Discretization and index-robust error analysis for constrained high-index saddle dynamics on high-dimensional sphere

TL;DR

This paper develops and analyzes a robust numerical discretization method for constrained high-index saddle dynamics on high-dimensional spheres, addressing complexity and index-robustness issues.

Contribution

It introduces an error analysis framework for constrained high-index saddle dynamics, including an index-robust error estimate in an averaged norm.

Findings

Derived error estimates for numerical discretization.

Provided index-robust error analysis with relaxation parameters.

Supported the accuracy of numerical computations for high-dimensional saddle points.

Abstract

We develop and analyze numerical discretization to the constrained high-index saddle dynamics, the dynamics searching for the high-index saddle points confined on the high-dimensional unit sphere. Compared with the saddle dynamics without constraints, the constrained high-index saddle dynamics has more complex dynamical forms, and additional operations such as the retraction and vector transport are required due to the constraint, which significantly complicate the numerical scheme and the corresponding numerical analysis. Furthermore, as the existing numerical analysis results usually depend on the index of the saddle points implicitly, the proved numerical accuracy may be reduced if the index is high in many applications, which indicates the lack of robustness with respect to the index. To address these issues, we derive the error estimates for numerical discretization of the constrained high-index saddle dynamics on high-dimensional sphere, and then improve it by providing an index-robust error analysis in an averaged norm by adjusting the relaxation parameters. The developed results provide mathematical supports for the accuracy of numerical computations.