多重网格重整化
Multigrid Renormalization
摘要：
我们结合多重网格（MG）方法与国家的最先进的概念，从变分制定的数值重整化群。由此产生的MG重整化（MGR）方法是一个自然推广的MG方法求解偏微分方程。当寻求$N$点网格上的解时，我们的MGR方法具有计算成本缩放为$\mathcal{O}(\log(N))$，而不是最佳标准MG方法的$\mathcal{O}(N)$。因此，MGR可以指数级地加速标准MG计算。为了说明我们的方法，我们开发了一种新的算法的基态计算的非线性薛定谔方程。我们的算法变分张量产品和更新张量一个接一个地解决一个局部非线性优化问题。我们比较了几种不同的方法的非线性张量更新，发现牛顿法是最有效的，以及精确的。MGR与我们的非线性基态算法的组合产生精确的结果的非线性薛定谔方程的$N = 10^{18}$网格点在三维空间。
英文摘要：
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 Schrödinger 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 Schrödinger equation on $N = 10^{18}$ grid points in three spatial dimensions.
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