Mathematical and Numerical Analysis to Shrinking-Dimer Saddle Dynamics with Local Lipschitz Conditions (CSIAM Transactions on Applied Mathematics, 2023)
We present a mathematical and numerical investigation to the shrinking-dimer saddle dynamics for finding any-index saddle points in the solution landscape.Due to the dimer approximation of Hessian in saddle dynamics, the local Lipschitz assumptions and the strong nonlinearity for the saddle dynamics, it remains challengesfor delicate analysis, such as the boundedness of the solutions and the dimer error. Weaddress these issues to bound the solutions under proper relaxation parameters, basedon which we prove the error estimates for numerical discretization to the shrinking-dimer saddle dynamics by matching the dimer length and the time step size. Furthermore, the Richardson extrapolation is employed to obtain a high-order approximation. The inherent reason of requiring the matching of the dimer length and the timestep size lies in that the former serves a different mesh size from the later, and thusthe proposed numerical method is close to a fully-discrete numerical scheme of somespace-time PDE model with the Hessian in the saddle dynamics and its dimer approximation serving as a "spatial operator" and its discretization, respectively, which inturn indicates the PDE nature of the saddle dynamics.