电子科技大学：《材料设计与计算 Materials Design and Computation》课程教学资源（课件讲稿）MDC 13 Monte Carlo

Uncertainties Efficiency Scoring Monte Carlo PDFs Sampling RNGs

Monte Carlo Uncertainties Efficiency PDFs RNGs Sampling Scoring

(Pseudo)Random numbers Non-correlated sequences of numbers generated by an iterative equation. Repeatability after a very long number of random numbers. Non-uniform sequence. 0.015 ·Reproducible:“seed', 10k 0.014 1M siue. 100M 0.013 Ij+1=(aI;+c)mod m 0.012 where 0.011 a =663608941 0.01 0.009 C =0 0.008 m 232 0.007 0 0.2 0.4 0.6 0.8 X

(Pseudo) Random numbers • Non-correlated sequences of numbers generated by an iterative equation. • Repeatability after a very long number of random numbers. • Non-uniform sequence. • Reproducible: “seed”. Ij+1 = (aIj + c) mod m where: a = 663608941 c = 0 m = 232

Random numbers >Uniformly distributed numbers in [0,1] Most useful method for obtaining random numbers for computer use is a pseudo random number generator >How random are these pseudo random numbers? Anyone who considers arithmetical methods of producing random digits is,of course,in a state of sin. John von Neumann(1951)

Random Numbers  Uniformly distributed numbers in [0,1]  Most useful method for obtaining random numbers for computer use is a pseudo random number generator  How random are these pseudo random numbers? Anyone who considers arithmetical methods of producing random digits is, of course, in a state of sin. John von Neumann (1951)

Next: Using a computer to generate random events: We need to be able to generate random numbers X with any given probability function f(x),or a given cumulative distribution c(x). 1)Uniformly distributed random numbers 2)General random numbers:can be obtained from a sequence of independent uniform random numbers

Next: Using a computer to generate random events: We need to be able to generate random numbers X with any given probability function f(x), or a given cumulative distribution c(x) . 1) Uniformly distributed random numbers 2) General random numbers: can be obtained from a sequence of independent uniform random numbers

Random number generation Random numbers,uniform distribution f(x) Generation of uniform random numbers 1/(b-a) Definition: a b a)The random variable X is uniformly distributed on the interval [a,b], PX =R [a,b] :distribution density:f()=1/(b-a)x[a,6],E R b)A sequence f(xi),iE I C N,xiE R}is called a sequence of inde- pendent identically distributed uniform random numbers :=Xi(w),with (Xi)ier with p=R [0,1],and all Xi are independent

a b f(x) 1/(b-a) Random number generation

Random numbers,uniform distribution,cont. We will tactically assume that such sequences can be generated on our computers. The issue of producing such(reproducible (!)for code verification)se- quences leads into number theory,and too far away from the applicati- ons. Typical random number generators used today have a near perfect uni- form distribution,are near independent,and have a periodicity determi- ned by the largest integer which can be represented on a given machi- ne. (Note:in massively parallel computing,with many tens of millions samp- les drawn rapidly,this periodicity may require special attention

Algorithms for producing uniform distribution: > Mid-square method > Linear congruential method Nonlinear congruential method > Inversive congruential method > Quadratic congruence method > Cubic congruence method BBS method Fibonacci method > Delayed Fibonacci Method Shift register method > Decimal method

Algorithms for producing uniform distribution:  Mid-square method  Linear congruential method  Nonlinear congruential method  Inversive congruential method  Quadratic congruence method  Cubic congruence method  BBS method  Fibonacci method  Delayed Fibonacci Method  Shift register method  Decimal method

We will see next Any continuous distribution can be generated from uniform random numbers on [0,1] Any discrete distribution can be generated from uniform random numbers on [0,1] Hence: Any given distribution can be generated from uniform random numbers on [0,1]

We will see next: Any continuous distribution can be generated from uniform random numbers on [0,1] Any discrete distribution can be generated from uniform random numbers on [0,1] Hence: Any given distribution can be generated from uniform random numbers on [0,1]

Random numbers,general distributions Transformation from uniform random numbers to random numbers with other distributions (e.g.:Maxwellian distribution,Poisson distr.,etc...) i.e..R[0,1]-P,with distribution P specified. Measure-theory:each "measure"can be decomposed into a weighted sum of three parts: part 1.)has a continuous distribution (with a probability density) part 2.)has a discrete distribution part 3.)is a "pathological"contribution,which is required for the ab- stract mathematical case only (general measurable spaces witho- algebras,....),but which does not occur in practical Monte-Carlo app- lications

Random numbers,general distributions 1.Direct Method: 1)let-oo<a<b<oo R[0,1]:=(b-a)z+a R[a,b] Proof:transformation of random variables,see:Mathematical theory of probability. 2)Discrete uniform distribution ("Laplace distribution")on {j,j+ 1,j+n-1},j∈N R[0,1]X:=[nz]+j is Laplace distributed on {j,j+1,…j+n-1} i.e.P(X=i)=1/n for all i from fj,j+1,....j+n-1 especially:j=0,n=2:coin;j=1,n=6:dice;j=0,n=37:Roulette

1. Direct Method：