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
