# Some principles for mountain pass algorithms, and the parallel distance

@article{Brereton2012SomePF, title={Some principles for mountain pass algorithms, and the parallel distance}, author={Justin Brereton and Chin How Jeffrey Pang}, journal={arXiv: Numerical Analysis}, year={2012} }

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…

## Figures from this paper

## References

SHOWING 1-10 OF 31 REFERENCES

Constrained mountain pass algorithm for the numerical solution of semilinear elliptic problems

- MathematicsNumerische Mathematik
- 2004

A new numerical algorithm for solving semilinear elliptic problems is presented, based on the deformation lemma and the mountain pass theorem in a constrained setting, which finds new numerical solutions in various applications.

Computing mountain passes and transition states

- Computer ScienceMath. Program.
- 2004

The elastic string algorithm is proposed for computing mountain passes in finite-dimensional problems and the convergence properties and numerical performance of this algorithm are analyzed for benchmark problems in chemistry and discretizations of infinite-dimensional variational problems.

An Efficient and Stable Method for Computing Multiple Saddle Points with Symmetries

- Computer ScienceSIAM J. Numer. Anal.
- 2005

An efficient and stable numerical algorithm for computing multiple saddle points with symmetries is developed by modifying the local minimax method, and a principle of invariant criticality is proved for the generalization.

A New Branch of Mountain Pass Solutions for the Choreographical 3-Body Problem

- Mathematics
- 2006

AbstractWe prove the existence of a new branch of solutions of Mountain Pass type for the periodic 3-body problem with choreographical constraint. At first we describe the variational structure of…

Methods for Finding Saddle Points and Minimum Energy Paths

- Chemistry
- 2002

The problem of finding minimum energy paths and, in particular, saddle points on high dimensional potential energy surfaces is discussed. Several different methods are reviewed and their efficiency…

Minimax methods in critical point theory with applications to differential equations

- Mathematics
- 1986

An overview The mountain pass theorem and some applications Some variants of the mountain pass theorem The saddle point theorem Some generalizations of the mountain pass theorem Applications to…

A Local Minimax-Newton Method for Finding Multiple Saddle Points with Symmetries

- Mathematics, Computer ScienceSIAM J. Numer. Anal.
- 2004

It is proved that the Newton method is invariant to symmetries and that such an invariance is insensitive to numerical error and the Newton direction can be easily solved in an invariant subspace.

Numerical mountain pass solutions of Ginzburg-Landau type equations

- Mathematics, Physics
- 2008

We study the numerical solutions of a system of Ginzburg-Landau type equations arising in the thin film model of superconductivity. These solutions are obtained by the Mountain Pass algorithm that…

Cylinder Buckling: The Mountain Pass as an Organizing Center

- MathematicsSIAM J. Appl. Math.
- 2006

The classical problem of the buckling of a long thin axially compressed cylindrical shell is revisited and an estimate of the sensitivity of the shell to imperfections is deduced by examining the energy landscape of the perfect cylinder.