The objective of this paper is to establish that RBF-PUM is viable for parabolic PDEs of convection-diffusion type. Therefore, research is currently directed towards localized RBF approximations such as the RBF partition of unity collocation method (RBF-PUM) proposed here. For global RBF methods, one of the major disadvantages is the computational cost associated with the dense linear systems that arise. Numerical experiments illustrating the use of our algorithms are included.read more read lessĪbstract: Meshfree methods based on radial basis function (RBF) approximation are of interest for numerical solution of partial differential equations (PDEs) because they are flexible with respect to geometry, they can provide high order convergence, they allow for local refinement, and they are easy to implement in higher dimensions. The former method can be viewed as an efficient alternative to ridge regression or smoothing spline approximation, while the latter forms an extension of the classical polynomial pseudo-spectral approach. We discuss extensions of this approach that can be applied in the setting of iterated approximate moving least squares approximation of function value data and for RBF pseudo-spectral methods for the solution of partial differential equations. The method of cross validation has long been used in the statistics literature, and the special case of leave-one-out cross validation forms the basis of the algorithm for choosing an optimal value of the shape parameter proposed by Rippa in the setting of scattered data interpolation with RBFs. In most papers the authors end up choosing this shape parameter by trial and error or some other ad hoc means. Abstract: Many radial basis function (RBF) methods contain a free shape parameter that plays an important role for the accuracy of the method.
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