Monte Carlo Simulation - YouTube

Hi and welcome to a video about Monte Carlo simulations,

a type of simulation that i really enjoy because

it's a super simple concept and it still allows to solve

incredibly complex problems. So after watching this video you will know

exactly what Monte Carlo simulations are and what types of problems you can solve

with them. So what are Monte Carlo simulations? Well

the name Monte Carlo actually refers to the

city in Monaco which is known for its casinos and gambling,

and in the context of simulations "Monte Carlo" is basically used as a synonym for

randomness, like the randomness in gambling.

So, Monte Carlo simulations are simulations evolving

randomly, and this might seem counterintuitive at first because

how can something useful come out of a randomly evolving simulation?

but we will see why randomness can be actually very useful in a simulation in

just a moment. So let's take a look at the first

example. In the simulation we have a sort of

marble dropping device that moves around randomly above this

rectangular table and drops marbles, and on the table there

are two bowls one with a square cross section and one

with a circular cross section, and each of these bowls is placed on

some sort of scale which displays how many marbles are in

each bowl at this time, and if we let the

simulation evolve for a while

and then divide the number of marbles in the circular bowl

by the number of marbles in the square bowl the result happens to be

roughly pi. Without any advanced math knowledge,

simply by randomly dropping marbles into two bowls,

we can estimate pi. Okay so let's unpack what happened in this simple example of

a Monte Carlo simulation. When we drop a marble in a uniformly

random location then the probability for this marble to

end up in one of the bowls is proportional to the bowls

cross-section area, and if we repeat this process over and

over again then also the number of marbles ending

up in this bowl will be proportional to the bowl's

cross-section area. The area of the square bowl in this

example is its edge length a squared and the area

of the circular bowl is pi times a squared, and that's how we get

pi as the fraction of the two areas and

consequently as the fraction of marbles in each bowl.

So in this example of a monte carlo simulation, we essentially determine an

area by taking random samples. With each random sample we probe whether

this specific location is inside or outside of some area, and by

taking enough samples we get a good idea of how big an area is.

This idea of obtaining random samples is probably quite familiar to you if you

think about how real world studies are designed. Let's

say we wanted to find out the average height of all people

worldwide. Then, in principle we would need to

measure the height of each person worldwide, and then take the

average to get an accurate result, but of course we can't go out and

measure the height of billions of people. So what can we do? Well we can measure

the height of some smaller group of people, and hope

that their average height is a good estimate for the average

height of all people worldwide, but there are two things that we need to

consider. Firstly the selection of the group of people

needs to be unbiased. We can't just measure the

height of the next five people we meet as we might live in an area with

especially tall or short people. A better way to get an unbiased sample

group would be to randomly select people

worldwide. Randomness helps us to make an unbiased

selection. Secondly, the group of measured

people must not be too small. If we only measure the height

of let's say five people we might have just

coincidentally picked five taller or shorter people,

and we can be more and more confident about the average height

the more people we measure. In probability theory this is called

the "law of large numbers" which essentially states that

an average tends to come closer to its expected value

the more samples we have, and the exact same is also true for Monte Carlo

simulations. The core idea of a Monte Carlo simulation is that we can

get an unbiased, representative group of samples

from some large ocean of possibilities if we allow the simulation to evolve

randomly. In the marble dropping example, in principle we would need to test for

every possible location whether a marble ends up inside or

outside the bowl to determine the bowl's cross-section

area precisely, just like we in principle would need to

measure the height of each person worldwide

to determine the average height accurately.

Instead, we can rely on randomly selected samples, and according to the

law of large numbers we can be more and more confident about

the result the more samples we take. We can see this

if we plot the fraction of marbles in both bowls

over time. Initially, the value is fluctuating

heavily, but with more and more samples generated

by the randomly evolving simulation, the fluctuations become smaller and

smaller, and we can see that the fraction approaches the expected

value pi

Now I hope that using a Monte Carlo simulation to determine pi

was an illustrative example, but it's certainly not really relevant,

but actually this animation of dropping marbles

has a second example of a monte carlo simulation somewhat

hidden in it. So let's take a look at this hidden more relevant example of a

Monte Carlo simulation in the last part of this video.

Rendering such animations or the simpler example scene

is all about simulating the flow of light, and to get an accurate image you

need to find out how much light hits different areas of

the scene. This is not an easy task if you

consider that diffuse surfaces can scatter light in

virtually any direction. So let's say we want to find out how

much light hits this marked area. Then, in principle we

would need to evaluate all possible light paths to find out

which fraction of them ends up in the marked area. Now clearly

it's impossible to follow all the possible

light paths. So what can we do if we can't follow all

possible paths? Well I hope that after watching the

video to this point, the answer is quite obvious. If we can't

simulate all possible light paths then the best

we can do is to simulate a representative group of

sample paths, and what is a good way to get an

unbiased representative group of samples? We can employ randomness! So

whenever we hit a diffuse surface we simply pick,

randomly, one of the infinitely many directions

in which the light could be scattered. One randomly simulated light path on its

own is not valuable, but if we generate many

sample paths then we get a good idea of the

illumination of the scene. We will see areas that are hit by many

of the randomly generated light rays (brighter areas) and we'll see other areas

that are hit only by few of the randomly generated rays

(darker areas). Actually if you think about it,

counting the number of randomly generated light rays

hitting some area is really not that different from

counting the number of randomly dropped marbles

hitting some bowl, and in this marble dropping

attempt to determine pi we saw that the result gets more

accurate the more samples we take, and the same is also true for

Monte Carlo path tracing. These four images

show the same scene rendered with a different number of

randomly simulated light paths. We can clearly see

that with more sampled paths the result becomes

more accurate, leading to a less noisy image.

Okay so let's wrap this up. A Monte Carlo simulation is a randomly evolving

simulation, and in this video we have looked at two

examples how such a randomly evolving simulation can be useful. We have

determined pi by randomly dropping marbles into bowls,

and we have simulated the flow of light by generating random light paths.

Both these examples and many other examples where

Monte Carlo simulations are useful, they have one thing in common.

They involve an unfeasibly large number of possibilities,

and instead of trying to go through all these possibilities,

with monte carlo simulations we deliberately only explore

a random subset of them, and according to the law of large numbers we can still

get away with these random samples if we only gather enough of them.

Okay so that's it for this video. I hope you found it useful. If you did,

please leave a like and maybe subscribe and i hope to see

you in the next video! Bye.

you