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We present a mathematical and numerical investigation to the shrinkingdimersaddle dynamics for finding any-index saddle points in the solution landscape.Due to the dimer approximation of Hessian in saddle dynamics, the localLipschitz assumptions and the strong nonlinearity for the saddle dynamics, itremains challenges for delicate analysis, such as the the boundedness of thesolutions and the dimer error. We address these issues to bound the solutionsunder proper relaxation parameters, based on which we prove the error estimatesfor numerical discretization to the shrinking-dimer saddle dynamics by matchingthe dimer length and the time step size. Furthermore, the Richardsonextrapolation is employed to obtain a high-order approximation.  The inherent reason of requiring the matching of the dimer length and thetime step size lies in that the former serves a different mesh size from thelater, and thus the proposed numerical method is close to a fully-discretenumerical scheme of some spacetime PDE model with the Hessian in the saddledynamics and its dimer approximation serving as a "spatial operator" and itsdiscretization, respectively, which in turn indicates the PDE nature of thesaddle dynamics.

Mathematical and numerical analysis to shrinking-dimer saddle dynamics with local Lipschitz conditions