhC.120...13O}}</ref> is a method for [[variance reduction\|reducing error]] in [[Monte Carlo simulation]]s by using a known or approximate [[probability distribution]] function to concentrate the search in those areas of the [[integrand]] that make the greatest contribution to the final [[integral]].		The '''VEGAS algorithm''', due to [[G. Peter Lepage]],<ref name=Lepage1978>{{cite journal\|last=Lepage\|first=G.P.\|title=A New Algorithm for Adaptive Multidimensional Integration\|journal=Journal of Computational Physics\|date=May 1978\|volume=27\|issue=2\|pages=192–203\|doi=10.1016/0021-9991(78)90004-9\|bibcode=1978JCoPh..27..192L}}</ref><ref name=Lepage1980>{{cite journal\|last=Lepage\|first=G.P.\|title=VEGAS: An Adaptive Multi-dimensional Integration Program\|journal=Cornell Preprint\|volume=CLNS 80-447\|date=March 1980}}</ref><ref name=Ohl1999>{{cite journal\|last=Ohl\|first=T.\|title=Vegas revisited: Adaptive Monte Carlo integration beyond factorization\|journal=Computer Physics Communications\|date=July 1999\|volume=120\|issue=1\|pages=13–19\|doi=10.1016/S0010-4655(99)00209-X\|arxiv=hep-ph/9806432\|bibcode=1999CoPhC.120...13O\|s2cid=18194240}}</ref> is a method for [[variance reduction\|reducing error]] in [[Monte Carlo simulation]]s by using a known or approximate [[probability distribution]] function to concentrate the search in those areas of the [[integrand]] that make the greatest contribution to the final [[integral]].

	The VEGAS algorithm is based on [[importance sampling]]. It samples points from the probability distribution described by the function <math>\|f\|,</math> so that the points are concentrated in the regions that make the largest contribution to the integral. The [[GNU Scientific Library]] (GSL) provides a [https://www.gnu.org/software/gsl/doc/html/montecarlo.html?highlight=vegas VEGAS routine].		The VEGAS algorithm is based on [[importance sampling]]. It samples points from the probability distribution described by the function <math>\|f\|,</math> so that the points are concentrated in the regions that make the largest contribution to the integral. The [[GNU Scientific Library]] (GSL) provides a [https://www.gnu.org/software/gsl/doc/html/montecarlo.html?highlight=vegas VEGAS routine].