Monte Carlo method - Revision history

Overfloater: /* Introduction and history */

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‎Introduction and history

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‎Introduction and history

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‎Introduction and history

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‎Introduction and history

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‎Introduction and history

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‎Introduction and history

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‎Introduction and history

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‎Introduction and history

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‎Introduction and history

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 |}
 Monte Carlo method is a stochastic method using pseudorandom numbers. We can also say it is a class of algorithms for the simulation of systems. We can use it for solving mathematical problems as calculating integrals, particularly multidimensional where conventional methods are not effective. This method has wide range of application - from simulation experiments through computation of definite integrals up to for example solving differential equations. The basic idea of this method is very simple. We want to determine the mean value of the variable that is the result of random action. Let's imagine creation of a computer model of that action and after a sufficient amount of simulations, data may be processed by conventional statistical methods, for example to determine the average and standard deviation.
-Monte Carlo method was formulated and also used during World War II by scientists John von Neumann and Stanislav Ulam in the United States in the development of atomic bomb.  A research of neutron behavior and its feasibility of penetration various substances raised the problem of how to determine the percentage of neutrons in a spray that penetrates for example the water tank of certain dimensions. It wasn't possible to find a practical method to predict further behavior of individual neutron, to predict the entire further "history of his life." Neumann and Ulam used for modeling the prediction of "history of neutron life " Monte Carlo method inspired in roulette wheel technique (from here was the idea for the name Monte Carlo). Using this method it is possible to predict the trajectory of each neutron of the given bundle. Simulation of neutron motion wandering in water and randomly colliding with atoms of hydrogen and oxygen was done as follows:<ref name="slabejova1">SLABEJOVÁ, Danica. Metoda Monte Carlo [online]. 2007 [cit. 2013-02-01]. Bakalářská práce. Masarykova univerzita, Přírodovědecká fakulta. Vedoucí práce Gejza Wimmer. Dostupné z: <http://is.muni.cz/th/151336/prif_b/></ref>
+Monte Carlo method was formulated and also used during World War II by scientists John von Neumann and Stanislaw Ulam in the United States in the development of atomic bomb.  A research of neutron behavior and its feasibility of penetration various substances raised the problem of how to determine the percentage of neutrons in a spray that penetrates for example the water tank of certain dimensions. It wasn't possible to find a practical method to predict further behavior of individual neutron, to predict the entire further "history of his life." Neumann and Ulam used for modeling the prediction of "history of neutron life " Monte Carlo method inspired in roulette wheel technique (from here was the idea for the name Monte Carlo). Using this method it is possible to predict the trajectory of each neutron of the given bundle. Simulation of neutron motion wandering in water and randomly colliding with atoms of hydrogen and oxygen was done as follows:<ref name="slabejova1">SLABEJOVÁ, Danica. Metoda Monte Carlo [online]. 2007 [cit. 2013-02-01]. Bakalářská práce. Masarykova univerzita, Přírodovědecká fakulta. Vedoucí práce Gejza Wimmer. Dostupné z: <http://is.muni.cz/th/151336/prif_b/></ref>

@@ Line 1: / Line 1: @@
 =Introduction and history=
 {| class="wikitable" align=left width="220"
@@ Line 44: / Line 41: @@
 * [http://en.wikipedia.org/wiki/University_of_Florida/ University of Florida]
 |}
 ''"Is known that random event - that neutron is absorbed by hitting the hydrogen atom - will occur in one of hundred possible cases on average. To map the route of neutron is spinned the roulette wheel divided into one hundred parts, and among them only one part is indicating the "neutron absorption by a hydrogen atom." If the roulette wheel stops on that part, it indicates the "end of life" of neutron. In opposite case is possible by the another roulette wheel to determine the direction and speed of the neutron after collision. Then with another roulette wheel is possible to determine the trajectory that neutron will take before the next collision occurs. Simulation of "history of life" of neutron is performed as long until the neutron is absorbed, or until it lose such energy that the final exit of the neutron is no longer interesting for us, or until the neutron manages to escape without damage from the tank. When we will make a large amount of such simulations, we get more or less accurate information about percentage of neutrons that have managed to escape from the tank. The accuracy of this method is determined by mathematical-statistical methods of estimation theory."'' František Fabian, Zdenek Kluiber, Metoda Monte Carlo a možnosti jejího uplatnění, Praha: Prospektrum, 1998, page.13-14 <ref name="fabian">FABIAN, František. Metoda Monte Carlo a možnosti jejího uplatnění. 1. vyd. Praha: Prospektrum, spol.s.r.o., 1998, 148 s. ISBN 80-717-5058-1</ref>

