期刊
JOURNAL OF COMPUTATIONAL PHYSICS
卷 372, 期 -, 页码 587-602出版社
ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.jcp.2018.06.065
关键词
Multigrid methods; Numerical renormalization group; Density matrix renormalization group; Variational renormalization group methods; Matrix product states; Quantics tensor trains
资金
- NQIT (Networked Quantum Information Technologies) Hub of the UK National Quantum Technology Programme
- EPSRC Tensor Network Theory Grant [EP/K038311/1]
- EPSRC [EP/M013243/1, EP/K038311/1, EP/P009565/1] Funding Source: UKRI
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 O(log(N)), as opposed to 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 Schrodinger 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 Schrodinger equation on N = 10(18) grid points in three spatial dimensions. (C) 2018 Elsevier Inc. All rights reserved.
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