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What is the advantage of this Monte Carlo approach over a typical numerical integration method (like Runge-Kutta)?

by eig 10mo ago

MengerSponge 10mo ago
Typical numerical methods are faster and way cheaper for the same level of accuracy in 1D, but it's trivial to integrate over a surface, volume, hypervolume, etc. with Monte Carlo methods.
- jgalt212 10mo ago
  The writer would have been well served to discuss why he chose Monte Carlo over than summing up all the small trapezoids.
- adrianN 10mo ago
  At least if you can sample the relevant space reasonably accurately, otherwise it becomes really slow.
kens 10mo ago
I was wondering the same thing, but near the end, the article discusses using statistical techniques to determine the standard error. In other words, you can easily get an idea of the accuracy of the result, which is harder with typical numerical integration techniques.
- ogogmad 10mo ago
  Numerical integration using interval arithmetic gets you the same thing but in a completely rigorous way.
- fph 10mo ago
  With many quadrature rules (e.g. trapezoidal rule, Simpson's rule) you have a very cheap error estimator obtained by comparing the results over n and 2n subdivision points.
edschofield 10mo ago
Numerical integration methods suffer from the “curse of dimensionality”: they require exponentially more points in higher dimensions. Monte Carlo integration methods have an error that is independent of dimension, so they scale much better. See, for example, https://ww3.math.ucla.edu/camreport/cam98-19.pdf https://ww3.math.ucla.edu/camreport/cam98-19.pdf
a-dub 10mo ago
as i understand: numerical methods -> smooth out noise from sampling/floating point error/etc for methods that are analytically inspired that are computationally efficient where monte carlo -> computationally expensive brute force random sampling where you can improve accuracy by throwing more compute at the problem.

What is the advantage of this Monte Carlo approach over a typical numerical integration method (like Runge-Kutta)?

by eig 10mo ago

What is the advantage of this Monte Carlo approach over a typical numerical integration method (like Runge-Kutta)?

Typical numerical methods are faster and way cheaper for the same level of accuracy in 1D, but it's trivial to integrate over a surface, volume, hypervolume, etc. with Monte Carlo methods.

jgalt212 10mo ago
The writer would have been well served to discuss why he chose Monte Carlo over than summing up all the small trapezoids.
adrianN 10mo ago
At least if you can sample the relevant space reasonably accurately, otherwise it becomes really slow.

kens 10mo ago

I was wondering the same thing, but near the end, the article discusses using statistical techniques to determine the standard error. In other words, you can easily get an idea of the accuracy of the result, which is harder with typical numerical integration techniques.

ogogmad 10mo ago
Numerical integration using interval arithmetic gets you the same thing but in a completely rigorous way.
fph 10mo ago
With many quadrature rules (e.g. trapezoidal rule, Simpson's rule) you have a very cheap error estimator obtained by comparing the results over n and 2n subdivision points.

edschofield 10mo ago

Numerical integration methods suffer from the “curse of dimensionality”: they require exponentially more points in higher dimensions. Monte Carlo integration methods have an error that is independent of dimension, so they scale much better. See, for example, https://ww3.math.ucla.edu/camreport/cam98-19.pdf https://ww3.math.ucla.edu/camreport/cam98-19.pdf

a-dub 10mo ago

as i understand: numerical methods -> smooth out noise from sampling/floating point error/etc for methods that are analytically inspired that are computationally efficient where monte carlo -> computationally expensive brute force random sampling where you can improve accuracy by throwing more compute at the problem.