Comparison of Moving Least Squares and RBF+poly for Interpolation and Derivative Approximation

Bayona, Víctor

doi:10.1007/s10915-019-01028-8

Cited by 32 publications

(16 citation statements)

References 40 publications

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“…Santos et al [48] compared the performance of stabilized flat Gaussian RBFs with PHS-RBF for two dimensional Poisson equation and observed that the PHS approach is more robust and computationally more efficient than the stabilized Gaussian RBFs. Bayona [49] compared a local weighted least squares approach with polynomial basis and the PHS-RBF method for interpolation and derivative approximation. The choice of weighting function is a difficulty in the least squares approach and was found to fail at high polynomial degrees.…”

Section: Introductionmentioning

confidence: 99%

A High-Order Accurate Meshless Method for Solution of Incompressible Fluid Flow Problems

Shahane,

Radhakrishnan,

Vanka

2020

Preprint

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Meshless solution to differential equations using radial basis functions (RBF) is an alternative to grid based methods commonly used. Since it does not need an underlying connectivity in the form of control volumes or elements, issues such as grid skewness that adversely impact accuracy are eliminated.Gaussian, Multiquadrics and inverse Multiquadrics are some of the most popular RBFs used for the solutions of fluid flow and heat transfer problems. But they have an additional shape parameter that has to be fine tuned for accuracy and stability. Moreover, they also face stagnation error when the point density is increased for accuracy. Recently, Polyharmonic splines (PHS) with appended polynomials have been shown to solve the above issues and give rapid convergence of discretization errors with the degree of appended polynomials. In this research, we extend the PHS-RBF method for the solution of incompressible Navier-Stokes equations. A fractional step method with explicit convection and explicit diffusion terms is combined with a pressure Poisson equation to satisfy momentum and continuity equations. Systematic

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Section: Introductionmentioning

confidence: 99%

A High-Order Accurate Meshless Method for Solution of Incompressible Fluid Flow Problems

Shahane,

Radhakrishnan,

Vanka

2020

Preprint

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“…Previous works [9][10][11][12] have looked to find point locations for finite-difference methods but do not formulate piecewise-defined Lebesgue constants and have not focused on RBF-FD methods using PHSs and polynomials. A few works, however, have considered Lebesgue constants for RBF-FD methods for other purposes [21,22].…”

Section: Node Sampling For Rbf-fd Methodsmentioning

confidence: 99%

Node Generation for RBF-FD Methods by QR Factorization

Liu

Platte

2021

Mathematics

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Polyharmonic spline (PHS) radial basis functions (RBFs) have been used in conjunction with polynomials to create RBF finite-difference (RBF-FD) methods. In 2D, these methods are usually implemented with Cartesian nodes, hexagonal nodes, or most commonly, quasi-uniformly distributed nodes generated through fast algorithms. We explore novel strategies for computing the placement of sampling points for RBF-FD methods in both 1D and 2D while investigating the benefits of using these points. The optimality of sampling points is determined by a novel piecewise-defined Lebesgue constant. Points are then sampled by modifying a simple, robust, column-pivoting QR algorithm previously implemented to find sets of near-optimal sampling points for polynomial approximation. Using the newly computed sampling points for these methods preserves accuracy while reducing computational costs by mitigating stencil size restrictions for RBF-FD methods. The novel algorithm can also be used to select boundary points to be used in conjunction with fast algorithms that provide quasi-uniformly distributed nodes.

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“…Flatter functions give higher accuracy but result in coefficient matrices of high condition numbers [39][40][41][42][43][44][45][46][47][48][49]. Appending polynomials to the RBF has been shown to give high accuracy, depending on the degree of the highest monomial appended [19][20][21][22][50][51][52][53][54][55][56]. Ideally, high order of accuracy can be achieved by appending suitably high order monomials.…”

Section: Multiquadrics (Mq)mentioning

confidence: 99%

Application of a High Order Accurate Meshless Method to Solution of Heat Conduction in Complex Geometries

Naman¹,

Shahane²,

Roy³

et al. 2021

Preprint

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In recent years, a variety of meshless methods have been developed to solve partial differential equations in complex domains. Meshless methods discretize the partial differential equations over scattered points instead of grids. Radial basis functions (RBFs) have been popularly used as high accuracy interpolants of function values at scattered locations. In this paper, we apply the polyharmonic splines (PHS) as the RBF together with appended polynomial and solve the heat conduction equation in several geometries using a collocation procedure. We demonstrate the expected exponential convergence of the numerical solution as the degree of the appended polynomial is increased. The method holds promise to solve several different governing equations in thermal sciences.

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Comparison of Moving Least Squares and RBF+poly for Interpolation and Derivative Approximation

Cited by 32 publications

References 40 publications

A High-Order Accurate Meshless Method for Solution of Incompressible Fluid Flow Problems

A High-Order Accurate Meshless Method for Solution of Incompressible Fluid Flow Problems

Node Generation for RBF-FD Methods by QR Factorization

Application of a High Order Accurate Meshless Method to Solution of Heat Conduction in Complex Geometries

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