Error estimate for semi-implicit method of sphere-constrained high-index saddle dynamics

TL;DR

This paper establishes error estimates for a semi-implicit numerical scheme used in high-dimensional saddle dynamics constrained on spheres, addressing complexities from the scheme's coupled computations and providing theoretical and experimental validation.

Contribution

The paper introduces and proves error estimates for a novel semi-implicit scheme in sphere-constrained saddle dynamics, overcoming analytical challenges from coupled computations and orthonormalization.

Findings

Error estimates are rigorously derived for the semi-implicit scheme.

Numerical experiments confirm the theoretical error bounds.

The method effectively finds saddle points in high-dimensional constrained systems.

Abstract

We prove error estimates for the semi-implicit numerical scheme of sphere-constrained high-index saddle dynamics, which serves as a powerful instrument in finding saddle points and constructing the solution landscapes of constrained systems on the high-dimensional sphere. Due to the semi-implicit treatment and the novel computational procedure, the orthonormality of numerical solutions at each time step could not be fully employed to simplify the derivations, and the computations of the state variable and directional vectors are coupled with the retraction, the vector transport and the orthonormalization procedure, which significantly complicates the analysis. We address these issues to prove error estimates for the proposed semi-implicit scheme and then carry out numerical experiments to substantiate the theoretical findings.