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Poisson's Equation

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Re: Poisson's Equation

#121

Post by mbauman »

hasmanean wrote: post #97
mbauman wrote: post #47

Earlier quoted context omitted.

It's not some obscure symbol; it's just division. It just so happens that we typically "divide" matrices from the left when solving equations like this (and matrix "division" isn't commutative) so instead of `a/b` it's `b\a`.

Yes I know. It’s a solve operator…it’s equivalent to division only in the infinite precision world. The \ operator differs from the / operator in that it doesn’t compute an inverse … it solves the system of equations. Solver algorithms are more numerically stable ( in that you’re much less likely to have large errors due to wacky input data).

Yes, I know. :)

I'd just say that solving the system of equations is the best way to divide by a matrix — that's why I put air quotes around "divide" above. In Julia, right-dividing matrices (with `A/B`) actually does the smart adjoint-commuting thing to left-divide it and do the solve (with `(B'\A')'`).

Re: Poisson's Equation

#122

Post by yngvizzle »

enriquto wrote: post #2

Since you have written it as a symmetric, positive definite, sparse linear system, why don't use a standard solver like CHOLMOD which is available in Julia? (and behind octave's anti-slash operator). It should be faster than the ad-hoc single-scale Gauss-Seidel.

While CHOLMOD is great, you cannot always use the Cholesky factorisation when you solve PDEs. For real-world simulations, we often have to solve systems with hundreds of millions, if not billions, of equations and in then case, even a highly optimised direct solver like CHOLMOD fails. The fill-in simply becomes too large.

For these small test cases, however, simply using CHOLMOD (or any other sparse solver) would do the trick perfectly.

Re: Poisson's Equation

#123

Post by [deleted] »

jbay808 wrote: post #22

A friend of mine broke a badminton racket during a match, and I was struck by how the sharply bent and twisted metal rim was transformed into a smooth, continuously double-curved surface by the racket weave. I looked closely at the balance of tension in the woven cord, thought of how it resembles Poisson's equation, and suddenly it all made sense. Edit - it looked something like this: https://thumbs.dreamstime.com/b/…

[deleted]

Re: Poisson's Equation

#124

Post by _Nat_ »

wildmanx wrote: post #9
zitterbewegung wrote: post #3

Sorry but can there be more context to why it is a powerful tool?

This is a typical issue with HN posts. Some poor soul wrote a somewhat competent and maybe even lengthy blog post / article about something they really care about and are knowledgeable about. It may be directed at a specific audience, or maybe just screaming into the void to record down some insight they had for themselves to read again later, or similar. And they use a more-than-necessary general title like "the bes…

> For an aerospace engineer this all makes a lot of sense and is a super great tool, I'm sure.

The article seems like a decent introduction to the Poisson-equation. But the original title was very misleading.

Because the Poisson-equation isn't some super-useful tool so much as a Day-1 topic discussed in intro-level classes. It's a really simple equation compared to others used in Engineering, Math, and Physics, so it's often introduced as a starting point.

Re: Poisson's Equation

#126

Post by MauranKilom »

> From here we could use Bernoulli's equation to find the pressure distribution, which we could integrate over the surface to find drag and lift and so on. With a few tweaks we could simulate rotational flow, vortex panels, real wing profiles, and so on.

> With just a few simple building block we're already edging up on real computational fluid dynamics. All this just by adding up some matrices!

Correct me if I'm wrong, but the only "real" CFD you could solve with this are incompressible potential flows [0]. Solving Navier Stokes is clearly not just "a few tweaks away" from the Laplace equation, but I would be curious which tweaks would take you to e.g. rotational flows.

[0]: https://en.wikipedia.org/wiki/Potential_flow

Re: Poisson's Equation

#127

Post by H8crilA »

SavantIdiot wrote: post #56

I could have used this 32(!) years ago when I was struggling in college. (This and 3b1b.) It amazes me just how many key topics were so inaccessible to the majority of the class at engineering school. I base this on observations from group study sessions and the hyper-aggressive test curves. I knew lots of people who never got the hang of div/grad/curl, or a Jacobians, or eignenvectors, or Z-transforms... These are k…

What's a "test curve"? How does it make 23/100 be an A grade, is it some kind of CDF-based transformation that makes certain % of students pass the test?

Re: Poisson's Equation

#128

Post by TrackerFF »

MontyCarloHall wrote: post #95
SavantIdiot wrote: post #56

I could have used this 32(!) years ago when I was struggling in college. (This and 3b1b.) It amazes me just how many key topics were so inaccessible to the majority of the class at engineering school. I base this on observations from group study sessions and the hyper-aggressive test curves. I knew lots of people who never got the hang of div/grad/curl, or a Jacobians, or eignenvectors, or Z-transforms... These are k…

> rather than add a curve to a test that makes a 23 out of 100 an "A" grade. For people wondering why fresh college grads they interview somehow have 4.0 averages yet can’t code FizzBuzz during an interview, this is the answer. It’s also why good hiring managers do not even bother to look at GPAs listed on CVs. They are so inflated as to be totally meaningless.

Obviously depends on the school, and how much grade inflation (or deflation) they've gone through. When I studied Electrical Engineering, class average was 2.8 or so. The few students that actually had anything close to a 4.0 were terrific students.

I remember in our real analysis class, the professor started with a comment (to the class) in the lines of:

"This is a demanding class. Top performing students usually spend 25-35 hours a week on the problem sets alone, and top grades are rarely awarded - some years there are zero A's. Please take the weekend to consider if you really need or want to take this class."

FWIW, this was no top University - but then again, grade inflation is not that bad in STEM, from my experience.

Re: Poisson's Equation

#129

Post by mrmuagi »

H8crilA wrote: post #127
SavantIdiot wrote: post #56

I could have used this 32(!) years ago when I was struggling in college. (This and 3b1b.) It amazes me just how many key topics were so inaccessible to the majority of the class at engineering school. I base this on observations from group study sessions and the hyper-aggressive test curves. I knew lots of people who never got the hang of div/grad/curl, or a Jacobians, or eignenvectors, or Z-transforms... These are k…

What's a "test curve"? How does it make 23/100 be an A grade, is it some kind of CDF-based transformation that makes certain % of students pass the test?

That sounds like what it is from my experience as well but it's really up to professor.

One exam was really hard such that your new grade was sqrt(#correct/#total). Another was to make new total the value of highest correct score, e.g. everybody in class got 0-20 out of 100, new total is 20.

Re: Poisson's Equation

#130

Post by gpsx »

jbay808 wrote: post #65
codethief wrote: post #55

Earlier quoted context omitted.

Hmmm, this looks more like a minimal surface, i.e. a solution to the minimal-surface equation[0], than a solution to Poisson's equation. Then again, both equations are of elliptic type. Some links for people who've never heard of minimal surfaces: https://en.wikipedia.org/wiki/Minimal_surface https://minimalsurfaces.blog/ (lots of illustrations) https://makmanx.github.io/math3435s18/talks/MSE.pdf (brief intro with hi…

Isn't Poisson's equation basically describing a minimal surface for small z? I'm not saying the badminton racket follows exactly a (discrete) 2D Poisson equation. But it's certainly related enough to be more than a surface similarly. The cords are under high tension, which means that any curvature along x (that is, dz^2/dx^2) will result in a net z-axis tension force unless balanced by an oppositely curved cord runni…

I think that is good intuition. The transverse force is the second derivative of the transverse position, to first order. But then there are higher order effects in it. I think that is where that math breaks down.
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