# Level set methods for finding critical points of mountain pass type

@article{Lewis2009LevelSM, title={Level set methods for finding critical points of mountain pass type}, author={Adrian S. Lewis and Chin How Jeffrey Pang}, journal={Nonlinear Analysis-theory Methods \& Applications}, year={2009}, volume={74}, pages={4058-4082} }

## Figures and Tables from this paper

## 8 Citations

### Level set methods for finding saddle points of general Morse index

- Mathematics, Computer Science
- 2010

The convergence of the algorithms to find saddle points of general Morse index in the nonsmooth case are proved, and the local superlinear convergence of another algorithm in the smooth finite dimensional case is proved.

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

- Computer Science, Mathematics
- 2011

An algorithm to find saddle points of mountain pass type to find the bottlenecks of optimal mountain passes by using quadratic models on affine spaces similar to the strategy in the conjugate gradient algorithm is proposed.

### On mountain pass type algorithms

- Mathematics
- 2013

We consider constructive proofs of the mountain pass lemma, the saddle point theorem and a linking type theorem. In each, an initial “path” is deformed by pushing it downhill using a (pseudo)…

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

- Computer Science
- 2012

This work points out that a good global mountain pass algorithm should have good local and global properties, and shows how to design algorithms for the mountain pass problem based on perturbing parameters of the parallel distance, and that methods based on the paralleldistance have midrange local andglobal properties.

### On the Mountain-pass algorithm for the quasi-linear Schrodinger equation

- Mathematics
- 2012

We discuss the application of the Mountain Pass algorithm to the so-called quasi-linear Schrodinger equation, which is naturally associated with a class of nonsmooth functionals so that the classical…

### ACCOMPLISHMENTS AND PLAN PANG CHIN HOW

- Mathematics
- 2011

1. Set-valued analysis and applications in optimization 1 1.1. Background: Marginal functions in optimization 2 1.2. Generalized differentiation of set-valued maps [Pan11b] 2 1.3. Characterizing the…

### Machine learning , model reduction and multiphysics simulations of matter

- Physics
- 2013

The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact…

## References

SHOWING 1-10 OF 47 REFERENCES

### 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.

### 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.

### 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 characterization for computing multiple nonsmooth saddle critical points

- MathematicsMath. Program.
- 2005

A local minimax characterization for multiple nonsmooth saddle critical points in either a Hilbert space or a reflexive Banach space is established in this paper to provide a mathematical justification for numerical algorithm design.

### A Minimax Method for Finding Multiple Critical Points and Its Applications to Semilinear PDEs

- MathematicsSIAM J. Sci. Comput.
- 2001

Based on the local theory, a new local numerical minimax method for finding multiple saddle points is developed and implemented successfully to solve a class of semilinear elliptic boundary value problems for multiple solutions on some nonconvex, non star-shaped and multiconnected domains.

### A bisection algorithm for the numerical Mountain Pass

- Computer Science
- 2004

A constructive proof for the Ambrosetti-Rabinowitz Mountain Pass Theorem is proposed providing an algorithm, based on a bisection method, for its implementation, which improves the one currently used and proposed by Y.S. Choi and P.J. McKenna.

### Convergence Results of a Local Minimax Method for Finding Multiple Critical Points

- Computer ScienceSIAM J. Sci. Comput.
- 2003

First Step 5 in the algorithm is modified with the design of a new stepsize rule that is easier to implement practically and with which convergence results of the numerical minimax method are established for isolated and nonisolated critical points.

### Critical Point Theory and Hamiltonian Systems

- Mathematics
- 1989

1 The Direct Method of the Calculus of Variations.- 2 The Fenchel Transform and Duality.- 3 Minimization of the Dual Action.- 4 Minimax Theorems for Indefinite Functional.- 5 A Borsuk-Ulam Theorem…