Roninkoi @roninkoi@mathstodon.xyz

Solve the 2D Poisson's equation
$$
\frac{\partial^2}{\partial x^2} f(x, y) + \frac{\partial^2}{\partial y^2} f(x, y) = g(x, y)
$$
in parallel using OpenMPI: https://github.com/Roninkoi/ParPS
The problem is parallelized using domain decomposition and each piece solved using successive over-relaxation.This can be run on a single PC or a multi-node compute cluster #hpc

@dosnostalgic Thought this looked like Wings, but it actually predates it by 8 years!

RIES (https://mrob.com/pub/ries/index.html) is such a fascinating tool. Sometimes it can even find closed-form solutions from the numerical values of integrals that Mathematica can't do!

Two VGA mode X triangle filling strategies I've been considering. 1. fill one plane at a time 2. fill edges one plane at a time and fill center using all planes at once. The second approach should be faster for large fills, while requiring only one extra plane change

Building a small Ryzen 9 5950X system for light compute workloads

Raymarching gives mirror effects for free, so I've been playing around with reflection. Here's a #shader that draws the main mirror the James Webb Space Telescope #JWST https://www.shadertoy.com/view/sstcW7

Heard someone was wreaking havoc on our department's compute cluster. Super relieved it wasn't my fault 😅

@henryseg Interesting idea! It would probably help to some extent, but eventually it would get noisy again. The noise seems to get worse with large \(K\)

If you're into chaos, you might enjoy this visualization of the standard map
\begin{align}
p_{n+1} = p_n + K \sin \theta_n, \\
\theta_{n+1} = \theta_n + p_{n+1},
\end{align}
with varying \(K\). It's pretty hypnotic https://www.shadertoy.com/view/WdjcW3

@tpfto Thanks! Your posts have helped me many times over the years

As it turns out, there is a better approach using Clenshaw's algorithm! A review of different methods is given in arXiv:1410.1748 [physics.chem-ph]. Now I could get up to high values of \(\ell\) and \(m\) without everything blowing up! From the ALPs we can then simply obtain the SHs as
\[
Y_\ell^m(\theta, \phi) = \sqrt{\frac{(2 \ell + 1) (\ell - m)!}{4\pi (\ell + m)!}} P_\ell^m (\cos \theta) e^{i m \phi}.
\]
The final code can be viewed at https://www.shadertoy.com/view/3dKGWG, which borrows from iq's shader

In 2019, I wanted have spherical harmonics \(Y_\ell^m\) in shader, solved in real time instead of from precomputed functions. After looking at some papers, one approach was to solve them using the hypergeometric function, which gives \(P_\ell^m\) as
\[
P_\ell^m(x) = \frac{1}{\Gamma(1-m)} \frac{(1+z)^{m/2}}{(1-z)^{m/2}} {_2F_1}\left( -n, n + 1, 1-m, \frac{1-x}{2} \right).
\]
This requires computing a series consisting of many factorials, which is slow and results in precision problems