Finding saddle points of mountain pass type with quadratic models on affine spaces

TL;DR

This paper introduces a quadratic model-based algorithm for efficiently finding mountain pass saddle points in numerical PDEs and chemistry, with convergence analysis and numerical validation.

Contribution

It presents a novel quadratic model approach for saddle point computation on affine spaces, including convergence results and algorithm enhancements.

Findings

Algorithm successfully finds saddle points in numerical experiments.

Convergence results support the method's reliability.

Numerical tests demonstrate the method's effectiveness.

Abstract

The problem of computing saddle points is important in certain problems in numerical partial differential equations and computational chemistry, and is often solved numerically by a minimization problem over a set of mountain passes. We propose an algorithm to find saddle points of mountain pass type to find the bottlenecks of optimal mountain passes. The key step is to minimize the distance between level sets by using quadratic models on affine spaces similar to the strategy in the conjugate gradient algorithm. We discuss parameter choices, convergence results, and how to augment the algorithm to a path based method. Finally, we perform numerical experiments to test the convergence of our algorithm.