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Poisson's Equation

mattferraro.dev

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Re: Poisson's Equation

#21

Post by burnished »

infogulch wrote: post #8
enriquto wrote: post #7

Earlier quoted context omitted.

Sure! But it's a bit surprising that they do not use the language-provided linear solver and write simply f=A\b

Wow! It seems you have the magical capacity to ingest the reference to an equation and instantly derive an intuition for how it works and what it's useful for. Learning technology this advanced has never been seen before by humankind, I hope you share it with the rest of us!

Everything being referenced is stuff you'd learn in a typical technical education as part of either calculus or statistics. Your reply comes off as of defensive in a way that implies that you're shocked some one would know any of this stuff.

Re: Poisson's Equation

#22

Post by jbay808 »

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/broken-badminton-racket-phot...

Re: Poisson's Equation

#23

Post by wildmanx »

RicoElectrico wrote: post #16
wildmanx wrote: post #9

Earlier quoted context omitted.

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…

I'd rather have more submissions like this on the HN frontpage than the slew of entrepreneurship advice (good or bad), programming-languages-du-jour and disguised content marketing. Anyway, the premise that "everything I see has to appeal to me" would best stay on YouTube where it originated.

I didn't intend to demand "everything I see has to appeal to me". I was merely pointing out that the meaning and impact of a headline are relative to the target audience and this was an example that illustrated the difference. With the effect that somebody somewhat rightfully asked "eh what is this about again"?

The HN crowd has a different average toolbox than the article's author, so the apparent mismatch between headline and article content was confusing.

Re: Poisson's Equation

#24

Post by [deleted] »

enriquto wrote: post #7
phkahler wrote: post #5

Earlier quoted context omitted.

At the end the author mentions there are a large number of methods available for solving. Also mentions there are a much larger set of applications that those discussed.

Sure! But it's a bit surprising that they do not use the language-provided linear solver and write simply f=A\b

[deleted]

Re: Poisson's Equation

#25

Post by burnished »

wildmanx wrote: post #11
pgustafs wrote: post #4

Great post, one nitpick -- I wouldn't say that a matrix is a "sparsely defined" function, but rather a function defined on a finite grid. It might also be worth pointing out that same approach works for any graph, not just a grid.

Also, what's confusing is that algebra usually uses matrices to describe linear functions from n-dimensional to m-dimensional vector spaces. Matrix has n rows, m columns, you give it an n-dim vector and after matrix multiplication you get back an m-dim vector. The author uses a matrix quite differently. You give it two integer coordinates i and j and it gives you the value at position (i, j) back. That's a valid use,…

Can you link to context for this? I learned both in linear algebra, so it seems like either would be just as 'expected'.

Re: Poisson's Equation

#26

Post by xbar »

davidkuhta wrote: post #15
pm90 wrote: post #6

This was a very interesting read even as someone who probably has no practical use for these tools.

> probably can lead you to some fun places.

I am intrigued, but I think I still need a bridge to help me get to practical uses in my my fields involving (largely) non-physical systems.

Re: Poisson's Equation

#27

Post by Jeff_Brown »

EDIT: I'm leaving this here to help anyone else who might have been confused by this, which I imagine is likely.

What confused me is that the author is not treating the matrix as a function from vectors to vectors, as is the customary way to treat matrices as functions. Rather, they're using the matrix to represent a sparse, regular sampling of a function from vectors to scalars.

---

This article makes no sense right off the bat. Here's the first substantive passage:

"[Laplace's Equation means] Find me a function f where every value everywhere is the average of the values around it ... In this post, when we talk about a function f we mean a 2D matrix where each element is some scalar value like temperature or pressure or electric potential ... If it seems weird to call a matrix a function, just remember that all matrices map input coordinates (i,j) to output values f(i,j). Matrices are functions that are just sparsely defined. This particular matrix does satisfy Laplace's equation because each element [of the matrix] is equal to the average of its neighbors."

The values of a function are the outputs it maps its inputs to. The elements of the matrix are neither inputs nor outputs.

Re: Poisson's Equation

#28

Post by freeone3000 »

Jeff_Brown wrote: post #27

EDIT: I'm leaving this here to help anyone else who might have been confused by this, which I imagine is likely. What confused me is that the author is not treating the matrix as a function from vectors to vectors, as is the customary way to treat matrices as functions. Rather, they're using the matrix to represent a sparse, regular sampling of a function from vectors to scalars. --- This article makes no sense right…

The matrix is the function. The elements are the outputs. The coordinates are the inputs.

Take, for example, f(x) = x*x. Its matrix would be: f = [0, 1, 4, 9, 16].

Re: Poisson's Equation

#29

Post by MontyCarloHall »

rawtxapp wrote: post #13
wildmanx wrote: post #9

Earlier quoted context omitted.

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…

Just to point out, it seems like the author themselves submitted the story to HN. Still I agree with your reasoning, but I think slight clickbaity titles do get clicks which is why we keep seeing them.

Yup. This wouldn’t get nearly the same amount of attention if the title were “Poisson’s equation is the most powerful tool in your toolbox for finding steady-state solutions to the heat equation with arbitrarily placed sources”
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