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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.
As a hobbyist, I'm playing with analog computer circuits right now. If you can match your curve with a similar voltage profile, a simple analog integrator (an op-amp with a capacitor connected in feedback) will also give you the area under the curve (also as a voltage of course). Analog circuits (and op-amps just generally) are surprising cool. I know, kind of off on a tangent here but I have integration on the brain…
[0] https://www.amazon.com/Electronic-Analog-Computers-D-c/dp/B0...
What is the advantage of this Monte Carlo approach over a typical numerical integration method (like Runge-Kutta)?
See, for example, https://ww3.math.ucla.edu/camreport/cam98-19.pdf
Conversely, good symbolic integration is hard, because you can get stuck and have to try another route through a combinatoric maze. Good symbolic differentiation is easy, because just applying the next obvious operation usually converges.
Huh.
Mandatory XKCD: [1]
What is the advantage of this Monte Carlo approach over a typical numerical integration method (like Runge-Kutta)?
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.
As a hobbyist, I'm playing with analog computer circuits right now. If you can match your curve with a similar voltage profile, a simple analog integrator (an op-amp with a capacitor connected in feedback) will also give you the area under the curve (also as a voltage of course). Analog circuits (and op-amps just generally) are surprising cool. I know, kind of off on a tangent here but I have integration on the brain…
Yep. This is also how you solve differential equations with analog computers. (You need to recast them as integral equations because real-world differentiators are not well-behaved, but it still works.) https://i4cy.com/analog_computing/
Good numerical integration is easy, because summing smooths out noise. Good numerical differentiation is hard, because noise is amplified. Conversely, good symbolic integration is hard, because you can get stuck and have to try another route through a combinatoric maze. Good symbolic differentiation is easy, because just applying the next obvious operation usually converges. Huh. Mandatory XKCD: [1] [1] https://xkcd.…
- Differenting a function composed of simpler pieces always "converges" (the process terminates). One just applies the chain rule. Among other things, this is why automatic differentiation is a thing.
- If you have an analytic function (a function expressible locally as a power series), a surprisingly useful trick is to turn differentiation into integration via the Cauchy integral formula. Provided a good contour can be found, this gives a nice way to evaluate derivatives numerically.
What is the advantage of this Monte Carlo approach over a typical numerical integration method (like Runge-Kutta)?