[onalark]

That's because direct solvers can't scale. If you want to solve a large (distributed over hundreds of nodes) sparse linear algebra problem as fast as possible, decades of research have been poured into efficient techniques (Krylov methods, Multigrid, preconditioners) for solving them iteratively.

[math_and_stuff]

Can't scale in a weak, strong, or asymptotic complexity sense? And for what sorts of problems (I assume you're thinking of 2D and 3D PDEs discretized with local basis functions)?

[onalark]

Yes, I'm thinking of discretizations of elliptic 2D/3D PDEs. They don't scale in the weak or strong sense, and they can't hold O(n log n) asymptotic complexity due to fill-in from Cholesky/LU-style factorizations.