Notes

Currently, we (or at least I) don’t know how to do multigrid on general problems, so we’re stuck using conjugate gradient. The problem with conjugate gradient is that it is fundamentally about linear systems: given $\mathrm{Ax}=b$, construct the Krylov subspace $b,\mathrm{Ax},A^{2}x,\dots$ and pick out the best available linear combination. It’s all in terms of linear spaces.

Interesting human scale physics is mostly not about linear spaces: it’s about half-linear subspaces, or linear spaces with inequality constraints. As we start cramming more complicated collision handling into algorithms, these inequalities play a larger and larger role in the behavior, and linearizing everything into a CG solve hurts.

Enter multigrid. Yes, it’s theoretically faster, but the more important thing is that the intuition for why multigrid works is less dependent on linearity: start with a fine grid, smooth it a bit, and then coarsen to handle the large scale cheaply. Why shouldn’t this work on a system that’s only half linear? There are probably a thousand reasons, but happily I haven’t read enough multigrid theory yet to know what they are.

So, I predict that eventually someone will find a reasonably flexible approach to algebraic multigrid that generalizes to LCPs, and we’ll be able to advance beyond the tyranny of linearization.

After extensive testing with glasses of water, I’ve determined that it’s impossible to stably balance one infinitely thin circle on top of another. This is true whether the circles are the same size or different, and does not depend on the mass distribution of the circles. This is because two circles intersect at either zero or two points (unless they are the same, and that still isn’t a stable configuration).

Interestingly, two circles seem to be the only two dimensional simple closed curves for which this property holds; any other pair of curves can be scaled or rotated until they intersect at four points, and I believe these intersections can be arranged so that their convex hull contains the center of mass (for balance). I spent a while on the plane trying to prove this, but no luck yet.

Here are some incredibly beautiful images:

http://www.ted.com/index.php/talks/george_smoot_on_the_design_of_the_universe.html

It feels like I didn’t quite realize how large the universe was. I was imagining that it was very large but mostly empty, trailing off out beyond the galaxies. The universe does not look empty in those pictures; it looks full. Somehow a full universe seems much larger than an empty universe.