Monte Carlo Techniques

Transcription

1 Physics Part III: Monte Carlo Methods 40 Monte Carlo Techniques Monte Carlo refers to any procedure that makes use of random numbers. Monte Carlo methods are used in: Simulation of natural phenomena Simulation of experimental appartus Numerical analysis Random Numbers What is a random number? Is 3? No such thing as a single random number. A sequence of random numbers is a set of numbers that have nothing to do with the other numbers in the sequence.

2 Physics Part III: Monte Carlo Methods 4 In a uniform distribution of random numbers in the range [0,], every number has the same chance of turning up. Note that is just as likely as 0000 How to generate a sequence of random numbers. Use some chaotic system. (like balls in a barrel - Lotto 6-49). Use a process that is inherently random: radioactive decay thermal noise cosmic ray arrival Tables of a few million truely random numbers do exist, but this isn t enough for most applications. Hooking up a random machine to a computer is not a good idea. This would lead to irreproducable results, making debugging difficult.

3 Physics Random Number Generation 42 Random Number Generation Pseudo-Random Numbers These are sequences of numbers generated by computer algorithms, usally in a uniform distribution in the range [0,]. To be precise, the alogrithms generate integers between 0 and M, and return a real value: An early example : x n = I n / M Middle Square (John Von Neumann, 946) To generate a sequence of 0 digit integers, start with one, and square it amd then take the middle 0 digits from the answer as the next number in the sequence. eg = so the next number is given by The sequence is not random, since each number is completely determined from the previous. But it appears to be random.

4 Physics Random Number Generation 43 This algorothm has problems in that the sequence will have small numbers lumped together, 0 repeats itself, and it can get itself into short loops, for example: = = = = With more bits, long sequences are possible. 38 bits 750,000 numbers A more complex algorithm does not necessarily lead to a better random sequence. It is better to use an algorithm that is well understood.

5 Physics Random Number Generation 44 Linear Conguential Method (Lehmer, 948) I n+ = (a I n + c) mod m Starting value (seed) = I 0 a, c, and m are specially chosen a, c 0 and m > I 0, a, c A poor choice for the constants can lead to very poor sequences. example: a=c=i o =7, m=0 results in the sequence: 7, 6, 9, 0, 7, 6, 9, 0,... The choice c=0 leads to a somewhat faster algorithm, and can also result in long sequences. The method with c=0 is called: Multiplicative congruential.

6 Physics Random Number Generation 46 With c=0, one cannot get the full period, but in order to get the maximum possible, the following should be satisfied: i) I 0 is relatively prime to m ii) a is a primative element modulo m It is possible to obtain a period of length m-, but usually the period is around m/4. RANDU generator A popular random number generator was distributed by IBM in the 960 s with the algorithm: I n+ = (65539 I n ) mod 2 3 This generator was later found to have a serious problem...

7 Physics Random Number Generation 47 Results from Randu: D distribution Random number Looks okay

8 Physics Random Number Generation 48 Results from Randu: 2D distribution Still looks okay

9 ' Physics $ Random Number Generation 49 Results from Randu: 3D distribution Problem seen when observed at the right angle! Dean Karlen/Carleton University Rev /99

10 Physics Random Number Generation 50 The Marsaglia effect In 968, Marsaglia published the paper, Random numbers fall mainly in the planes (Proc. Acad. Sci. 6, 25) which showed that this behaviour is present for any multiplicative congruential generator. For a 32 bit machine, the maximum number of hyperplanes in the space of d-dimensions is: d= d= d= 6 20 d=0 4 The RANDU generator had much less than the maximum. The replacement of the multiplier from to improves the performance signifigantly.

11 Physics Random Number Generation 52 One way to improve the behaviour of random number generators and to increase their period is to modify the algorithm: I n = (a I n- + b I n-2 ) mod m Which in this case has two initial seeds and can have a period greater than m. RANMAR generator This generator (available in the CERN library, KERNLIB, requires 03 initial seeds. These seeds can be set by a single integer from to 900,000,000. Each choice will generate an independat series each of period, This seems to be the ultimate in random number generators!

12 Physics Simulating General Distributions 68 Simulating General Distributions The simple simulations considered so far, only required a random number sequence that is uniformly distributed between 0 and. More complicated problems generally require random numbers generated according to specic distributions. For example, the radioactive decay of a large number of nuclei (say 0 23 ), each with a tiny decay probability, cannot be simulated using the methods developed so far. It would be far too inecient and require very high numerical precision. Instead, a random number generated according to a Poisson distribution could be used to specify the number of nuclei that disintigrate in some time T. Random numbers following some special distributions, like the Poisson distribution, can be generated using special purpose algorithms, and ecient routines can be found in various numerical libraries. If a special purpose generator routine is not available, then use a general purpose method for generating random numbers according to an arbitrary distribution.

13 Physics Inversion Technique 73 Inversion Technique This method is only applicable for relatively simple distribution functions: First normalize the distribution function, so that it becomes a probability distribution function (PDF). Integrate the PDF analytically from the minimum x to an aritrary x. This represents the probability of chosing avalue less than x. Equate this to a uniform random number, and solve for x. The resulting x will be distributed according to the PDF. In other words, solve the following equation for x, given a uniform random number, : Z x Z xmax x min f(x) dx x min f(x) dx = This method is fully ecient, since each randomnumber gives an x value.

14 Physics Inversion Technique 74 Examples of the inversion technique ) generate x between 0 and 4 according to f(x) =x ; 2 : R x 0 x; 2 dx R 4 0 x; 2 dx = 2x 2 = ) generate x according to x =4 2 2) generate x between 0 and according to f(x) =e ;x : R x 0 e;x dx R 0 e ;x dx = ; e ;x = ) generate x according to x = ; ln( ; ) Note that the simple rejection technique would not work for either of these examples.

15 Physics Rejection Technique 69 Rejection Technique Problem: Generate a series of random numbers, x i,which follow a distribution function f(x). In the rejection technique, a trial value, x trial is chosen at random. It is accepted with a probability proportional to f(x trial ). Algorithm: Choose trial x, given a uniform random number : x trial = x min +(x max ; x min ) Decide whether to accept the trial value: if f(x trial ) > 2 f big then accept where f big f(x) for all x, x min x x max. Repeat the algorithm until a trial value is accepted.

16 Physics Rejection Technique 70 This algorithm can be visualized as throwing darts: f big f(x) x min x max This procedure also gives an estimate of the integral of f(x): I = Z xmax x min f(x) dx n accept n trial f big (x max ; x min )

17 Physics Rejection Technique 72 The rejection algorithm is not ecient if the distribution has one or more large peaks (or poles). In this case trial events are seldomly accepted: f big f(x) x min x max In extreme cases, where there is a pole, f big cannot be specied. This algorithm doesn't work when the range of x is (; +). A better algorithm is needed...