Lesson 1: Introduction to Monte Carlo

1. Lesson 1: Introduction to Monte Carlo • Go over outline and syllabus • Background and course overview • Statistical Formulae: • Mean • SD of population • SD of mean • A little practice

2. Background and course overview • Monte Carlo methods are a branch of mathematics that involves the stochastic solution of problems. • Experimental approach to solving a problem. (“Method of Statistical Trials”) • When the analyst is trying to use a Monte Carlo approach to estimate a physically measurable variable, the approach breaks itself down into two steps: • Devise a numerical experiment whose expected value would correspond to the desired measurable value . • Run the problem to determine an estimate to this variable. We call the estimate . • Many (probably most) Monte Carlo problems are of the “Hit or Miss” category, which finds the probability of some event occurring. (e.g., hitting a straight flush, neutron escaping through a surface, etc.)

3. BG and Overview (2) • The first step can either be very simple or very complicated, based on the particulars features of the problem. • If the problem is itself stochastic, the experimental design step is very simple: Let the mathematical simulation simulate the problem. This is an analog simulation, since the calculation is a perfect analog to the problem. • Lucky for us, the transport of neutral particles is a stochastic situation. All we HAVE to do to get a guess at a measurable effect from a transport situation is to simulate the stochastic "decisions" that nature makes.

4. BG and Overview (3) • For processes that are NOT inherently stochastic, the experimental design is more complex and generally requires that the analyst: • Derive an equation (e.g., heat transfer equation, Boltzmann transport equation) from whose solution an estimate of the effect of interest can be inferred. • Develop a Monte Carlo method to solve the equation. • In this course we will do BOTH approaches: • 3 weeks of event-based mathematical basic • 4 weeks of optimization of event-based transport methods • 3 weeks function-based, using a more formal “functional analysis” approach to the solution of integral and differential equations (with the BTE as an example)

5. Our First Example: Finding p • Our first example will be a numerical estimation of , based on use of a “hit or miss” approach. We know that the ratio of the area of circle to the area of the square that (just barely) encloses it is: • Knowing this, we can design an experiment that will deliver an expected value of p

6. Our First Example (2) • Choose a point at random inside a 2x2 square by: • Choosing a random number (x1) between -1 and 1 for the x coordinate, and • Choosing a random number (x2) between -1 and 1 for the y coordinate. NOTE: By doing this you have made an implicit “change of variable” to this (which is called a “unit hypercube of order 2”):

7. Our First Example (3) • Score the result of a the trial: Consider a "hit" (score = 4) to be the situation when the chosen point is inside the circle, i.e., a "miss" scoring 0. (Why does a success score 4?) • Run the experiment a large number (N) of times, with the final estimate of the circle's area being an average of the results:

8. Coding • This course will require lots of coding. You need to be able to write code in SOME language. In order of my preference: • FORTRAN • Java • C or C++ • BASIC or QBASIC • MatLab • If you have no better option, program in QBASIC • Very simple syntax • QBASIC available on the public area of the course • Get QBASIC.EXE, QBASIC.HLP, MCBASE.BAS

9. Basic view of MC process • Our basic view of a Monte Carlo process is a black box that has a stream of random numbers (between 0 and 1) as input and a stream of estimates of the effect of interest as output: • Sometimes the estimates can be quite approximate, but with a long enough stream of estimates, we can get a good sample.

10. 3 Formulae • There are three statistical formulae that we will be using over and over in this course: • Our estimate of the expected value, • Our estimate of the variance of the sample. • Our estimate of the variance of the expected value. • You must be able to tell them apart

11. Estimate of the expected value • The first, and most important, deals with the how we gather from the stream of estimates the BEST POSSIBLE estimate of the expected value. The resulting formula for is: • Thus, our best estimate is the unweighted average of the individual estimates. This is not surprising, of course. • Let’s compare with a couple of exact formulae.

12. Mean of continuous distribution • For a continuous distribution, p(x), over a range (a,b) (i.e., x=a to x=b). the true mean, , is the first moment of x: • where we have assumed that p(x) is a true probability density function (pdf), obeying the following:

13. Mean of discrete distribution • For a discrete distribution we choose one of M choices, each of which probability • The equation for the mean is: • Again, we have limitations on the probabilities:

14. Example: p problem • For our example of finding , we were dealing with a binomial distribution (i.e., two possible outcomes): • Outcome 1 = Hit the circle: • Outcome 2 = Miss the circle: • Therefore, the expected value is:

15. Estimate of the sample variance • Variance = Expected squared error • MC estimate of variance:

16. Sample standard deviation • The standard deviation is the square root of the variance. • The same is true of our estimate of it: (Many texts call the estimate the “standard error”) • Recall that we have been talking about properties of the sample distribution: How much the individual estimates differ from each other

17. Example: p problem • Using the same outcomes as before: • Very non-normal distribution

18. Estimate of the variance of mean • Turn our attention to the variance and standard deviation of the mean. • How much confidence we have in the mean that we obtained from N samples • We could estimate this by making many estimates of the mean (each using N independent samples) and do a statistical analysis on these estimates. • To our great relief, we can streamline this process and get an estimate of the mean from a single set of N samples

19. Variance of mean (cont’d) • The text has a derivation showing that the variance of the mean is related to the variance of the distribution by: • Since we do not know the actual variance, we have to use our estimate:

20. Example: p problem • Back to our example of finding , using the probabilities from the previous example, the standard deviation of the mean for a sample of N=10,000 would be: • This brings us to the famous principle of Monte Carlo: Each extra digit of accuracy requires that the problem be run with 100 times as many histories.

21. Markov inequality • Most distributions are not normal • What can we say about the probability of a selection being with 1 s when it is NOT normal? • An upper bound is given by the Chebyshev inequality, but before attacking it, we need to build a tool we will use: The Markov inequality • Thought experiment: If I tell you that a group of people has an average weight of 100 pounds, what can you say about the number that weigh more than 200 pounds?

22. Markov inequality (2)

23. Chebyshev inequality • The Chebyshev applies this to the variance:

24. Chebyshev inequality (2) • The resulting statements you can say are not very satisfying to us (especially since we are used to normal distributions): • Normal: 68.3% within 1 svsChebyshev: ? • Normal: 95.4% within 2 svsChebyshev: ? • Normal: 99.7% within 3 svsChebyshev: ? • But, Chebyshev is extremely valuable to theoretical mathematicians because it proves that the integral over the “tails” is guaranteed to decrease with n, with a limit of 0. • Replace n with its square (it’s just a number!)

25. Law of Large Numbers • Theoretical basis of Monte Carlo is the Law of Large Numbers • LLN: The weighted average value of the function, : • This relates the result of a continuous integration with the result of a discrete sampling. All MC comes from this.

26. Law of Large Numbers (2) • At first glance, this looks like this would be useful for mathematicians trying to estimate integrals, but not particularly useful to us—We are not performing integrations we are simulating physical phenomena • This attitude indicates that you are just not thinking abstractly enough—All Monte Carlo processes are (once you dig down) integrations over a domain of “all possible outcomes” • Our values of “x” are over all possible lives that a particle might lead

27. Central limit theorem • The second most important (i.e., useful) theoretical result for Monte Carlo is the Central Limit Theorem • CLT: The sum of a sufficiently large number of independent identically distributed random variables (i.i.d.) eventually becomes normally distributed as N increases • This is useful for us because we can draw useful conclusions from the results from a large number of samples (e.g., 68.7% within one standard deviation, etc.) • This relates the result of a continuous integration with the result of a discrete sampling. All MC comes from this.