Multigrid is the future

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 $Ax = b$, construct the Krylov subspace $b, Ax, A^2 x, \ldots$ 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.

Visualizing collisions