Communications in Mathematical Sciences

Volume 22 (2024)

Number 4

Scalar auxiliary variable approach in iterative minimization formulation for saddle point search

Pages: 977 – 997

DOI: https://dx.doi.org/10.4310/CMS.2024.v22.n4.a5

Authors

Shuting Gu (Shenzhen Technology University, Shenzhen, China)

Chenxi Wang (Southern University of Science and Technology (SUSTech), Shenzhen, China)

Zhen Zhang (Southern University of Science and Technology (SUSTech), Shenzhen, China)

Abstract

Saddle points have been extensively investigated in the study of activated process in gradient flow driven by free energy. This paper aims to use the iterative minimization formulation (IMF) coupled with scalar auxiliary variable (SAV) approach to locate the transition states of activated processes in the $H^{-1}$ gradient flow, i.e., index-1 saddle points of the corresponding energy in $H^{-1}$ metric. In each cycle of the IMF, we introduce the SAV approach to minimize the auxiliary functional. A general principle of constructing linear, efficient and robust energy stable schemes for this approach is presented. This new SAV based IMF method improves the efficiency of saddle point search and can be implemented easily for different free energies. By conducting some numerical experiments for the Ginzburg-Landau and the Landau-Brazovskii free energies, the efficient performance of the proposed method is validated.

Keywords

saddle points, transition states, scalar auxiliary variable, iterative minimization formulation

2010 Mathematics Subject Classification

65K05, 82B05

Full Text (PDF format)

Received 1 March 2022

Received revised 27 February 2023

Accepted 2 October 2023

Published 12 July 2024