Stable Computations with Gaussian Radial Basis Functions

Fornberg, Bengt; Larsson, Elisabeth; Flyer, Natasha

doi:10.1137/09076756x

Cited by 352 publications

(290 citation statements)

References 30 publications

Supporting

Mentioning

284

Contrasting

Unclassified

Order By: Relevance

“…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

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: 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

“…This uncertainty principle, however, is tied directly to the use of the standard ("bad") basis, and we demonstrate below how it can be circumvented by choosing a better -orthonormal -basis. The following discussion is motivated by the recent work of Bengt Fornberg and his collaborators [14,15] in which they have proposed a so-called RBF-QR algorithm which allows for stable RBF computations. In addition to this QR-based approach they have also proposed other stable algorithms such as the Contour-Padé algorithm [16].…”

Section: Stable Computationmentioning

confidence: 99%

“…Since the eigenvalues of the Gaussian kernel decay very quickly we are now able to directly follow the QR-based strategy suggested in [14] -without the need for any additional transformation to Chebyshev polynomials.…”

Section: The Rbf-qr Algorithmmentioning

confidence: 99%

“…The QR-based algorithm of [14] corresponds to the following. Starting with an expansion of the basis functions centered at x j , j = 1, .…”

Section: The Rbf-qr Algorithmmentioning

confidence: 99%

“…Aside from some scattered work on preconditioning of RBF systems, only Bengt Fornberg together with his co-workers has tackled this problem with some success. We are especially motivated by their RBF-QR idea [14,15] and will provide some of our own thoughts on this approach in Section 4.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

Fasshauer

2011

Springer Proceedings in Mathematics

View full text Add to dashboard Cite

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.

show abstract

Recent developments of the meshless radial point interpolation method for time‐domain electromagnetics

Kaufmann

Engström

et al. 2012

Int J Numerical Modelling

View full text Add to dashboard Cite

Meshless methods are a promising new field in computational electromagnetics. Instead of relying on an explicit mesh topology, a numerical solution is computed on an unstructured set of collocation nodes. This allows to model fine geometrical details with high accuracy and facilitates the adaptation of node distributions for optimization or refinement purposes. The radial point interpolation method (RPIM) is a meshless method based on radial basis functions. In this paper, the current state of the RPIM in electromagnetics is reviewed. The localized RPIM scheme is summarized, and the interpolation accuracy is discussed in dependence of important parameters. A time-domain implementation is presented, and important time iteration aspects are reviewed. New formulations for perfectly matched layers and waveguide ports are introduced. An unconditionally stable RPIM scheme is summarized, and its advantages for hybridization with the classical RPIM scheme are discussed in a practical example. The capabilities of an adaptive time-domain refinement strategy based on the experiences on a frequency-domain solver are discussed. Figure 2. Randomly disturbed node arrangements for different extents of the support domain. Number of neighbors are (a) n As ¼ 4, (b) n As ¼ 8, (c)n As ¼ 16, (d) n As ¼ 24, and (e) n As ¼ 40. 472T. KAUFMANN ET AL.

show abstract

Stable Computations with Gaussian Radial Basis Functions

Cited by 352 publications

References 30 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

Recent developments of the meshless radial point interpolation method for time‐domain electromagnetics

Contact Info

Product

Resources

About