Multigrid renormalization

JOURNAL OF COMPUTATIONAL PHYSICS

卷:372

Article

Lubasch, Michael¹  Moinier, Pierre²  Jaksch, Dieter^1,3,4 

[1] Univ Oxford, Clarendon Lab, Parks Rd, Oxford OX1 3PU, England

[2] BAE Syst MAI, Computat Engn, Buckingham House,FPC 267 POB 5, Bristol BS34 7QW, Avon, England

[3] Natl Univ Singapore, Ctr Quantum Technol, 3 Sci Dr 2, Singapore 117543, Singapore

[4] Univ Oxford, Keble Coll, Parks Rd, Oxford OX1 3PG, England

关键词: Multigrid methods;    Numerical renormalization group;    Density matrix renormalization group;    Variational renormalization group methods;    Matrix product states;    Quantics tensor trains;   

DOI  :  10.1016/j.jcp.2018.06.065

来源: Elsevier

【 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 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.

【 授权许可】

Free