  We combine the multigrid (MG) method with state-of-the-art concepts from the
variational formulation of the numerical renormalization group. The resulting
MG renormalization (MGR) method is a natural generalization of the MG method
for solving partial differential equations. When the solution on a grid of $N$
points is sought, our MGR method has a computational cost scaling as
$\mathcal{O}(\log(N))$, as opposed to $\mathcal{O}(N)$ for the best standard MG
method. Therefore MGR can exponentially speed up standard MG computations. To
illustrate our method, we develop a novel algorithm for the ground state
computation of the nonlinear Schr\"{o}dinger equation. Our algorithm acts
variationally on tensor products and updates the tensors one after another by
solving a local nonlinear optimization problem. We compare several different
methods for the nonlinear tensor update and find that the Newton method is the
most efficient as well as precise. The combination of MGR with our nonlinear
ground state algorithm produces accurate results for the nonlinear
Schr\"{o}dinger equation on $N = 10^{18}$ grid points in three spatial
dimensions.
