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									Random Number
 Random Number Generators
 Based  upon specific mathematical
 Which are repeatable and sequential
 Truly   Random
  – Exhibiting true randomness
 Pseudorandom
  – Appearance of randomness but having a
    specific repeatable pattern
 Quasi-random
  – Having a set of non-random numbers in
    a randomized order
 Difficult   to isolate
  – Often need to replace current generator
  – Require
      Knowledge of current generator
      Sometimes in-depth understanding of
       random number generators themselves
 Largescale tests cause most
  – Needing sometimes millions or billions
    of random numbers
      Desirable Properties
 When performing Monte Carlo
 – Attributes of each particle should be
   independent of those attributes of any
   other particle
 – Fill the entire attribute space in a
   manner which is consistent with the
     Random Number Cycle
 Basis
  – sequence of pseudorandom integers
     Some   exceptions
 Integers   (“Fixed”)
  – Manipulated arithmetically to yield
    floating point (“real”)
 Canbe presented in either Integer or
 Real numbers
   What Does This Show Us?
          of pseudorandom
 Properties
 sequences of integers
  – The sequence has a finite number of
  – The sequence gets traversed in a
    particular order
  – The sequence repeats if the period of
    the generator is exceeded
 Most   commonly used RNG
  – Linear Congruential Generator
     Requires initial “seed” denoted as X0
     Appears random because of Modulo function

     Next “random” number depends heavily on
      previous X
         – Typical of linear, congruential generators
         – Restricts period
Equations - LCG
                      Using LCG
 Choosing Correct Input is Key
 LCG (a,c,m,X0)
    – LCG (5, 1, 16, 1)
       Yields   –
                1,6,15,12,13,2,11,8,9,14,7,4,5,10,3,0,
                1,6,15,12,13,2,11,8,9,14,…
        When the next result depends upon only the
        previous integer, the longest period possible is P=M
       Odd/Even pattern

       lack of randomness results from using a power of two
        for M
Cycle LCG(5,1,16,1)
              Example #2
 LCG(5,0,16,1)
  – Yields - 1,5,9,13,1,5,9,13,…
  – M is a power of 2 (here: 2^4)
     C=0

     Maximum   period is going to be 2^(m-2)
  – Correlation (each differ by 4)
Cycle (5,0,16,1)
 Using   the date and time
  – Enter the date and time into an
    equations and return an integer then
    make sure it is odd
  – Standard seed for these equations
 Overflow & Negative Numbers
 Usinglarge values of a and large
 values of M are needed
  – Often 31 bits long
     On   32 bit machines
  – A*M results in 62 bit number
  – Overflow
     Can   result in 32nd bit being a negative
     N-Tuple Generalization
 Choose    R1 and R2
 – Choose Rn and R(n+1)
 – Then plot this point of interest in a
   surrounding area.
    Plotthese points in succession
    The area will be uniformly covered by the
     LCG in a “random” order
      – Covering of only part of the unit or certain areas
        of the unit would prove to be not useful for Monte
        Carlo Methods
      Embarrassingly Parallel'
       or no interprocessor
 Little
 Easy to code
               N Streams
N   Streams
 – N independent random numbers
 – N independent processes

 Needto find N seeds far away from
 each other on the cycle
                 Find Seeds
   Find Seeds
    – LCG rule successively applied:
    Lagged Fibonacci Generators
   Increasingly popular
    – Lags are k and l
    – M is power of 2
        With proper choice of k and L
        Period of Generator can be
          – [(2^L)-1] * [2^(m-1)]
 Computationally   simple
  – Integer add
  – Logical AND
  – Decrement of 2 array pointers

  – Must keep L words current in memory
  – LCG needs only one
                  LFG (cont)
 LFG are an attempt to improve LCG
 Similar to Combined LCG
    – Take 2 previous numbers in the sequence to
      produce a new number
    – Where p and q are the “lags”
    – Some arithmetic computation is performed
    – Then mod that answer for the next number
Monte Carlo Methods

