A Stability Boundary Based Method for Finding Saddle Points on Potential Energy Surfaces

Abstract

The task of finding saddle points on potential energy surfaces plays a crucial role in understanding the dynamics of a micromolecule as well as in studying the folding pathways of macromolecules like proteins. The problem of finding the saddle points on a high dimensional potential energy surface is transformed into the problem of finding decomposition points of its corresponding nonlinear dynamical system. This paper introduces a new method based on TRUST-TECH (TRansformation Under STability reTained Equilibria CHaracterization) to compute saddle points on potential energy surfaces using stability boundaries. Our method explores the dynamic and geometric characteristics of stability boundaries of a nonlinear dynamical system. A novel trajectory adjustment procedure is used to trace the stability boundary. Our method was successful in finding the saddle points on different potential energy surfaces of various dimensions. A simplified version of the algorithm has also been used to find the saddle points of symmetric systems with the help of some analytical knowledge. The main advantages and effectiveness of the method are clearly illustrated with some examples. Promising results of our method are shown on various problems with varied degrees of freedom.

TRUST-TECH-Based Expectation Maximization for Learning Finite Mixture Models

“…Originally, the basic idea of our algorithm was to find decomposition points on the practical stability boundary. Since each decomposition point connects two local maxima uniquely, it is important to obtain the saddle points from the given local maximum and then move to the next local maximum through this decomposition point [25]. Although this procedure gives a guarantee that the local maximum is not revisited, the computational expense for tracing the stability boundary and identifying the decomposition point is high compared to the cost of applying the EM algorithm directly using the exit point without considering the decomposition point.…”

“…4a). One can use the saddle point tracing procedure described in [25] for applications where the local methods like EM are more expensive.…”

“…The EM algorithm converges to a local maximum ðLM i Þ. By tracing the stability boundary, one can find a decomposition point ð 1 Þ using the procedure described in [25]. The eigenvector direction corresponding to the positive eigenvalue of the Jacobian evaluated at 1 will connect two local maxima.…”

Stability Region Based Expectation Maximization for Model-based Clustering

“…Originally, the basic idea of our algorithm was to find decomposition points on the practical stability boundary. Since, each decomposition point connects two local maxima uniquely, it is important to obtain the saddle points from the given local maximum and then move to the next local maximum through this decomposition point [14]. Though, this procedure gives a guarantee that the local maximum is not revisited, the computational expense for tracing the stability boundary and identifying the decomposition point is high compared to the cost of applying the EM algorithm directly using the exit point without considering the decomposition point.…”

“…Though, this procedure gives a guarantee that the local maximum is not revisited, the computational expense for tracing the stability boundary and identifying the decomposition point is high compared to the cost of applying the EM algorithm directly using the exit point without considering the decomposition point. One can use the saddle point tracing procedure described in [14] for applications where the local methods like EM are much expensive.…”