@@ Line 1: / Line 1: @@
 =Introduction and history=
 Monte Carlo method is a stochastic method using pseudorandom numbers. We can also say it is a class of algorithms for the simulation of systems. We can use it for solving mathematical problems as calculating integrals, particularly multidimensional where conventional methods are not effective. This method has wide range of application - from simulation experiments through computation of definite integrals up to for example solving differential equations. The basic idea of this method is very simple. We want to determine the mean value of the variable that is the result of random action. Let's imagine creation of a computer model of that action and after a sufficient amount of simulations, data may be processed by conventional statistical methods, for example to determine the average and standard deviation.
 Monte Carlo method was formulated and also used during World War II by scientists John von Neumann and Stanislav Ulam in the United States in the development of atomic bomb.  A research of neutron behavior and its feasibility of penetration various substances raised the problem of how to determine the percentage of neutrons in a spray that penetrates for example the water tank of certain dimensions. It wasn't possible to find a practical method to predict further behavior of individual neutron, to predict the entire further "history of his life." Neumann and Ulam used for modeling the prediction of "history of neutron life " Monte Carlo method inspired in roulette wheel technique (from here was the idea for the name Monte Carlo). Using this method it is possible to predict the trajectory of each neutron of the given bundle. Simulation of neutron motion wandering in water and randomly colliding with atoms of hydrogen and oxygen was done as follows:<ref name="slabejova1">SLABEJOVÁ, Danica. Metoda Monte Carlo [online]. 2007 [cit. 2013-02-01]. Bakalářská práce. Masarykova univerzita, Přírodovědecká fakulta. Vedoucí práce Gejza Wimmer. Dostupné z: <http://is.muni.cz/th/151336/prif_b/></ref>
@@ Line 42: / Line 22: @@
 * [http://en.wikipedia.org/wiki/Institute_for_Advanced_Study/ Institute for Advanced Study]
 * [http://en.wikipedia.org/wiki/Los_Alamos_National_Laboratory/ Site Y, Los Alamos]
 |}
 ''"Is known that random event - that neutron is absorbed by hitting the hydrogen atom - will occur in one of hundred possible cases on average. To map the route of neutron is spinned the roulette wheel divided into one hundred parts, and among them only one part is indicating the "neutron absorption by a hydrogen atom." If the roulette wheel stops on that part, it indicates the "end of life" of neutron. In opposite case is possible by the another roulette wheel to determine the direction and speed of the neutron after collision. Then with another roulette wheel is possible to determine the trajectory that neutron will take before the next collision occurs. Simulation of "history of life" of neutron is performed as long until the neutron is absorbed, or until it lose such energy that the final exit of the neutron is no longer interesting for us, or until the neutron manages to escape without damage from the tank. When we will make a large amount of such simulations, we get more or less accurate information about percentage of neutrons that have managed to escape from the tank. The accuracy of this method is determined by mathematical-statistical methods of estimation theory."'' František Fabian, Zdenek Kluiber, Metoda Monte Carlo a možnosti jejího uplatnění, Praha: Prospektrum, 1998, page.13-14 <ref name="fabian">FABIAN, František. Metoda Monte Carlo a možnosti jejího uplatnění. 1. vyd. Praha: Prospektrum, spol.s.r.o., 1998, 148 s. ISBN 80-717-5058-1</ref>

