Mathematical Modelling Lecture 13 Randomness

Transcription

1 Lecture 13 Randomness

2 Overview of Course Model construction dimensional analysis Experimental input fitting Finding a best answer optimisation Tools for constructing and manipulating models networks, differential equations, integration Tools for constructing and simulating models randomness Real world difficulties chaos and fractals A First Course in Mathematical Modeling by Giordano, Weir & Fox, pub. Brooks/Cole. Today we re in chapters 5 and 6.

3 Aim Introduction To study how random numbers can be generated To see how random numbers can be used

4 In the last lecture we talked about Monte Carlo methods of integration. This uses the fact that: b a f (x)dx = (b a) < f (x) > Often random numbers will come in the range 0 x < 1 so we ll need to rescale our function and integral: b a f (x)dx = 1 0 g(u)du

5 choose a random N-point sample calculate f (x) at each of the N points compute the sample average of f this is our estimate for the integral!

6 How can we get random numbers? Measure a random process e.g. timing of radioactive decay tabulate results every time you use a random number, cross it off table e.g. ERNIE

7 How can we get random numbers? Measure a random process Pseudo-random number generator e.g. linear congruence method, x n+1 = (ax n + b) mod c e.g. a=16807, b=2836, c= gives 2 31 numbers before it repeats To make it 0 x n < 1 just divide by c before use

8 Random processes We can also use random numbers when simulating random events. E.g. tossing a coin: Choose random number 0 x i < 1 if 0 x i < 1 2 heads if 1 2 x i < 1 tails The range of random numbers we assign to each event is given by the cumulative frequencyor cumulative probability.

9 A biased die Introduction A slightly more complex example: the score on an unfair die. score prob. cumulative prob

10 General procedure Calculate or measure the frequency distribution Convert into a cumulative frequency distribution Simulate This kind of problem often occurs as a submodel of a larger problem.

11 Example Petrol station Task is to minimise cost of storing and delivering petrol at a petrol station to meet fluctuating demand. The daily cost is a function of three variables: Storage costs assume constant up to some max. capacity Delivery costs assume constant up to some max. capacity Demand fluctuates need three submodels. We ll concentrate on the demand model.

12 Example Petrol station demand Measure demand over a period of time, e.g days Demand is continuous, so discretise (e.g. 10 intervals) Compute frequency and hence probability Compute cumulative probability Model cumulative probability with empirical methods, e.g. linear splines

13 Example Petrol station demand score freq. (days) prob. cumulative prob

14 Example Petrol station demand

15 Example Petrol station demand

16 Example Petrol station demand Model cumulative frequency with empirical methods, e.g. linear splines Invert empirical model (e.g. splines) to give demand for given probability Run simulations and use inverse model to map random no. 0 x i < 1 to a daily demand

17 Example Petrol station demand

18 Example Petrol station demand Model cumulative frequency with empirical methods, e.g. linear splines Invert empirical model (e.g. splines) to give demand for given probability Run simulations and use inverse model to map random no. 0 x i < 1 to a daily demand Investigate different storage/delivery strategies to Minimise cost (min. excess fuel) Avoid running out of fuel

19 Brownian motion is a simple example of a random process. How can we model it? One approach is as a random walk. e.g. toss a coin, heads move left, tails move right After N steps, how far R N have we moved? The mean displacement will be zero, < R N >= 0 The mean distance is not! < R 2 N > 0

20 1D Example Introduction First, let s look at a directed walk (i.e. not random) Suppose we move distance s per step (s = 1 in our example) R N = Ns, so < R N >= Ns in our example then < R N >= N RN 2 = N2 s 2, so < RN 2 >= N2 s 2 in our example then < RN 2 >= N2

21 1D Example Introduction So for a directed walk it s easy. What about a random walk? Let s simulate it: Use random numbers 0 z < 1 R 0 = 0 If 0 z < 1 2, R N+1 = R N 1 If 1 2 z < 1, R N+1 = R N + 1 We run the simulation for different final values of N

22 1D Example Introduction N < RN 2 >

23 1D Example Introduction How can we model this? Let s suppose < RN 2 > CNp, i.e. ln < RN 2 > = ln C + p ln N Should give a straight(-ish) line on log-log plot In fact least-squares fitting gives c = , p = 1.8 Is that it? No random process need to repeat and average

24 1D example Introduction N < RN 2 > 1 < RN 2 > 10 < RN 2 > 100 < RN 2 > c p So it seems as we average over more runs, c 1 and p 1.

25 2D example Introduction What about 2D? Square lattice, now can move either up/down or left/right each turn (but not both). N < RN 2 > So again it seems, c 1 and p 1.

26 Einstein Introduction In 1905 Einstein showed Brownian motion is related to diffusion Thus < RN 2 > N (i.e. p = 1) is exact See e.g. Feynmann Vol. 1 for proof

27 Onward and upward This model is very flexible can do lots of interesting physics with it Can often model things easily that are v. difficult to solve mathematically e.g. QMC/DMC techniques use this to solve Schrödinger s equation e.g. non-crossing random walks

28 Non-crossing random walks Random walk, but cannot visit any location more than once Boring in 1D, interesting in higher dimensions Model of a polymer excluded volume effect Path now has some kind of memory of where it s been

29 Example polymer 0 z < 1 3 turn right 1 3 z < 2 3 keep going z < 1 turn left 2 3 Very difficult maths Very easy simulation! < R 2 N > N1.4

30 Extensions Introduction There are lots of other things we could do, e.g. Can also vary the step length Model for electron conduction resistance due to scattering off defects However always remember the results have some statistical error σ due to the sampling you need to be careful. σ 1 N

31 Summary Introduction Can get random numbers from tables or pseudorandom generators Used for Monte carlo integration Used for random walks e.g. brownian motion, polymers, ties etc.