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Squaring VEGAS -- Multidimensional Integration Using 2D Correlations

Siemund, Philip LU (2021) FYTK02 20202
Theoretical Particle Physics
Abstract
Multidimensional integrals are used in many areas of physics; electrodynamics, quantum mechanics and statistical physics, to name a few. Many integrals can not be solved analytically, but can instead be approximated numerically. VEGAS is a Monte Carlo integration algorithm, which specializes it in approximating multidimensional integrals. By treating each integration variable independently, VEGAS only offers good approximations if the characteristic regions of the integrand's graph align with the coordinate axes. This thesis presents a modified VEGAS called VEGAS squared, which tracks possible correlations between the integration variables pairwise. When integrating Gaussian functions in two and three dimensions, for which the central... (More)
Multidimensional integrals are used in many areas of physics; electrodynamics, quantum mechanics and statistical physics, to name a few. Many integrals can not be solved analytically, but can instead be approximated numerically. VEGAS is a Monte Carlo integration algorithm, which specializes it in approximating multidimensional integrals. By treating each integration variable independently, VEGAS only offers good approximations if the characteristic regions of the integrand's graph align with the coordinate axes. This thesis presents a modified VEGAS called VEGAS squared, which tracks possible correlations between the integration variables pairwise. When integrating Gaussian functions in two and three dimensions, for which the central parts of the graphs misalign with the coordinate axes, VEGAS squared produces a standard error by a factor of 2.5 less than what VEGAS does. (Less)
Popular Abstract
A definite integral represents the area under the graph of the function, known as the integrand, that is to be integrated. A double definite integral represents a volume and anything beyond represents a hypervolume. The latter integrals are referred to as multidimensional integrals and can be found in many areas of physics; electrodynamics, quantum mechanics and statistical physics, to name a few. Many integrals do not have an exact solution in which the value of the integral can be expressed with a finite number of symbols. These integrals can instead be approximated numerically; through solutions that can be shown to be almost exact.

Monte Carlo integration, named after the Monte Carlo Casino in Monaco, is a numerical integration... (More)
A definite integral represents the area under the graph of the function, known as the integrand, that is to be integrated. A double definite integral represents a volume and anything beyond represents a hypervolume. The latter integrals are referred to as multidimensional integrals and can be found in many areas of physics; electrodynamics, quantum mechanics and statistical physics, to name a few. Many integrals do not have an exact solution in which the value of the integral can be expressed with a finite number of symbols. These integrals can instead be approximated numerically; through solutions that can be shown to be almost exact.

Monte Carlo integration, named after the Monte Carlo Casino in Monaco, is a numerical integration technique in which the integral is approximated by the mean of random evaluations of the integrand. More generally, the method of using random sampling as a means to estimate a deterministic quantity, such as a hypervolume, is known as the Monte Carlo method. This method is said to originate from scientists working at the Manhattan project, where they were challenged with testing their theories about neutron diffusion in fissionable material while simultaneously not affording to run countless experiments. Since then it has become a very popular tool in making theoretical predictions and its applications reach beyond sciences.

The Monte Carlo method uses a brute force approach. Consider rolling two dice. To find out the different probabilities of the outcomes, one can construct a simple table. There will be thirty-six outcomes in total and finding the probabilities from the table is straightforward. This is the mathematical approach and no dice were rolled. Another approach is to actually roll the dice, say ten thousand times, and make a note of the outcome with each roll. This is what the Monte Carlo method does; your results will be approximate, meaning there will be an uncertainty, and they can always be improved by continuing rolling the dice.

The uncertainty in the approximation indicates how well a numerical integration technique performs. There are different strategies to reduce the uncertainty when using Monte Carlo integration. VEGAS is a Monte Carlo integration algorithm that tries to reduce the uncertainty by evaluating the integrand where it is large. This strategy has made VEGAS very efficient in higher dimensions compared to other Monte Carlo integration algorithms using other strategies. However, VEGAS only offers good approximations if the characteristic regions of the integrand’s graph align with the coordinate axes of a given coordinate system. The purpose of this thesis is to modify VEGAS to improve its approximation of integrals for which it performs less well and thereby possibly contribute to a more accurate prediction somewhere. The modified VEGAS is called VEGAS squared. (Less)
Please use this url to cite or link to this publication:
author
Siemund, Philip LU
supervisor
organization
course
FYTK02 20202
year
type
M2 - Bachelor Degree
subject
keywords
statistics, numerical integration, Monte Carlo integration, VEGAS
language
English
id
9042520
date added to LUP
2021-04-01 14:14:19
date last changed
2021-04-01 14:14:19
@misc{9042520,
  abstract     = {{Multidimensional integrals are used in many areas of physics; electrodynamics, quantum mechanics and statistical physics, to name a few. Many integrals can not be solved analytically, but can instead be approximated numerically. VEGAS is a Monte Carlo integration algorithm, which specializes it in approximating multidimensional integrals. By treating each integration variable independently, VEGAS only offers good approximations if the characteristic regions of the integrand's graph align with the coordinate axes. This thesis presents a modified VEGAS called VEGAS squared, which tracks possible correlations between the integration variables pairwise. When integrating Gaussian functions in two and three dimensions, for which the central parts of the graphs misalign with the coordinate axes, VEGAS squared produces a standard error by a factor of 2.5 less than what VEGAS does.}},
  author       = {{Siemund, Philip}},
  language     = {{eng}},
  note         = {{Student Paper}},
  title        = {{Squaring VEGAS -- Multidimensional Integration Using 2D Correlations}},
  year         = {{2021}},
}