Numerical mountain pass solutions of Ginzburg-Landau type equations

Glotov, Dmitry; McKenna, Patrick H.

doi:10.3934/cpaa.2008.7.1345

Cited by 1 publication

(1 citation statement)

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“…The original paper of a mountain pass algorithm to solve partial differential equations is [CM93], and it contains several semilinear elliptic problems. Particular applications in numerical partial differential equations include finding periodic solutions of a boundary value problem modeling a suspension bridge [Fen94] (introduced by [LM91]), studying a system of Ginzburg-Landau type equations arising in the thin film model of superconductivity [GM08], the choreographical 3-body problem [ABT06], and cylinder buckling [HLP06]. Other notable works in computing saddle points for solving numerical partial differential equations include the use of constrained optimization [Hor04], extending the mountain pass algorithm to find saddle points of higher Morse index [DCC99,LZ01], extending the mountain pass algorithm to find nonsmooth saddle points [YZ05], and using symmetry [WZ04,WZ05].…”

Section: Introductionmentioning

confidence: 99%

Some principles for mountain pass algorithms, and the parallel distance

Brereton¹,

Pang²

2012

Preprint

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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.

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