Preconditioning Stochastic Galerkin Saddle Point Systems

Semi-infinite minimax fractional programming problems with both inequality and equality constraints are considered. The sets of parametric saddle point conditions are established for a new class of nonconvex differentiable semi-infinite minimax fractional programming problems under(?,?)-invexity assumptions. With the reference to the said concept of generalized convexity, we extend some results of saddle point criteria for a larger class of nonconvex semi-infinite minimax fractional programming problems in comparison to those ones previously established in the literature.

Download Full-text

Bimolecular Reactions, Transition-State Theory

10.1093/oso/9780198805014.003.0006 ◽

2018 ◽

Author(s):

Niels Engholm Henriksen ◽

Flemming Yssing Hansen

Keyword(s):

Transition State ◽

Saddle Point ◽

Transition State Theory ◽

Classical Mechanics ◽

Small Degree ◽

State Theory ◽

Reaction Coordinate ◽

Partition Functions ◽

Bimolecular Reactions ◽

Activated Complex

This chapter discusses an approximate approach—transition-state theory—to the calculation of rate constants for bimolecular reactions. A reaction coordinate is identified from a normal-mode coordinate analysis of the activated complex, that is, the supermolecule on the saddle-point of the potential energy surface. Motion along this coordinate is treated by classical mechanics and recrossings of the saddle point from the product to the reactant side are neglected, leading to the result of conventional transition-state theory expressed in terms of relevant partition functions. Various alternative derivations are presented. Corrections that incorporate quantum mechanical tunnelling along the reaction coordinate are described. Tunnelling through an Eckart barrier is discussed and the approximate Wigner tunnelling correction factor is derived in the limit of a small degree of tunnelling. It concludes with applications of transition-state theory to, for example, the F + H2 reaction, and comparisons with results based on quasi-classical mechanics as well as exact quantum mechanics.

Download Full-text

Quantum Hall effect in a saddle-point system

Physica E Low-dimensional Systems and Nanostructures ◽

10.1016/s1386-9477(98)00108-8 ◽

1998 ◽

Vol 2 (1-4) ◽

pp. 523-526

Author(s):

M.V Budantsev ◽

Z.D Kvon ◽

A.G Pogosov ◽

E.B Olshanetskii ◽

D.K Maude ◽

...

Keyword(s):

Saddle Point ◽

Hall Effect ◽

Quantum Hall Effect ◽

Quantum Hall ◽

Point System ◽

Saddle Point System

Download Full-text

The Improvement of Newton Method and the Validation of Optimization Idea of Blind Walking Repeatedly

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.250-253.4061 ◽

2011 ◽

Vol 250-253 ◽

pp. 4061-4064

Author(s):

Chun Ling Zhang

Keyword(s):

Saddle Point ◽

Newton Method ◽

Optimization Problem ◽

Initial Point ◽

Optimization Method ◽

Maximum Point ◽

Optimal Result ◽

Feasible Domain ◽

Parallel Arithmetic ◽

Newton Direction

The existence of maximum point, oddity point and saddle point often leads to computation failure. The optimization idea is based on the reality that the optimum towards the local minimum related the initial point. After getting several optimal results with different initial point, the best result is taken as the final optimal result. The arithmetic improvement of multi-dimension Newton method is improved. The improvement is important for the optimization method with grads convergence rule or searching direction constructed by grads. A computational example with a saddle point, maximum point and oddity point is studied by multi-dimension Newton method, damped Newton method and Newton direction method. The importance of the idea of blind walking repeatedly is testified. Owing to the parallel arithmetic of modernistic optimization method, it does not need to study optimization problem with seriate feasible domain by modernistic optimization method.

Download Full-text