Kernel techniques: From machine learning to meshless methods

Schaback, Robert; Wendland, Holger

doi:10.1017/s0962492906270016

Cited by 238 publications

(203 citation statements)

References 208 publications

Supporting

Mentioning

198

Contrasting

Order By: Relevance

“…Without further elaboration, we refer the interested reader to [35,32] and consider other types of solutions to the trade-off dilemma. Indeed, a preconditioning strategy for Lagrange interpolation by polyharmonic splines is developed in [2], while for infinitely smooth RBFs, it is now well-known that the accuracy of the interpolation depends strongly on the choice of the shape parameter ε.…”

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 is the standard setup in machine learning, and depending on the loss function that is used to assess the fidelity to the data, different approaches turn out to be optimal. Schaback and Wendland [69] discuss the role of kernel methods in machine learning. The approach that is most closely related to the kernel interpolants ("splines") in Section 2 is the concept of smoothing splines, which corresponds to a quadratic loss function.…”

Section: Generalizations and Related Problemsmentioning

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

“…Such kernels were provided bu Z.M. Wu [22] and H. Wendland [18], and the books [5,19] together with the survey [17] contain a fairly complete account of the background information on kernels.…”

Section: Kernels and Convolutionsmentioning

confidence: 99%

“…They are special cases of kernel-based techniques which arise in many other areas as well [17]. Similar computational methods were introduced for problems in weak form [1,2,3] but they still deserve a thorough theoretical analysis.…”

Section: Introductionmentioning

confidence: 99%

Kernel techniques: From machine learning to meshless methods

Cited by 238 publications

References 208 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?

Recovery of functions from weak data using unsymmetric meshless kernel-based methods

Contact Info

Product

Resources

About