 Introduction

 History

 Examples

 Applications

 Real   Life practices
 Define   Monte Carlo Method
  – The Monte Carlo method is a numerical
    method for solving mathematical
    problems using stochastic sampling.
  – It performs simulation of any process
    whose development is influenced by
    random factors, but also if the given
    problem involves no chance, the method
    enables artificial construction of a
    probabilistic model.
            Introduction cont…
   Similarly, Monte Carlo methods randomly
    select values to create scenarios of a
    problem. These values are taken from
    within a fixed range and selected to fit a
    probability distribution [e.g. bell curve,
    linear distribution, etc.]. This is like rolling
    a dice. The outcome is always within the
    range of 1 to 6 and it follows a linear
    distribution - there is an equal opportunity
    for any number to be the outcome.
         Introduction cont…
 MC method is often referred to as the
  “method of last resort”, as it is apt to
  consume large computing resources;
 Characteristics:
   – consuming vast computing resources
   – have historically had to be executed
     upon the fastest computers available at
     the time
   – and employ the most advanced
   – implemented with substantial
     programming acumen.
          Introduction cont…
      components of Monte Carlo
 Major
  – Probability distribution functions
  – Random number generator
  – Scoring
  – Error estimation
  – Variance reduction techniques
  – Parallelization and vectorization
   Where does Monte Carlo method come
    from? When? Who?
    – The name "Monte Carlo" comes from the city
      of Monte Carlo in the principality of Monaco,
      famous for its gambling house
    – Birth date of the Monte Carlo method is 1949,
      when an articale entitled "The Monte Carlo
      Method"( by N. Metropolis and S. Ulam )
    – The American mathematicians J. Neyman and
      S. Ulam are considered its originators.
History cont…
            History cont…
 The theoretical foundation of the method
  had been known long before first articles
  were published.
 Well before 1949 certain problems in
  statistics were sometimes solved by
  means of random sampling
 However, simulation of random variables
  by hand is a laborious process
 Use of the Monte Carlo method as a
  universal numerical technique became
  practical only with the advent of
  computers and high-quality
  pseudorandom number generators
               History cont…
   Buffon's needle problem
    – In 1768 Buffon, a French mathematician,
      experimentally determined a value of π by
      casting a needle on a ruled grid
 Lord Rayleigh even delved into this field
  near the turn of the century.
 Fredericks and Levy in 1928 showed how
  the method could be used to solve
  boundary value problems
 Enrico Fermi in the 1930's used Monte
  Carlo in the calculation of neutron
  diffusion (involving nuclear reactors )
           History cont…
 In the 1940's, a formal foundation
  for the Monte Carlo method was
  developed by von Neumann (PDE)
 Stanislaw Ulam realized the
  importance of the digital computer in
  the implementation of the approach
  from collaboration results of the work
  on the Manhattan project during
  World War II
       Example to Understand:
 Simple
 computing the area of a plane figure
  – completely arbitary figure with a
    curvilinear boundary, given graphically
    or analytically, connected or consisting
    of several pieces
  – assume that it is contained completely
    within the unit square.
            Examples cont…

Figure S in the unit square, being covered with sampling points
          Examples cont…
         Randomness to the
 Applying
  – Choose at random N points in the
    square and designate the number of
    points lying inside S by N'. It is
    geometrically obvious that the area of S
    is approximately equal to the ratio N'/N.
    The greater the N, the greater the
    accuracy of this estimate.
          Examples cont…
 Buffon's   Needle:
  – A simple Monte Carlo method for the
    estimation of the value of π, 3.1415926
  – Assumptions:
     Suppose   you have a tabletop with a number
      of parallel lines drawn on it, which are
      equally spaced (say the spacing is 1 inch,
      for example).
     Suppose you also have a pin or needle,
      which is also an inch long.
             Examples cont…
   Dropping needles on the tablet:
    – The needle crosses or touches one of the lines
    – The needle crosses no lines
 Keep dropping this needle over and over
  on the table
 Record the statistics.
    – Keep track of both the total number of times
      that the needle is randomly dropped on the
      table N, and the number of times that it
      crosses a line N’.
          Examples cont…
 Findings:
  – 2N/N’=π
  – Because, the probability on any given
    drop of the needle that it should cross a
    line is given by 2/pi
  – After many tries, N/N’ will approach the
    probability number.
 Monte  Carlo methods can help in
  design and prediction of behavior of
  systems in nuclear applications and
  radiation physics
 The use of MC in the area of nuclear
  power has undergone an important
  evolution. Notable are the extensions
  to compute burnup in reactor cores,
  and full core neutronic simulations.
         Applications cont…
 helpresearchers understand the
 probability of the occurrence of an
 adverse effect associated with
 exposures to chemicals. Monte Carlo
 sampling simulates the distribution
 of total exposures, by simulating
 random samples of factors
 associated with each exposure route
 and accumulating them to arrive at
 an individual total exposure.
          Applications cont…
   The use of MC methods to model physical
    problems allows us to examine more
    complex systems than we otherwise can.
    Solving equations which describe the
    interactions between two atoms is fairly
    simple; solving the same equations for
    hundreds or thousands of atoms is
    impossible. With MC methods, a large
    system can be sampled in a number of
    random configurations, and that data can
    be used to describe the system as a
       Applications cont…
 Random   numbers generated by the
  computer are used to simulate
  naturally random processes
 many previously intractable
  thermodynamic and quantum
  mechanics problems have been
  solved using Monte Carlo techniques
        Real Life Practice
 Quantum   Monte Carlo
 – The microscopic world is described by
   quantum mechanics. We need to use
   simulation techniques to “solve” many-
   body quantum problems.
 – Both the wavefunction and expectation
   values are determined by the
 – QMC gives most accurate method for
   general quantum many-body systems.
    Real Life Practice cont…

• Weather
 Equipment Productivity

 Soil Conditions

Projects are often associated with a
  high degree of uncertainty resulting
  from the unpredictable nature of
       Real Life Practice cont…
   Risk Analysis and Risk Management
    – Monte Carlo Simulation is a valuable modeling
      tool that generates multiple scenarios
      depending upon the data and the assumptions
      fed into the model.
    – Simulation calculates multiple scenarios by
      repeatedly inserting different sampling values
      from probability distribution for the uncertain
      variables into the computerized spread-sheet.
    – probability or percentage chance that a
      particular forecast value will fall within a
      certain specified range.
Why has the Monte Carlo method
     become so popular?

         methods tend to be
 Analytic
 prohibitive (but some very difficult
 problems have finally been solved
 using MC)
 Monte Carlo is somewhat intuitive
 (and several good books have now
 been written on the subject)
 Computers continue to get faster
 and cheaper

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