Some principles for mountain pass algorithms, and the parallel distance

TL;DR

This paper discusses principles for mountain pass algorithms, introduces the parallel distance concept, and proposes algorithms that leverage its properties for improved saddle point computation in PDEs and chemistry.

Contribution

It introduces the parallel distance and its quadratic property, guiding the design of mountain pass algorithms with balanced local and global features.

Findings

Parallel distance squared exhibits quadratic behavior.

Algorithms based on parallel distance have balanced local and global properties.

Proposed methods improve saddle point computation efficiency.

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 point out that a good global mountain pass algorithm should have good local and global properties. Next, we define the parallel distance, and show that the square of the parallel distance has a quadratic property. We show how to design algorithms for the mountain pass problem based on perturbing parameters of the parallel distance, and that methods based on the parallel distance have midrange local and global properties.