@@ Line 23: / Line 23: @@
 Monte Carlo method is a stochastic method using pseudorandom numbers. We can also say it is a class of algorithms for the simulation of systems. We can use it for solving mathematical problems as calculating integrals, particularly multidimensional where conventional methods are not effective. This method has wide range of application - from simulation experiments through computation of definite integrals up to for example solving differential equations. The basic idea of this method is very simple. We want to determine the mean value of the variable that is the result of random action. Let's imagine creation of a computer model of that action and after a sufficient amount of simulations, data may be processed by conventional statistical methods, for example to determine the average and standard deviation.
 Monte Carlo method was formulated and also used during World War II by scientists John von Neumann and Stanislav Ulam in the United States in the development of atomic bomb.  A research of neutron behavior and its feasibility of penetration various substances raised the problem of how to determine the percentage of neutrons in a spray that penetrates for example the water tank of certain dimensions. It wasn't possible to find a practical method to predict further behavior of individual neutron, to predict the entire further "history of his life." Neumann and Ulam used for modeling the prediction of "history of neutron life " Monte Carlo method inspired in roulette wheel technique (from here was the idea for the name Monte Carlo). Using this method it is possible to predict the trajectory of each neutron of the given bundle. Simulation of neutron motion wandering in water and randomly colliding with atoms of hydrogen and oxygen was done as follows:<ref name="slabejova1">SLABEJOVÁ, Danica. Metoda Monte Carlo [online]. 2007 [cit. 2013-02-01]. Bakalářská práce. Masarykova univerzita, Přírodovědecká fakulta. Vedoucí práce Gejza Wimmer. Dostupné z: <http://is.muni.cz/th/151336/prif_b/></ref>
 {| class="wikitable" align=left width="220"
@@ Line 45: / Line 43: @@
 * [http://en.wikipedia.org/wiki/Los_Alamos_National_Laboratory/ Site Y, Los Alamos]
 |}
 Despite fact this is a very simplified example the nature of the Monte Carlo method is described close enough. The simulation would be of course very time consuming, in case it is actually necessary always spin the roulette. However, the authors worked at a time when the development of information technology was already in full progress, therefore they could use for these simulations simple computers (compared to current ones) that simulation time significantly shortened.
 The basic principle of Monte Carlo method has been spoken in 1777. In this year the French natural scientist Georges de Buffon formulated his famous needle problem - known as Buffon's task. It determines the value of the number π by random throwing a needle on paper with painted equidistant parallels. Very simply, it is an experiment, when he throws a needle on sheet of paper. Paper is divided by parallel lines and the needle is the same size as the distance between the lines. The result is that the probability of the needle crosses one of the lines is 2 / π. From this can be deduced π value. In 1901 Lazzerini made 34 080 needle throws and got for number π value 3,1415929 which was really amazingly good result.

