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Poisson's Equation

mattferraro.dev

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Re: Poisson's Equation

#41

Post by chris_st »

prionassembly wrote: post #38

My (unorthodox and somewhat rickety) note-taking gizmo uses Poisson's equation to classify (continuously) entries. Basically the note-taking gizmo is a graph. Nodes are given conceptual masses either through pagerank or betweenness centrality (i.e. either through how many random walks or how many shortest paths cross a node). Then we calculate a potential energy (gravity potential) if we by inverting the graph laplac…

That looks really cool -- I presume it has some way to enter more than a single word/phrase in a node? Not sure linking individual words is useful for me :-) I should just try it, of course...

Re: Poisson's Equation

#42

Post by chris_st »

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…

Except prionassembly's post here shows it's WAY more useful than the original "model physical things" of the original article. Also glad this is here, if only to surface such interesting things!

Re: Poisson's Equation

#43

Post by shoto_io »

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…

Seems like we need a disallow robots.txt equivalent!

Re: Poisson's Equation

#44

Post by mixedmath »

I think this is a beautiful article. There's code, there's math. There are many plotted examples using a variety of different plotting techniques (in total, representing a 2d array as data, or in black/white, or with a terrible 'jet' colormap, or as a 3d terrain.

I very much appreciate this sort of post.

Re: Poisson's Equation

#45

Post by icegreentea2 »

enriquto wrote: post #14
icegreentea2 wrote: post #12

Earlier quoted context omitted.

Wait, how does this work? I'm really rusty, what would the f, A and b correspond to here?

It's the common way to solve a linear system in octave, matlab and julia. You have an invertible square matrix A, a vector b of the same dimension, and you want to find a vector x such that "A*x=b". Then you write "x=A\b", which is like "x=A^(-1)*b" but does not get to compute the full inverse matrix (which is useless).

Sorry, yes I know about the syntax. I'm just struggling with what exactly you would be plugging in. Like with respect to any of the example problems given, what would A and x and B be?

Re: Poisson's Equation

#46

Post by pphysch »

The notion that the simple Laplace solver can scale to grids of arbitrary size, without modification, is a bit misleading for practical purposes. Computational performance will tank or zero out if memory hierarchy constraints are not considered. The author does mention high-performance solutions like multigrid. However, even a basic successive overrelaxation algorithm like the one shown can be partitioned and parallelized, and it is a very good programming exercise to implement a partitioning scheme using MPI or even a low-performance messaging library (or even just optimize for cache sizes on a single device, with no network transport).

Like the transition from the elegant, 5-character Laplace equation to the relatively verbose and complex numerical Julia solver, there is an additional and necessary step in making the numerical solution further scalable with present technology. In particular, the notion that the computational boundaries map nicely to the physical boundaries must be thrown out, because now we must respect the layer of "virtual boundaries" between the partitions.

Re: Poisson's Equation

#47

Post by mbauman »

hasmanean wrote: post #18
enriquto wrote: post #14

Earlier quoted context omitted.

It's the common way to solve a linear system in octave, matlab and julia. You have an invertible square matrix A, a vector b of the same dimension, and you want to find a vector x such that "A*x=b". Then you write "x=A\b", which is like "x=A^(-1)*b" but does not get to compute the full inverse matrix (which is useless).

Don’t forget APL. I can’t say for sure but I imagine this \ operator came from there.

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`.

Re: Poisson's Equation

#48

Post by leephillips »

mbustamanter wrote: post #20
zitterbewegung wrote: post #3

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

It is easier to study from a theoretical point of view (easier that the heat or the wave equation: [1], Ch. 2), and it's easier to implement a solving method. When you are learning the finite element method, this is one of the first examples that people use to test if they got the right implementation. Now, I wonder if the author regards it important in his particular area (aerospace engineering), I'm new to the fiel…

If you take the divergence of the incompressible Navier-Stokes equations you get a Poisson equation for the pressure, with a function of velocity on the RHS. So it’s important in fluid dynamics.

Re: Poisson's Equation

#49

Post by burnished »

mixedmath wrote: post #40
burnished wrote: post #25

Earlier quoted context omitted.

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

Here's a concrete example. The first matrix in the post is f = [[1, 1, 1], [1, 1, 1], [1, 1, 1]]. In linear algebra, we would interpret this as a linear map. A true equation would be f([1, 2, 3]^T) = [6, 6, 6]^T (where I'm using ^T to mean "transpose to a column vector"). But here, the author means f(1, 2) = 1, i.e. the (1,2) coordinate of the matrix is 1.

Thank you! Yes, I agree, thank you for explaining that to me.
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