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We develop and analyze numerical discretization to the constrained high-indexsaddle dynamics, the dynamics searching for the high-index saddle pointsconfined on the high-dimensional unit sphere. Compared with the saddle dynamicswithout constraints, the constrained high-index saddle dynamics has morecomplex dynamical forms, and additional operations such as the retraction andvector transport are required due to the constraint, which significantlycomplicate the numerical scheme and the corresponding numerical analysis.Furthermore, as the existing numerical analysis results usually depend on theindex of the saddle points implicitly, the proved numerical accuracy may bereduced if the index is high in many applications, which indicates the lack ofrobustness with respect to the index. To address these issues, we derive theerror estimates for numerical discretization of the constrained high-indexsaddle dynamics on high-dimensional sphere, and then improve it by providing anindex-robust error analysis in an averaged norm by adjusting the relaxationparameters. The developed results provide mathematical supports for theaccuracy of numerical computations.

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