Monte Carlo Methods and Area Estimates - Cornell … · Monte Carlo Methods and Area Estimates CS3220 - Summer 2008 Jonathan Kaldor. Monte Carlo Methods ... as a Monte Carlo simulation

Monte Carlo Methods and Area Estimates

CS3220 - Summer 2008Jonathan Kaldor

Monte Carlo Methods

• In this course so far, we have assumed (either explicitly or implicitly) that we have some clear mathematical problem to solve

• Model to describe some physical process (linear or nonlinear, maybe with some simplifying assumptions)

Monte Carlo Methods

• Suppose we don’t have a good model for the overall process, though (or we wish to validate our model against the process). How can we go about this?

• Suppose we can imitate an experiment (trial, etc). Can we use this to draw conclusions about the overall process?

Monte Carlo Methods

• If we can simulate the experiment on a computer, we can change the data inputs and observe the effects on the results

• In particular, if the experiment involves some random chance, we can run it a bunch of times and accumulate statistics on the outputs

Monte Carlo Methods

• When we simulate a process on a computer that involves random chance, that is known as a Monte Carlo simulation

• One simulation run: particular choices for each of the random choices.

Uses

• Whenever the underlying model is unknown / difficult to compute

• Whenever the inputs are unknown (or are known with a large amount of uncertainty)

Applications

• Particle Physics

• Physical Chemistry

• Finance

• Computer Graphics

Physically Based Rendering

• Rendering an image requires computing the light arriving at each point in the camera

• Light can arrive at the camera after bouncing any number of times - we can look at one set of bounces as a path of light

• Model can be expressed as a complex differential-integral equation that can be very difficult to evaluate

Physically Based Rendering

• We can simulate it using a Monte Carlo approach:

• Simulate a particular path for light

• Do this a bunch (a bunch!) of times

• In the limit, we will end up with the average distribution of light in the scene

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[Moon and Marschner, 2006]

A Simple Example

• Suppose we want to verify some basic rules about probability with rolling a die

• In particular, what are the odds of rolling two 1s in a row?

• Basic probability: 1/36 (1/6 chance of rolling the first 1, another 1/6 chance for the second 1), or p = 0.0278 chance

A Simple Example

• Monte Carlo experiment: roll a die twice, count how many times we roll 1 twice in a row. Divide the number of occurrences by the total number of trials to give us the probability of occurrence

• For 100 trials: 0.0200 For 1000 trials: 0.0310For 10000 trials: 0.0293For 100000 trials: 0.0278

A Simple Example

• We can already see some general properties of Monte Carlo simulations

• Simulating each trial is relatively quick

• ... but we typically need a lot of trials to converge to the correct answer (with only 4 digits of precision to boot!)

Computing Random Numbers

• Monte Carlo simulations require a good source of randomness

• What is randomness to begin with?

• Intuitively: no pattern in the numbers

• May also have some constraints on distribution (either uniform or normally distributed)


• On (almost all) computers, the best we can do is a pseudorandom sequence

• Called so because the process for determining the next number is entirely deterministic, although the resulting sequence “looks” random

• We typically can provide a seed which controls the initial starting point


• That’s not to say that the pseudorandom sequences we get are not random

• They can pass several tests of randomness

• Much more common problem: misuse of random numbers by programmers that makes them less random


• An example: suppose we want to generate random numbers uniformly distributed in a circle with unit diameter

• Uniformly distributed: probability of landing in a particular region depends only on the area of the region (equal area regions will have approximately equal numbers of points inside)


• Here are two algorithms

• 1.) Generate two random numbers r1, r2 uniformly in the unit square. If sqrt(r12 + r22) ≤ 1, return the numbers; otherwise, repeat.

• 2.) Generate two random numbers r1 and r2 uniformly. Return r1/2 cos(2πr2) and r2/2 sin(2πr2)


• These algorithms look like they will both generate points randomly in the circle... but their characteristics are quite different

50% of points

50% of points

Not 50% of area!


• When using randomness in our algorithms, we need to be careful that we are using it correctly

• In particular, that we dont destroy randomness or distribution properties of our sequence when manipulating them

Estimating Areas Via Monte Carlo

• The previous discussion leads to a method for estimating the area / volume of an arbitrarily shaped object

• We only need to be able to test whether or not a point is inside the object


• Procedure: generate a uniformly distributed random point in some enclosing rectangle of area A and test whether it is inside or outside the object

• After n trials, we have p of our points inside our object. The estimated area is then p*A/n



• Why do we need uniformly distributed points in order to get a good estimate of the area / volume?

Estimating Area Via Monte Carlo

• We can use this to compute an estimate for the value of π in a similar way as well

• 10,000 trials: 3.1144100,000 trials: 3.1385

Estimating Integrals Via Monte Carlo

• We can also use Monte Carlo simulation to estimate the value of integrals

• ∫01 f(x) dx ≈ 1/N ∑ f(xi)where we have N uniformly distributed random points in [0, 1]


• Note that this is only over integral bounds from 0 to 1. In general, we have an integral over a to b

• 1/(b-a) ∫ab f(x) dx ≈ 1/N ∑ f(xi)

• Moving the weight to the other side, we get:∫ab f(x) dx ≈ (b-a)/N ∑ f(xi)


• This can be directly extended to multiple integrals:

∫cd ∫ab f(x,y) dx dy ≈ (b-a)(c-d)/N ∑ f(xi, yi)


• In general, this is an extremely slow way of getting “good” estimates (at least in terms of error)

• For integrals, the error is typically decreased with the square root of N (that is, the error is around 1/√N), which is usually nowhere near good quadrature algorithms)


• The benefit of Monte Carlo is with higher dimension multiple integrals, and with extremely complex integrals (like those in rendering)

• It also provides an easy (but slow!) ground truth to compare against approximations

• Rendering!

Documents

Monte Carlo Methods and Area Estimates - Cornell … · Monte Carlo Methods and Area Estimates CS3220 - Summer 2008 Jonathan Kaldor. Monte Carlo Methods ... as a Monte Carlo simulation