@@ Line 14: / Line 14: @@
 '''Institutions:'''
-* Institute for Advanced Study
+* [http://en.wikipedia.org/wiki/Institute_for_Advanced_StudyInstitute/ for Advanced Study]
-* Harvard University
+* [http://en.wikipedia.org/wiki/Harvard_University/ Harvard University]
-* University of Wisconsin
+* [http://en.wikipedia.org/wiki/University_of_Wisconsin%E2%80%93Madison/ University of Wisconsin]
-* Los Alamos National Laboratory
+* [http://en.wikipedia.org/wiki/Los_Alamos_National_Laboratory/ Los Alamos National Laboratory]
-* University of Colorado
+* [http://en.wikipedia.org/wiki/University_of_Colorado_at_Boulder/ University of Colorado]
-* University of Florida
+* [http://en.wikipedia.org/wiki/University_of_Florida/ University of Florida]
 |}
 Monte Carlo method is a stochastic method using pseudorandom numbers. We can also say it is a class of algorithms for the simulation of systems. We can use it for solving mathematical problems as calculating integrals, particularly multidimensional where conventional methods are not effective. This method has wide range of application - from simulation experiments through computation of definite integrals up to for example solving differential equations. The basic idea of this method is very simple. We want to determine the mean value of the variable that is the result of random action. Let's imagine creation of a computer model of that action and after a sufficient amount of simulations, data may be processed by conventional statistical methods, for example to determine the average and standard deviation.
@@ Line 26: / Line 26: @@
 ''"Is known that random event - that neutron is absorbed by hitting the hydrogen atom - will occur in one of hundred possible cases on average. To map the route of neutron is spinned the roulette wheel divided into one hundred parts, and among them only one part is indicating the "neutron absorption by a hydrogen atom." If the roulette wheel stops on that part, it indicates the "end of life" of neutron. In opposite case is possible by the another roulette wheel to determine the direction and speed of the neutron after collision. Then with another roulette wheel is possible to determine the trajectory that neutron will take before the next collision occurs. Simulation of "history of life" of neutron is performed as long until the neutron is absorbed, or until it lose such energy that the final exit of the neutron is no longer interesting for us, or until the neutron manages to escape without damage from the tank. When we will make a large amount of such simulations, we get more or less accurate information about percentage of neutrons that have managed to escape from the tank. The accuracy of this method is determined by mathematical-statistical methods of estimation theory."'' František Fabian, Zdenek Kluiber, Metoda Monte Carlo a možnosti jejího uplatnění, Praha: Prospektrum, 1998, page.13-14 <ref name="fabian">FABIAN, František. Metoda Monte Carlo a možnosti jejího uplatnění. 1. vyd. Praha: Prospektrum, spol.s.r.o., 1998, 148 s. ISBN 80-717-5058-1</ref>
-{| class="wikitable" align=left width="300"
+{| class="wikitable" align=left width="220"
 ! scope="col" width="50px" |John von Neumann
 |-
@@ Line 40: / Line 40: @@
 '''Institutions:'''
-* University of Berlin
+* [http://en.wikipedia.org/wiki/University_of_Berlin/ University of Berlin]
-* Princeton University
+* [http://en.wikipedia.org/wiki/Princeton_University/ Princeton University]
-* Institute for Advanced Study
+* [http://en.wikipedia.org/wiki/Institute_for_Advanced_Study/ Institute for Advanced Study]
-* Site Y, Los Alamos
+* [http://en.wikipedia.org/wiki/Los_Alamos_National_Laboratory/ Site Y, Los Alamos]
 |}
 Despite fact this is a very simplified example the nature of the Monte Carlo method is described close enough. The simulation would be of course very time consuming, in case it is actually necessary always spin the roulette. However, the authors worked at a time when the development of information technology was already in full progress, therefore they could use for these simulations simple computers (compared to current ones) that simulation time significantly shortened.

@@ Line 5: / Line 5: @@
 | [[File:ulam.jpg]]
 |-
-| '''Born:''' April 13, 1909
+| '''Born:'''
-Lemberg, Austria-Hungary (now Lviv, Ukraine)
+* April 13, 1909
-'''Died:''' May 13, 1984 (aged 75)
+'''Died:'''
-Santa Fe, New Mexico
+* May 13, 1984 (aged 75)
-'''Institutions:''' Institute for Advanced Study
+'''Institutions:'''
-Harvard University
+* Institute for Advanced Study
-University of Wisconsin
+* Harvard University
-Los Alamos National Laboratory
+* University of Wisconsin
-University of Colorado
+* Los Alamos National Laboratory
-University of Florida
+* University of Colorado
 |}
 Monte Carlo method is a stochastic method using pseudorandom numbers. We can also say it is a class of algorithms for the simulation of systems. We can use it for solving mathematical problems as calculating integrals, particularly multidimensional where conventional methods are not effective. This method has wide range of application - from simulation experiments through computation of definite integrals up to for example solving differential equations. The basic idea of this method is very simple. We want to determine the mean value of the variable that is the result of random action. Let's imagine creation of a computer model of that action and after a sufficient amount of simulations, data may be processed by conventional statistical methods, for example to determine the average and standard deviation.
@@ Line 28: / Line 31: @@
 | [[File:neumann.gif]]
 |-
-| '''Born:''' December 28, 1903
+| '''Born:'''
-Budapest, Austria-Hungary
+* December 28, 1903
-'''Died:''' February 8, 1957 (aged 53)
+'''Died:'''
-Walter Reed General Hospital
+* February 8, 1957 (aged 53)
-Washington, D.C., United States
+* Walter Reed General Hospital Washington, D.C., United States
 |}
 Despite fact this is a very simplified example the nature of the Monte Carlo method is described close enough. The simulation would be of course very time consuming, in case it is actually necessary always spin the roulette. However, the authors worked at a time when the development of information technology was already in full progress, therefore they could use for these simulations simple computers (compared to current ones) that simulation time significantly shortened.

