Topics on Dynamics and Algorithms for Saddle Point Calculation

Description

Transition events appear in many important dynamical and physical systems subject to thermal fluctuations, such as chemical reactions and phase transitions. At low temperatures, by large deviation principle, the rare transitions can be described bytransition path crossing the relevant transition states. Such transition states connect two neighboring local minima as a bottleneck for transition paths and have their own geometric structures. Mathematically, these transition states are index-1 saddle points on the energy surface. Saddle point does not only arise in rare-event transitions, but also has been an important subject in other fields of applied maths, such as optimization, control theory, game theory etc. To streamline ideas and methodology without loss of generality, we first start by considering index-1 saddle points. While traditional gradient descent dynamics fails to find them, many numerical methods based on minimum mode of the Hessian have been developed and applied in the past decade, among which are some of the PI’s previous works, the continuous-time dynamicalsystem gentlest ascent dynamics (GAD) and the discrete. iterative minimization formulation (IMF). These methods have been successfully applied in problems such as atomic models and phase field models. However, new scenarios and challengesalso appear for computational efficiency and robustness, and they raise new critical questions. The main purpose of this project is to build a more general and holistic understanding in theory and construct new innovative methods in numerics for saddle point search problem. New connections and modelling of the existing saddle-point dynamics will be identified and new accelerating numerical methods will be produced. There are three main tasks: (1) To handle the case when the true energy function is too expensive to compute, we propose surrogate models such as Gaussian process and simultaneously combine the GAD, the sequential model training and the expensive data labelling. We attack the crucial question of selecting optimal locations forexpensive evaluation of true model by using innovative ideas from active learning. (2) The explicit connection between the iterative minimization formulation(IMF) for index-k saddle points and the di↵erential game will be identified and the saddle pointon energy surface will be identified as Nash equilibrium of the game. We develop an adjusted gradient method by using symplectic part of Jacobian to reduce the cycling behaviors in search of saddle point. (3) For saddle point of free energy defined onprobability measure space in describing the limit of infinite number of particles, we aim to develop the theory for GAD and IMF on the Wasserstein manifold by using sectional curvature and develop the corresponding numerical schemes.