Book contents
- Frontmatter
- Contents
- Introduction
- 1 Retrospective
- I First Steps Toward the Mountains
- II Reaching the Mountain Pass Through Easy Climbs
- III A Deeper Insight in Mountains Topology
- IV The Landscape Becoming Less Smooth
- V Speculating about the Mountain Pass Geometry
- VI Technical Climbs
- 22 Numerical MPT Implementations
- 23 Perturbation from Symmetry and the MPT
- 24 Applying the MPT in Bifurcation Problems
- 25 More Climbs
- A Background Material
- Bibliography
- Index
22 - Numerical MPT Implementations
Published online by Cambridge University Press: 04 September 2009
- Frontmatter
- Contents
- Introduction
- 1 Retrospective
- I First Steps Toward the Mountains
- II Reaching the Mountain Pass Through Easy Climbs
- III A Deeper Insight in Mountains Topology
- IV The Landscape Becoming Less Smooth
- V Speculating about the Mountain Pass Geometry
- VI Technical Climbs
- 22 Numerical MPT Implementations
- 23 Perturbation from Symmetry and the MPT
- 24 Applying the MPT in Bifurcation Problems
- 25 More Climbs
- A Background Material
- Bibliography
- Index
Summary
The particular form obtained by applying an analytical integration method may prove to be insuitable for practical purposes. For instance, evaluating the formula may be numerically instable (due to cancellation, for instance) or even impossible (due to division by zero).
A.R. Krommer and C.W. Ueberhuber, Computational integration, 1998.This chapter is devoted to some numerical implementations of the MPT. We first present the “mountain pass algorithm” by Choi and McKenna. Then we describe a partially interactive algorithm, also based on the MPT, by Korman for computing unstable solutions. A third algorithm used in quantum chemistry by Liotard and Penot [584] is described in the final notes.
The fact that the solutions obtained by the MPT are unstable makes them very interesting candidates for solving particular problems whose solutions are unstable in nature. But this instability, in addition to the global character of the inf max characterization (we “inf max” on functional spaces that are infinite dimensional), makes them hard to capture numerically. Moreover, the available discretization algorithms are not very useful with domains whose boundaries present some curvature. This may explain, in our opinion, the delayed appearance of a numerical algorithm for computing the solutions obtained by the MPT.
We will begin with the mountain pass algorithm of Choi and McKenna, an algorithm that is beginning to become known by nonlinear analysts and seems to be destined for a bright future.
- Type
- Chapter
- Information
- The Mountain Pass TheoremVariants, Generalizations and Some Applications, pp. 259 - 274Publisher: Cambridge University PressPrint publication year: 2003