87.10.61.143 at 10:39, 8 January 2015

2015-01-08T10:39:47Z

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	~~The '''VEGAS algorithm'''~~, ~~due to G~~. P. ~~Lepage{{Citation needed\|date=August 2013}}~~, 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 ~~graph that make~~ the ~~greatest contribution~~ to ~~the final [[integral]]~~.		Earlier than you decide whether or not stainless steel cookware is worth shopping for, lets first discuss what stainless steel cookware is. Chrome steel is fabricated from an alloy, or a mixture of metals. Most commonly, primary iron with chromium, nickel or another minor metals. The chromium provides rust safety and gives your cookware sturdiness. The nickel supplies rust safety as effectively, and adds a cultured look. Most properly made chrome steel cookware has copper or aluminum added to the bottom of the pan or pot. 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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.~~

	~~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>E_g(f; N)<~~/~~math>,~~

	~~:<math>E_g(f; N) = {1 \over N } \sum_i^N { f(x_i)}~~ / ~~g(x_i)~~ <~~/math~~>.



	The ~~[[variance]]~~ of the ~~new estimate is then~~

	~~:<math>\mathit{Var}_g(f; N) = \mathit{Var}(f/g; N)</math>~~

	~~where <math>\mathit{Var}(f;N)</math>~~ is the ~~variance~~ of the ~~original estimate,~~ <~~math~~>~~\mathit{Var}(f; N) = E(f^2; N)~~ - ~~(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>\mathit{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~~.

	The ~~VEGAS algorithm approximates~~ the ~~exact~~ distribution ~~by making~~ a ~~number~~ of ~~passes over~~ the ~~integration region while histogramming the function f~~. ~~Each histogram~~ is used to ~~define a sampling distribution~~ for ~~the next pass~~. ~~Asymptotically this procedure converges~~ to ~~the desired distribution~~. ~~In order~~ to ~~avoid~~ the ~~number of histogram bins growing like <math>K^d</math> the probability distribution~~ is ~~approximated by~~ a ~~separable function:~~ <~~math~~>~~g(x_1~~, ~~x_2~~, ~~\ldots) = g_1(x_1) g_2(x_2) \cdots</math> so that~~ the ~~number~~ of ~~bins required~~ is ~~only Kd~~. ~~This~~ is ~~equivalent to locating~~ the ~~peaks~~ of the ~~function from~~ the ~~projections of~~ the ~~integrand onto the coordinate axes~~. ~~The efficiency of VEGAS depends~~ on ~~the validity of this assumption~~. ~~It is most efficient~~ when the ~~peaks of the integrand are well-localized~~. ~~If an integrand can be rewritten in~~ a ~~form which~~ is ~~approximately separable this will increase the efficiency~~ of ~~integration with VEGAS.~~

	~~==See also==~~

	* [[Las Vegas algorithm]]
	* [[Monte Carlo integration]]

	~~== References ==~~

	* G.P. Lepage, ~~A New Algorithm~~ for ~~Adaptive Multidimensional Integration, Journal~~ of ~~Computational Physics 27, 192-203, (1978)~~
	* G.~~P. Lepage, VEGAS: An Adaptive Multi~~-~~dimensional Integration Program, Cornell preprint CLNS 80~~-~~447~~, ~~March 1980~~
	* The [http://www.~~gnu~~.~~org/software/gsl GNU Scientific Library] provides VEGAS routines~~

	~~[[Category:Monte Carlo methods]]~~
	~~[[Category:Computational physics]]~~
	~~[[Category:Statistical algorithms]]~~
	~~[[Category:Variance reduction]]~~

2A02:8071:92C:F01:88F:B730:A3FA:7354: /* General conservation law */ fixed reference to formula

2013-12-05T10:35:18Z

General conservation law: fixed reference to formula

New page

The '''VEGAS algorithm''', due to G. P. Lepage{{Citation needed|date=August 2013}}, 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 graph 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.

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>E_g(f; N)</math>,

:<math>E_g(f; N) = {1 \over N } \sum_i^N { f(x_i)} / g(x_i) </math>.

The [[variance]] of the new estimate is then

:<math>\mathit{Var}_g(f; N) = \mathit{Var}(f/g; N)</math>

where <math>\mathit{Var}(f;N)</math> is the variance of the original estimate, <math>\mathit{Var}(f; N) = E(f^2; N) - (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>\mathit{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.

The VEGAS algorithm approximates the exact distribution by making a number of passes over the integration region while histogramming the function f. Each histogram is used to define a sampling distribution for the next pass. Asymptotically this procedure converges to the desired distribution. In order to avoid the number of histogram bins growing like <math>K^d</math> the probability distribution is approximated by a separable function: <math>g(x_1, x_2, \ldots) = g_1(x_1) g_2(x_2) \cdots</math> so that the number of bins required is only Kd. This is equivalent to locating the peaks of the function from the projections of the integrand onto the coordinate axes. The efficiency of VEGAS depends on the validity of this assumption. It is most efficient when the peaks of the integrand are well-localized. If an integrand can be rewritten in a form which is approximately separable this will increase the efficiency of integration with VEGAS.

==See also==

* [[Las Vegas algorithm]]
* [[Monte Carlo integration]]

== References ==

* G.P. Lepage, A New Algorithm for Adaptive Multidimensional Integration, Journal of Computational Physics 27, 192-203, (1978)
* G.P. Lepage, VEGAS: An Adaptive Multi-dimensional Integration Program, Cornell preprint CLNS 80-447, March 1980
* The [http://www.gnu.org/software/gsl GNU Scientific Library] provides VEGAS routines

[[Category:Monte Carlo methods]]
[[Category:Computational physics]]
[[Category:Statistical algorithms]]
[[Category:Variance reduction]]

Finite volume method - Revision history

87.10.61.143 at 10:39, 8 January 2015

2A02:8071:92C:F01:88F:B730:A3FA:7354: /* General conservation law */ fixed reference to formula