Bounds on multivariate polynomials and exponential error estimates for multiquadric interpolation

Madych, W. R.; Nelson, S. A.

doi:10.1016/0021-9045(92)90058-v

Cited by 176 publications

(123 citation statements)

References 4 publications

Supporting

Mentioning

117

Contrasting

Unclassified

Order By: Relevance

“…We will consider two initial conditions (which are also the solution for all time) that are to be advected without distortion by the above steady wind (13)(14). As illustrated in Figure 7(a), the first is the classical test case in the literature [9], a cosine bell profile that is C 1 and centered at (λ c , θ c ):…”

Section: Numerical Test Case 1: Solid Body Rotationmentioning

confidence: 99%

“…completely independent of how the original spherical coordinate system was oriented in space. Thus, when implementing the advective operator with the wind specified by (13)- (14) using RBFs in spherical coordinates, the result is not only singularity free, but is also completely independent of the spherical coordinate orientation.…”

Section: Appendix A: Rotational Invariance Of D N For Solid Body Rotamentioning

confidence: 99%

“…[13]). On the other hand, the evidence strongly suggests that infinitely smooth RBFs will lead to spectral convergence [14,15]. Notice that the infinitely smooth RBFs depend on a shape parameter ε.…”

Section: Introduction To Radial Basis Functionsmentioning

confidence: 99%

See 2 more Smart Citations

Transport schemes on a sphere using radial basis functions

Flyer

Wright

2007

Journal of Computational Physics

133

View full text Add to dashboard Cite

This is an author-produced, peer-reviewed version of this article. AbstractThe aim of this work is to introduce the physics community to the high performance of radial basis functions (RBFs) compared to other spectral methods for modeling transport (pure advection) and to provide the first known application of the RBF methodology to hyperbolic partial differential equations on a sphere. First, it is shown that even when the advective operator is posed in spherical coordinates (thus having singularities at the poles), the RBF formulation of it is completely singularity-free. Then, two classical test cases are conducted: 1) linear advection, where the initial condition is simply transported around the sphere and 2) deformational flow (idealized cyclogenesis), where an angular velocity is applied to the initial condition, spinning it up around an axis of rotation. The results show that RBFs allow for a much lower spatial resolution (i.e. lower number of nodes) while being able to take unusually large time-steps to achieve the same accuracy as compared to other commonly used spectral methods on a sphere such as spherical harmonics, double Fourier series, and spectral element methods. Furthermore, RBFs are algorithmically much simpler to program.

show abstract

Section: Numerical Test Case 1: Solid Body Rotationmentioning

confidence: 99%

Section: Appendix A: Rotational Invariance Of D N For Solid Body Rotamentioning

confidence: 99%

See 1 more Smart Citation

Transport schemes on a sphere using radial basis functions

Flyer

Wright

2007

Journal of Computational Physics

133

View full text Add to dashboard Cite

show abstract

“…To tackle this ill-conditioning problem, a new class of compactly supported RBFs were constructed by [25]. For the theoretical developments of the RBFs in scattered data interpolation, Madych and Nelson [26,27] showed that the RBF-MQ interpolant employs exponential convergence with minimal semi-norm errors. Recently, Franke and Schaback [28,29] provided a theoretical proof in using the RBFs for the numerical solutions of PDEs.…”

Section: Introductionmentioning

confidence: 99%

A numerical method for solving the hyperbolic telegraph equation

Dehghan

Shokri

2007

Numerical Methods Partial

191

View full text Add to dashboard Cite

Recently, it is found that telegraph equation is more suitable than ordinary diffusion equation in modelling reaction diffusion for such branches of sciences. In this article, we propose a numerical scheme to solve the one-dimensional hyperbolic telegraph equation using collocation points and approximating the solution using thin plate splines radial basis function. The scheme works in a similar fashion as finite difference methods. The results of numerical experiments are presented, and are compared with analytical solutions to confirm the good accuracy of the presented scheme.

show abstract

“…Let Π k denote the space of all d-variate polynomials whose total degree is less or equal to k. It is known [7] Surface spline interpolation is a prominent member of a family of interpolants known as radial basis function interpolants. The approximation properties of these interpolants have received considerable attention in the literature (for a sampling see [8], [4], [26], [16], [9], [6], [19], [12], [22], [13], [23], [3], and the surveys [18], [5]). …”

Section: Introductionmentioning

confidence: 99%

The $L_{2}$-approximation order of surface spline interpolation

Johnson

2000

Math. Comp.

View full text Add to dashboard Cite

Abstract. We show that if the open, bounded domain Ω ⊂ R d has a sufficiently smooth boundary and if the data function f is sufficiently smooth, then the Lp(Ω)-norm of the error between f and its surface spline interpolant is O(δ γp +1/2 ) (1 ≤ p ≤ ∞), where γp := min{m, m − d/2 + d/p} and m is an integer parameter specifying the surface spline. In case p = 2, this lower bound on the approximation order agrees with a previously obtained upper bound, and so we conclude that the L 2 -approximation order of surface spline interpolation is m + 1/2.

show abstract

Bounds on multivariate polynomials and exponential error estimates for multiquadric interpolation

Cited by 176 publications

References 4 publications

Transport schemes on a sphere using radial basis functions

Transport schemes on a sphere using radial basis functions

A numerical method for solving the hyperbolic telegraph equation

The $L_{2}$-approximation order of surface spline interpolation

Contact Info

Product

Resources

About