The parameter R2 in multiquadric interpolation

Carlson, R.E.; Foley, Thomas A.

doi:10.1016/0898-1221(91)90123-l

Cited by 321 publications

(157 citation statements)

References 18 publications

Supporting

Mentioning

145

Contrasting

Order By: Relevance

“…The idea is to perform a series of interpolation experiments for various values of the shape parameter and identify the one which yields the minimum interpolation error. As already discussed, the optimal shape parameter may be different for different functions (see e.g., [5,30]), but our numerical examples show that using the same shape parameter for different problems typically yields good results. Exceptions may be needed for very small mesh sizes and high orders of the interpolants.…”

Section: On Choosing a Good Shape Parametermentioning

confidence: 89%

“…Examples include Burgers equations, the Buckley-Leverett equation, and the Euler equations; we have obtained Adapted grid with inital N = 2 4 Uniform grid with N = 2 4 Uniform grid with N = 2 5 Uniform grid with N = 2 6 Exact solution In the following we show that, for uniform one-dimensional meshes, the c C coefficients in (4.3) for a stencil of size n = k + 1 for k = 1 or k = 2 agree with the coefficients used in the classic polynomial WENO reconstructions for the same stencils sizes (see [33]) up to a term that depends on ε and ∆x. We conjecture that these results extend to k > 2.…”

Section: Discussionmentioning

confidence: 99%

See 1 more Smart Citation

Adaptive WENO Methods Based on Radial Basis Function Reconstruction

Bigoni

Hesthaven

2017

J Sci Comput

View full text Add to dashboard Cite

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.

show abstract

Section: On Choosing a Good Shape Parametermentioning

confidence: 89%

Section: Discussionmentioning

confidence: 99%

Adaptive WENO Methods Based on Radial Basis Function Reconstruction

Bigoni

Hesthaven

2017

J Sci Comput

View full text Add to dashboard Cite

show abstract

“…When it comes to deriving an optimal approximation of f , minimization of the norm of the error functional λ err = δ x 0 − λ u(x 0 ) for fixed x 0 ∈ T again amounts to the minimization the quadratic form Q in (6). In the general framework of conditionally positive definite kernels, however, λ err is not automatically in L P (T ), and the additional constraint…”

Section: Conditionally Positive Definite Kernelsmentioning

confidence: 99%

“…Consequently, more emphasis was put on the study of good configurations of the sampling locations [14,15,38] on the one hand, and edge correction strategies (see [25] for an overview) on the other hand to avoid the undesired oscillations near the boundaries that often come with smooth and flat kernels. Nevertheless several authors [6,27,26,62] have pointed out the big impact of the choice of e.g. the scaling parameter on the accuracy of the interpolant.…”

Section: Kernel Selection and Parameter Estimationmentioning

confidence: 99%

Interpolation of spatial data – A stochastic or a deterministic problem?

2013

View full text Add to dashboard Cite

Interpolation of spatial data is a very general mathematical problem with various applications. In geostatistics, it is assumed that the underlying structure of the data is a stochastic process which leads to an interpolation procedure known as kriging. This method is mathematically equivalent to kernel interpolation, a method used in numerical analysis for the same problem, but derived under completely different modelling assumptions. In this article we present the two approaches and discuss their modelling assumptions, notions of optimality and the different concepts to quantify the interpolation accuracy. Their relation is much closer than has been appreciated so far, and even results on convergence rates of kernel interpolants can be translated to the geostatistical framework. We sketch the different answers obtained in the two fields concerning the issue of kernel misspecification, present some methods for kernel selection, and discuss the scope of these methods with a data example from the computer experiments literature.

show abstract

“…In this work, it is denoted by β. Numerical experiments indicated that the optimal value of β depends on the function to be interpolated, the configuration of nodal points, the RBF type, and the machine precision [3,4,9,20,21,22,23]. The matrix condi-30 tion of the RBF method grows exponentially with the RBF width.…”

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

View full text Add to dashboard Cite

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.

show abstract

The parameter R2 in multiquadric interpolation

Cited by 321 publications

References 18 publications

Adaptive WENO Methods Based on Radial Basis Function Reconstruction

Adaptive WENO Methods Based on Radial Basis Function Reconstruction

Interpolation of spatial data – A stochastic or a deterministic problem?

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

Contact Info

Product

Resources

About