Communications in Mathematical Sciences

Volume 21 (2023)

Number 1

Global convergence of triangularized orthogonalization-free method

Pages: 195 – 218

DOI: https://dx.doi.org/10.4310/CMS.2023.v21.n1.a9

Authors

Weiguo Gao (School of Mathematical Sciences and School of Data Science, Fudan University, Shanghai, China; and Shanghai Artificial Intelligence Laboratory, Shanghai, China)

Yingzhou Li (School of Mathematical Sciences, Fudan University, Shanghai, China)

Bichen Lu (Shanghai Center for Mathematical Sciences, Fudan University, Shanghai, China)

Abstract

This paper proves the global convergence of a triangularized orthogonalization-free method (TriOFM). TriOFM, in general, applies a triangularization idea to the gradient of an objective function and removes the rotation invariance in minimizers. More precisely, in this paper, the TriOFM works as an eigensolver for sizeable sparse matrices and obtains eigenvectors without any orthogonalization step. Due to the triangularization, the iteration is a discrete-time flow in a non-conservative vector field. The global convergence relies on the stable manifold theorem, whereas the convergence to stationary points is proved in detail in this paper. We provide two proofs inspired by the noisy power method and the noisy optimization method, respectively.

Keywords

eigenvalue problem, orthogonalization-free, iterative eigensolver, full configuration interaction

2010 Mathematics Subject Classification

65F15

Received 18 October 2021

Received revised 5 March 2022

Accepted 20 April 2022

Published 27 December 2022