@@ Line 23: / Line 23: @@
-Define event '''''A''''' - randomly selected point of the unit square is located in a circular sector. Based on the geometric probability, we can write the probability of the event '''''A''''':
+Define event '''''A''''' - randomly selected point of the unit square is located in a circular sector. Based on the geometric probability, we can write the probability of the event '''''A''''':<ref name="fabian">FABIAN, František. Metoda Monte Carlo a možnosti jejího uplatnění. 1. vyd. Praha: Prospektrum, spol.s.r.o., 1998, 148 s. ISBN 80-717-5058-1</ref>
 <math>P(A)=\dfrac{\dfrac{\pi*r^2}{4}}{r^2} }=\frac{\pi}{4}</math>
@@ Line 82: / Line 82: @@
 ==Basic scheme of calculation by Monte Carlo method==
-General basic scheme of calculation by Monte Carlo method is as follows:
+General basic scheme of calculation by Monte Carlo method is as follows:<ref name="drimal">DŘÍMAL, Jiří, TRUNEC, David. Úvod do metody Monte-Carlo. 1. vyd. Brno: UJEP, 1989, 122 s. ISBN 80-210-0022-8</ref>
 * Generate random numbers <math>y_{i}</math> with uniform distribution on the interval (0,1).
 * Transform random numbers <math>y_{i}</math> into random numbers <math>z_{i}</math> with the required distribution.
@@ Line 103: / Line 103: @@
 For size of error we get the following relationship: <math>\epsilon=t_{\infty}\sqrt{\frac{p(1-p)}{n}}</math>, thence <math>n=\frac{p(1-p)}{\epsilon^2}t_{\infty}^2</math>.
-In practice, the value of '''''p''''' is unknown, so we determine number of required attempts '''''n''''' to achieve certain accuracy as follows: choose <math>n_{0}</math> as the value of '''''p = 0.01'''''. We realize <math>n_{0}</math> attempts and calculate approximate value of '''''p'''''. Substitute into the previous formula and calculate <math>n_{1}</math>. If <math>n_{1}</math> is greater than <math>n_{0}</math>, we'll make additional attempts and re-calculate the approximate value of '''''p'''''. If the previous estimate differs substantially from the new one, then specify the value of '''''n''''' again.
+In practice, the value of '''''p''''' is unknown, so we determine number of required attempts '''''n''''' to achieve certain accuracy as follows: choose <math>n_{0}</math> as the value of '''''p = 0.01'''''. We realize <math>n_{0}</math> attempts and calculate approximate value of '''''p'''''. Substitute into the previous formula and calculate <math>n_{1}</math>. If <math>n_{1}</math> is greater than <math>n_{0}</math>, we'll make additional attempts and re-calculate the approximate value of '''''p'''''. If the previous estimate differs substantially from the new one, then specify the value of '''''n''''' again.<ref name="slabejova1">SLABEJOVÁ, Danica. Metoda Monte Carlo [online]. 2007 [cit. 2013-02-01]. Bakalářská práce. Masarykova univerzita, Přírodovědecká fakulta. Vedoucí práce Gejza Wimmer. Dostupné z: <http://is.muni.cz/th/151336/prif_b/></ref>
 ==Monte Carlo method use==
@@ Line 126: / Line 126: @@
 ===Computer graphics===
-Especially Quasi Monte Carlo method is used for rendering 3D models. Graphics programs have a feature that creates light reflections from various surfaces, Monte Carlo is generating them randomly within a hemispherical distribution, Quasi Monte Carlo produces them regularly. Usage - computer games, graphics, animation, special effects, architecture, etc. One of rendering methods is radiosity, based on the law of conservation of energy.
+Especially Quasi Monte Carlo method is used for rendering 3D models. Graphics programs have a feature that creates light reflections from various surfaces, Monte Carlo is generating them randomly within a hemispherical distribution, Quasi Monte Carlo produces them regularly. Usage - computer games, graphics, animation, special effects, architecture, etc. One of rendering methods is radiosity, based on the law of conservation of energy.<ref name="fabian">FABIAN, František. Metoda Monte Carlo a možnosti jejího uplatnění. 1. vyd. Praha: Prospektrum, spol.s.r.o., 1998, 148 s. ISBN 80-717-5058-1</ref>
 =Conclusion=