Feynman vs. Computer
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Feynman vs. Computer
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Re: Feynman vs. Computer
#2What is the advantage of this Monte Carlo approach over a typical numerical integration method (like Runge-Kutta)?
Re: Feynman vs. Computer
#3As 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 lately. You say "4 lines of Python", and I say "1 op-amp".)
Re: Feynman vs. Computer
#4> I hear that in electronics and quantum dynamics, there are sometimes integrals whose value is not a number, but a function, and knowing that function is important in order to know how the thing it’s modeling behaves in interactions with other things.
I'd be interested in this. So finding classical closed form solutions is the actual thing desired there?
Re: Feynman vs. Computer
#5What 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.
Re: Feynman vs. Computer
#6An integral trick I picked up from a lecturer at university: if you know the result has to be of the form ax^n for some a that's probably rational and some integer n but you're feeling really lazy and/or it's annoying to simplify (even for mathematica), just plug in a transcendental value for x like Zeta[3].
Then just divide by powers of that irrational number until you have something that looks rational. That'll give you a and n. It's more or less numerical dimensional analysis.
It's not that useful for complicated integrals, but when you're feeling lazy it's a fucking godsend to know what the answer should be before you've proven it.
EDIT: s/irrational/transcendental/
Re: Feynman vs. Computer
#7As 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.)
Re: Feynman vs. Computer
#8> I hear that in electronics and quantum dynamics, there are sometimes integrals whose value is not a number, but a function, and knowing that function is important in order to know how the thing it’s modeling behaves in interactions with other things. I'd be interested in this. So finding classical closed form solutions is the actual thing desired there?
I think what the author was alluding to was the path integral formulation [of quantum mechanics] which was advanced in large part by Feynman.
It's not that finding closed form solutions is what matters (I don't think most path integrals would have closed form solutions), but that the integration is done over the space of functions, not over Euclidian space (or a manifold in Euclidian space, etc...)
Re: Feynman vs. Computer
#9What 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.
The writer would have been well served to discuss why he chose Monte Carlo over than summing up all the small trapezoids.
Re: Feynman vs. Computer
#10What 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.