Stable calculation of Gaussian-based RBF-FD stencils

Fornberg, Bengt; Lehto, Erik; Powell, Collin Eugene

doi:10.1016/j.camwa.2012.11.006

Cited by 186 publications

(112 citation statements)

References 21 publications

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“…The increasing flat region of 465 RBF is of particular interest since it often corresponds to the most accurate RBF approximations as shown in recent works [31,32]. In the paper, we have proposed an idea of using high-order IRBFs to construct combined compact approximations, which allows a more straightforward incorporation of nodal values of first-and second-order derivatives, and yields better accuracy over 470 compact approximations.…”

Section: Discussionmentioning

confidence: 99%

“…Fornberg and Wright [11] proposed the Contour-Padé algorithm which can stably compute the whole region of the shape parameter on the complex plane. Many different approaches to enhance the stability of DRBF methods have been proposed, for example [23,25,26,27,28,29,30,31,32] and their references therein. For IRBF approaches, Sarra [16] studied the case of global flat IRBFs.…”

Section: Introductionmentioning

confidence: 99%

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A numerical study of compact approximations based on flat integrated radial basis functions for second-order differential equations

Tien

Mai‐Duy

Tran

et al. 2016

Computers & Mathematics with Applications

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In this paper, we propose a simple but effective preconditioning technique to improve the numerical stability of Integrated Radial Basis Function (IRBF) methods. The proposed preconditioner is simply the inverse of a well-conditioned matrix that is constructed using non-flat IRBFs. Much larger values of the free shape parameter of IRBFs can thus be employed and better accuracy for smooth solution problems can be achieved. Furthermore, to improve the accuracy of local IRBF methods, we propose a new stencil, namely Combined Compact IRBF (CCIRBF), in which (i) the starting point is the fourth-order derivative; and (ii) nodal values of first-and second-order derivatives at side nodes of the stencil are included in the computation of first-and second-order derivatives at the middle node in a natural way. The proposed stencil can be employed in uniform/nonuniform Cartesian grids. The preconditioning technique in combination with the CCIRBF scheme employed with large values of the shape parameter are tested with elliptic equations and then applied to simulate several fluid flow problems governed by Poisson, Burgers, convectiondiffusion, and Navier-Stokes equations. Highly accurate and stable solutions are obtained. In some cases, the preconditioned schemes are shown to be several orders of magnitude more accurate than those without preconditioning.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A numerical study of compact approximations based on flat integrated radial basis functions for second-order differential equations

Tien

Mai‐Duy

Tran

et al. 2016

Computers & Mathematics with Applications

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“…Some of them are: Contour-Padé [9] and RBF-QR [10]. Related to the latter is the recent RBF-GA [11].…”

Section: Radial Basis Function Methodologymentioning

confidence: 99%

Applications of Radial Basis Function Schemes to Fractional Partial Differential Equations

Martínez¹,

Fuentes²

2017

Fractal Analysis - Applications in Physics, Engineering and Technology

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In modeling using diffusion equations, the relationship between fractal geometry and fractional calculus arises by modeling the conditions of the medium as a fractal whose fractional dimension determines the order of this equation. For this reason, it is very useful to have numerical methods that solve them and discretization of the domain is not determinant for the efficiency of the algorithm. In this work, it is proposed to show that meshless methods, in particular methods with radial basis functions (RBF), are an alternative to schemes in differences or structured meshes. We show that we can obtain numerical solutions to some fractional partial differential equations using collocation and RBF, over non equally distributed data.Keywords: fractional partial differential equations (FPDE), meshless methods, radial basis functions (RBF), Caputo derivative, Riemann-Liouville derivative, Riesz derivative, diffusion-convection 1. Background Radial basis function methodologyThe Hardy-based radial-based functions (RBF) methodology [1] arises from the need to apply multivariate interpolation to cartography problems, with randomly dispersed data (also known as collocation nodes). Micchelli [2], Powell et al. [3] gave it a great boost by proving non-singularity theorems. Later, Kansa [4,5] proposed to consider the analytical derivatives of the FBR to develop numerical schemes that deal with partial differential equations (PDE).Regarding PDE over spaces of dimension greater than one, we generally opt for finite element (FE) type discretizations on meshes, structured or not; also pseudo spectral (PS) methods through base functions such as Fourier or Chebyshev. The high degree of computational © 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. efficiency in the procedure raises the cost in the regularity constraints on the form of the computational domain.The FE methods involve decomposition of the domain; for example, in two dimensions rectangles are constructed with curvilinear mappings that allow the refinement of the mesh in critical areas. However, this type of implementation is complex and very close nodes are needed, mainly at the domain boundaries, which impairs stability conditions in time.For this reason, we look for numerical techniques that do not depend to a great extent on data distribution. RBF collocation methods belong to the "meshfree" methods; that is, they only require scattered collocation nodes on the domain and the boundary. They are also an alternative for dealing with problems in larger dimensions and irregular domains. Hence, in recent decades, such methods have attracted the attention of researchers in order to solve partial differential equations. See for example: Chen et al. [6].This type of technique approximates the solution by means of a linear...

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“…Furthermore, refining the patches in RBF-PUM for a fixed ε results in a decreasing 'effective' shape parameter value, that is, the shape parameter becomes smaller in relation to the patch size. However, the problem of ill-conditioning for small shape parameters can be avoided by employing stable evaluation methods such as the Contour-Padé approach [24], the RBF-QR method [21,23,32], or the RBF-GA method [22]. Here we employ the RBF-QR method which, simply put, corresponds to a change of basis from {φ j m } to {ψ j m } in the local problems.…”

Section: With the Global Vector Of Nodal Values Defined Bymentioning

confidence: 99%

“…Several methods have been proposed to eliminate these conditioning problems. The Contour-Padé algorithm [24] came first, and was then followed by the RBF-QR method for the sphere [23] and for Cartesian space [21] and, more recently, the RBF-GA method [22], see also [19]. All of these approaches compute the same end result as the ill-conditioned formulation, but through a stable reformulation.…”

Section: Introductionmentioning

confidence: 99%

Preconditioning for Radial Basis Function Partition of Unity Methods

et al. 2015

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Meshfree radial basis function (RBF) methods are of interest for solving partial differential equations due to attractive convergence properties, flexibility with respect to geometry, and ease of implementation. For global RBF methods, the computational cost grows rapidly with dimension and problem size, so localised approaches, such as partition of unity or stencil based RBF methods, are currently being developed. An RBF partition of unity method (RBF-PUM) approximates functions through a combination of local RBF approximations. The linear systems that arise are locally unstructured, but with a global structure due to the partitioning of the domain. Due to the sparsity of the matrices, for large scale problems, iterative solution methods are needed both for computational reasons and to reduce memory requirements. In this paper we implement and test different algebraic preconditioning strategies based on the structure of the matrix in combination with incomplete factorisations. We compare their performance for different orderings and problem settings and find that a no-fill incomplete factorisation of the central band of the original discretisation matrix provides a robust and efficient preconditioner.

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Stable calculation of Gaussian-based RBF-FD stencils

Cited by 186 publications

References 21 publications

A numerical study of compact approximations based on flat integrated radial basis functions for second-order differential equations

A numerical study of compact approximations based on flat integrated radial basis functions for second-order differential equations

Applications of Radial Basis Function Schemes to Fractional Partial Differential Equations

Preconditioning for Radial Basis Function Partition of Unity Methods

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