I remember learning Monte Carlo integration for the first time in high school and never thought about its actual applications in practice (specifically in graphics). I found this cool link online that shows how monte carlo integration as well as monte carlo simulation can be used in progressive "color rendering".
Gilbert-Han
How does choosing random samples give us any advantage over choosing samples at prespecified intervals? is this a solution to some kind of aliasing?
Carpetfizz
I think specifying an interval assumes a sampling frequency, so if you want to avoid aliasing effects you would want to sample above Nyquist. In order to find the Nyquist frequency of the signal you would have to compute the FFT. Even after you do that you could still be left with a bunch of samples you need to integrate over. The whole point of MCI is to make the computation really fast so I think random sampling is a good choice.
mnicoletti15
@Gilbert-Han Here is a naive attempt at answering this question: One way to view monte-carlo integration is that we average out a bunch of random samples of the function over the region we are integrating over, and then multiply by the area of the region in order to obtain the estimate of the integral of the function.
Because of some law of large numbers like the central limit theorem (a deep property of randomness), in high dimensions the empirical mean converges to the average value of the function faster than the trapezoidal approximation converges to the integral.
So really it may be better because fundamentally the approach is different, as we are truly estimating the average value of the function rather than the integral directly, and using randomness allows us to do this remarkably somewhat efficiently.
I remember learning Monte Carlo integration for the first time in high school and never thought about its actual applications in practice (specifically in graphics). I found this cool link online that shows how monte carlo integration as well as monte carlo simulation can be used in progressive "color rendering".
How does choosing random samples give us any advantage over choosing samples at prespecified intervals? is this a solution to some kind of aliasing?
I think specifying an interval assumes a sampling frequency, so if you want to avoid aliasing effects you would want to sample above Nyquist. In order to find the Nyquist frequency of the signal you would have to compute the FFT. Even after you do that you could still be left with a bunch of samples you need to integrate over. The whole point of MCI is to make the computation really fast so I think random sampling is a good choice.
@Gilbert-Han Here is a naive attempt at answering this question: One way to view monte-carlo integration is that we average out a bunch of random samples of the function over the region we are integrating over, and then multiply by the area of the region in order to obtain the estimate of the integral of the function.
Because of some law of large numbers like the central limit theorem (a deep property of randomness), in high dimensions the empirical mean converges to the average value of the function faster than the trapezoidal approximation converges to the integral.
So really it may be better because fundamentally the approach is different, as we are truly estimating the average value of the function rather than the integral directly, and using randomness allows us to do this remarkably somewhat efficiently.