Rescaled localized radial basis functions and fast decaying polynomial reproduction

openApproximating a set of data can be a difficult task but it is very useful in applications. Through a linear combination of basis functions we want to reconstruct an unknown quantity from partial information. We study radial basis functions (RBFs) to obtain an approximation method that is meshless, provides a data dependent approximation space and generalization to larger dimensions is not an obstacle. We analyze a rational approximation method with compactly supported radial basis functions (Rescaled localized radial basis function method). The method reproduces exactly the constants and the density of the interpolation nodes influences the support of the RBFs. There is a proof of the convergence in a quasi-uniform setting up to a conjecture: we can determine a lower bound for the approximant of the constant function 1 uniformly with respect to the size of the support of the kernel. We investigate the statement of the conjecture and bring some practical and theoretical results to support it. We study the Runge phenomenon on the approximant and obtain uniform estimates on the cardinal functions. We extend the distinguishing features of the method reproducing exactly larger polynomial spaces. We replace local polynomial reproduction with basis functions that decrease rapidly and approximate exactly a polynomial space. This change releases the basis functions from the compactness of the support and guarantees the same convergence rate (the oversampling problem does not appear). The rescaled localized radial basis function method can be interpreted in this new framework because the cardinal functions have global support even if the kernel has compact support. The decay of the basis functions undertake convergence and stability. In this analysis the smoothness of the approximant is not important, what matters is the "locality" provided by the fast decay. With a moving least squares approach we provide an example of a smooth quasi-interpolant. We continue trying to improve the performance of the method even when the weight functions do not have compact support. All the new theoretical results introduced in this work are also supported by numerical evidence.Approximating a set of data can be a difficult task but it is very useful in applications. Through a linear combination of basis functions we want to reconstruct an unknown quantity from partial information. We study radial basis functions (RBFs) to obtain an approximation method that is meshless, provides a data dependent approximation space and generalization to larger dimensions is not an obstacle. We analyze a rational approximation method with compactly supported radial basis functions (Rescaled localized radial basis function method). The method reproduces exactly the constants and the density of the interpolation nodes influences the support of the RBFs. There is a proof of the convergence in a quasi-uniform setting up to a conjecture: we can determine a lower bound for the approximant of the constant function 1 uniformly with respect to the size of the support of the kernel. We investigate the statement of the conjecture and bring some practical and theoretical results to support it. We study the Runge phenomenon on the approximant and obtain uniform estimates on the cardinal functions. We extend the distinguishing features of the method reproducing exactly larger polynomial spaces. We replace local polynomial reproduction with basis functions that decrease rapidly and approximate exactly a polynomial space. This change releases the basis functions from the compactness of the support and guarantees the same convergence rate (the oversampling problem does not appear). The rescaled localized radial basis function method can be interpreted in this new framework because the cardinal functions have global support even if the kernel has compact support. The decay of the basis functions undertake convergence and stability. In this analysis the smoothness of the approximant is not important, what matters is the "locality" provided by the fast decay. With a moving least squares approach we provide an example of a smooth quasi-interpolant. We continue trying to improve the performance of the method even when the weight functions do not have compact support. All the new theoretical results introduced in this work are also supported by numerical evidence

Error bound for radial basis interpolation in terms of a growth function

We suggest an improvement of Wu-Schaback local error bound for radial basis interpolation by using a polynomial growth function. The new bound is valid without any assumptions about the density of the interpolation centers. It can be useful for the localized methods of scattered data fitting and for the meshless discretization of partial differential equation

Analysis of moving least squares approximation revisited

In this article the error estimation of the moving least squares approximation is provided for functions in fractional order Sobolev spaces. The analysis presented in this paper extends the previous estimations and explains some unnoticed mathematical details. An application to Galerkin method for partial differential equations is also supplied.Comment: Journal of Computational and Applied Mathematics, 2015 Journal of Computational and Applied Mathematic

Enhancing SPH using moving least-squares and radial basis functions

In this paper we consider two sources of enhancement for the meshfree Lagrangian particle method smoothed particle hydrodynamics (SPH) by improving the accuracy of the particle approximation. Namely, we will consider shape functions constructed using: moving least-squares approximation (MLS); radial basis functions (RBF). Using MLS approximation is appealing because polynomial consistency of the particle approximation can be enforced. RBFs further appeal as they allow one to dispense with the smoothing-length -- the parameter in the SPH method which governs the number of particles within the support of the shape function. Currently, only ad hoc methods for choosing the smoothing-length exist. We ensure that any enhancement retains the conservative and meshfree nature of SPH. In doing so, we derive a new set of variationally-consistent hydrodynamic equations. Finally, we demonstrate the performance of the new equations on the Sod shock tube problem.Comment: 10 pages, 3 figures, In Proc. A4A5, Chester UK, Jul. 18-22 200

Modelling of radionuclide migration through the geosphere with radial basis function method and geostatistics

The modelling of radionuclide transport through the geosphere is necessary in the safety assessment of repositories for radioactive waste. A number of key geosphere processes need to be considered when predicting the movement of radionuclides through the geosphere. The most important input data are obtained from field measurements, which are not available for all regions of interest. For example, the hydraulic conductivity, as input parameter, varies from place to place. In such cases geostatistical science offers a variety of spatial estimation procedures. To assess the a long term safety of a radioactive waste disposal system, mathematical models are used to describe the complicated groundwater flow, chemistry and potential radionuclide migration through geological formations. The numerical solution of partial differential equations (PDEs) has usually been obtained by finite difference methods (FDM), finite element methods (FEM), or finite volume methods (FVM). Kansa introduced the concept of solving PDEs using radial basis functions (RBFs) for hyperbolic, parabolic and elliptic PDEs. The aim of this study was to present a relatively new approach to the modelling of radionuclide migration through the geosphere using radial basis functions methods and to determine the average and sample variance of radionuclide concentration with regard to spatial variability of hydraulic conductivity modelled by a geostatistical approach. We will also explore residual errors and their influence on optimal shape parameters