Meshless methods based on collocation with radial basis functions (RBFs) are investigated in detail in this paper. Both globally supported and compactly supported radial basis functions are used with collocation to solve partial differential equations (PDEs). Using RBFs as a meshless collocation method to solve PDEs possesses some advantages. It is a truly mesh-free method, and is space dimension independent. Furthermore, in the context of scattered data interpolation it is known that some radial basis functions have spectral convergence orders.This study shows that the accuracy of derivatives of interpolating functions are usually very poor on boundary of domain when a direct collocation method is used, therefore it will result in signi®cant error in solving a PDE with Neumann boundary conditions. Based on this fact, a Hermite type collocation method is proposed in this paper, in which both PDEs and prescribed traction boundary conditions are imposed on prescribed traction boundary. Numerical studies shows that the Hermite type collocation method improve the accuracy signi®cantly.
SUMMARYA ÿnite point method, least-squares collocation meshless method, is proposed. Except for the collocation points which are used to construct the trial functions, a number of auxiliary points are also adopted. Unlike the direct collocation method, the equilibrium conditions are satisÿed not only at the collocation points but also at the auxiliary points in a least-squares sense. The moving least-squares interpolant is used to construct the trial functions. The computational e ort required for the present method is in the same order as that required for the direct collocation, while the present method improves the accuracy of solution signiÿcantly. The proposed method does not require any mesh so that it is a truly meshless method. Three numerical examples are studied in detail, which show that the proposed method possesses high accuracy with low computational e ort.
We explore data-driven methods for gaining insight into the dynamics of a two-population genetic algorithm (GA), which has been effective in tests on constrained optimization problems. We track and compare one population of feasible solutions and another population of infeasible solutions. Feasible solutions are selected and bred to improve their objective function values. Infeasible solutions are selected and bred to reduce their constraint violations. Interbreeding between populations is completely indirect, that is, only through their offspring that happen to migrate to the other population. We introduce an empirical measure of distance, and apply it between individuals and between population centroids to monitor the progress of evolution. We find that the centroids of the two populations approach each other and stabilize. This is a valuable characterization of convergence. We find the infeasible population influences, and sometimes dominates, the genetic material of the optimum solution. Since the infeasible population is not evaluated by the objective function, it is free to explore boundary regions, where the optimum is likely to be found. Roughly speaking, the No Free Lunch theorems for optimization show that all blackbox algorithms (such as Genetic Algorithms) have the same average performance over the set of all problems. As such, our algorithm would, on average, be no better than random search or any other blackbox search method. However, we provide two general theorems that give conditions that render null the No Free Lunch results for the constrained optimization problem class we study. The approach taken here thereby escapes the No Free Lunch implications, per se. Ó 2007 Published by Elsevier B.V.
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