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

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