Multigrid Renormalization

TL;DR

This paper introduces a multigrid renormalization (MGR) method that significantly accelerates the solution of PDEs by combining multigrid techniques with variational renormalization, enabling efficient computation on extremely large grids.

Contribution

The paper presents a novel MGR method that reduces computational complexity to logarithmic scale and develops an efficient variational algorithm for nonlinear Schrödinger equation ground states.

Findings

MGR scales as O(log N), exponentially faster than standard MG.

The Newton method is most efficient for tensor updates.

Accurate solutions achieved on grids with 10^18 points in 3D.

Abstract

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.