Random Number Generators

 Based upon specific mathematical algorithms  Which are repeatable and sequential

Random  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

Problems  Difficult to isolate –Often need to replace current generator –Require  Knowledge of current generator  Sometimes in-depth understanding of random number generators themselves  Large scale tests cause most problems –Needing sometimes millions or billions of random numbers

Desirable Properties  When performing Monte Carlo Simulations –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 physics

Random Number Cycle  Basis –sequence of pseudorandom integers  Some exceptions  Integers (“Fixed”) –Manipulated arithmetically to yield floating point (“real”)  Can be presented in either Integer or Real numbers

Cycle

What Does This Show Us?  Properties of pseudorandom sequences of integers –The sequence has a finite number of integers –The sequence gets traversed in a particular order –The sequence repeats if the period of the generator is exceeded

LCG  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)

Table

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)

Prime Numbers  Using a prime as the divisor for the modulus can result in a period of m- 1 as maximum period  There are cases of prime moduli that fail bitwise testing  Case-by-case basis

Seed  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  Using large 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 32 nd 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.  Plot these 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'  Little or no interprocessor communication  Easy to code

N Streams  N Streams –N independent random numbers –N independent processes  Need to 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)]

LFG  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

Overview  Introduction  History  Examples  Applications  Real Life practices

Introduction  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 algorithms – implemented with substantial programming acumen.

Introduction cont…  Major components of Monte Carlo methods: –Probability distribution functions –Random number generator –Scoring –Error estimation –Variance reduction techniques –Parallelization and vectorization

History  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 ) appeared. –The American mathematicians J. Neyman and S. Ulam are considered its originators.

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

Examples  Simple Example to Understand: computing the area of a plane figure S. –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 randomly

Examples cont…  Applying Randomness to the example: –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, –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.

Applications  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…  help researchers 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 whole.

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 simulations. –QMC gives most accurate method for general quantum many-body systems.

Real Life Practice cont… Weather Weather  Equipment Productivity  Soil Conditions Projects are often associated with a high degree of uncertainty resulting from the unpredictable nature of events

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?  Analytic methods tend to be 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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