Interpolation in the limit of increasingly flat radial basis functions

Driscoll, Tobin A.; Fornberg, Bengt

doi:10.1016/s0898-1221(01)00295-4

Cited by 261 publications

(202 citation statements)

References 7 publications

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“…To derive the c C coefficients for all the cells in the stencil, we express α and β in terms of the cell averages of the solution corresponding to the underlying stencil by solving the linear system (3.9) and use (4.3) for direct comparison of the terms. Similar calculations, albeit with a different goal, can be found in [7], where the authors show that in 1D, under simple assumptions on the basis function, the interpolants converge to the Lagrange interpolating polynomial as ε → 0. …”

Section: Discussionsupporting

confidence: 53%

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Adaptive WENO Methods Based on Radial Basis Function Reconstruction

Bigoni

Hesthaven

2017

J Sci Comput

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We explore the use of radial basis functions (RBF) in the weighted essentially non-oscillatory (WENO) reconstruction process used to solve hyperbolic conservation laws, resulting in a numerical method of arbitrarily high order to solve problems with discontinuous solutions. Thanks to the mesh-less property of the RBFs, the method is suitable for non-uniform grids and mesh adaptation. We focus on multiquadric radial basis functions and propose a simple strategy to choose the shape parameter to control the balance between achievable accuracy and the numerical stability. We also develop an original smoothness indicator which is independent of the RBF for the WENO reconstruction step. Moreover, we introduce type I and type II RBF-WENO methods by computing specific linear weights. The RBF-WENO method is used to solve linear and nonlinear problems for both scalar and systems of conservation laws, including Burgers equation, the Buckley-Leverett equation, and the Euler equations. Numerical results confirm the performance of the proposed method. We finally consider an effective conservative adaptive algorithm that captures moving shocks and rapidly varying solutions well. Numerical results on moving grids are presented for both Burgers equation and the more complex Euler equations.

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Section: Discussionsupporting

confidence: 53%

“…In e.g. [7,12,14,13], the authors have introduced and further developed a series of so-called "stable algorithms" that overcome the trade-off principle. In such methods, the linear system (3.9) is not solved directly, thus allowing stable computations also when the shape parameter is very small.…”

Section: On Choosing a Good Shape Parametermentioning

confidence: 99%

Adaptive WENO Methods Based on Radial Basis Function Reconstruction

Bigoni

Hesthaven

2017

J Sci Comput

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“…In recent years so-called flat radial basis functions (RBFs) have received much attention in the case when the kernels are infinitely smooth (see, e.g., [5,16,21,22,23,33]). We begin by summarizing the essential insight gained in these papers, and then present some recent results from [38] that deal with radial kernels of finite smoothness in the next subsection.…”

Section: Infinitely Smooth Rbfsmentioning

confidence: 99%

Green’s Functions: Taking Another Look at Kernel Approximation, Radial Basis Functions, and Splines

Fasshauer

2011

Springer Proceedings in Mathematics

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The theories for radial basis functions (RBFs) as well as piecewise polynomial splines have reached a stage of relative maturity as is demonstrated by the recent publication of a number of monographs in either field. However, there remain a number of issues that deserve to be investigated further. For instance, it is well known that both splines and radial basis functions yield "optimal" interpolants, which in the case of radial basis functions are discussed within the so-called native space setting. It is also known that the theory of reproducing kernels provides a common framework for the interpretation of both RBFs and splines. However, the associated reproducing kernel Hilbert spaces (or native spaces) are often not that well understood -especially in the case of radial basis functions. By linking (conditionally) positive definite kernels to Green's functions of differential operators we obtain new insights that enable us to better understand the nature of the native space as a generalized Sobolev space. An additional feature that appears when viewing things from this perspective is the notion of scale built into the definition of these function spaces. Furthermore, the eigenfunction expansion of a positive definite kernel via Mercer's theorem provides a tool for making progress on such important questions as stable computation with flat radial basis functions and dimension independent error bounds.

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“…Since then, DRBF methods have been increasingly used for the 10 solution of elliptic, parabolic and hyperbolic PDEs which govern many engineering problems. In [7,8,9,10,11], practitioners demonstrated that the elliptic PDE solutions using DRBFs converge much faster than those based on polynomial approximations. Mai-Duy and Tran-Cong proposed the idea of using Indirect/Integrated RBFs (IRBFs) for the solution of PDEs [12,13].…”

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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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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Interpolation in the limit of increasingly flat radial basis functions

Cited by 261 publications

References 7 publications

Adaptive WENO Methods Based on Radial Basis Function Reconstruction

Adaptive WENO Methods Based on Radial Basis Function Reconstruction

Green’s Functions: Taking Another Look at Kernel Approximation, Radial Basis Functions, and Splines

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

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