ABSTRACT
This chapter explores innovative extensions of the Gentlest Ascent Dynamics (GAD) method, designed to efficiently locate transition states in complex rare event systems. The first extension, multiscale Gentlest Ascent Dynamics (MsGAD), is introduced to handle the effective dynamics of slow-fast systems, employing multiscale numerical techniques to ensure computational efficiency and accelerate convergence. A second extension focuses on non-gradient systems, where a simplified GAD variant reduces computational costs by using a single direction variable, thus avoiding the need for costly Jacobian matrix transpose operations. This variant is further adapted for multiscale non-gradient slow-fast systems to improve performance. Additionally, a linear projection operator is introduced to address numerical challenges arising from higher-order spatial derivatives in energy functionals within the H −1 metric. The chapter concludes with numerical experiments that validate the proposed methods, demonstrating their effectiveness and computational advantages across various scenarios involving stochastic ordinary differential equations and partial differential equations. These innovations offer substantial improvements in both accuracy and efficiency, making them valuable tools for rare event studies in complex systems.
