Kernel techniques: From machine learning to meshless methods

By Robert Schaback, Holger Wendland. Acta Numerica, 2006

347 citations recorded by the source corpus.

Kernels are valuable tools in various fields of numerical analysis, including approximation, interpolation, meshless methods for solving partial differential equations, neural networks, and machine learning. This contribution explains why and how kernels are applied in these disciplines. It uncovers the links between them, in so far as they are related to kernel techniques. It addresses non-expert readers and focuses on practical guidelines for using kernels in applications.

Subfields: Civil and Structural Engineering, Management, Monitoring, Policy and Law, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/a6bf288da1c2cdfc31004e44daaa1344aa50ebfb

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Civil and Structural Engineering

1. Ankit Srivastava
2. Chris J. Oates
3. Mark A. Girolami
4. Natarajan Sukumar
5. Natasha Flyer
6. Bengt Fornberg
7. Robert Schaback
8. E. Ooi

In Management, Monitoring, Policy and Law

1. Chris J. Oates
2. Mark A. Girolami
3. Natarajan Sukumar
4. Robert Schaback
5. Ankit Srivastava

In Mechanics of Materials

1. Natasha Flyer
2. Bengt Fornberg
3. Robert Schaback
4. Chris J. Oates
5. Mark A. Girolami
6. Natarajan Sukumar
7. Ankit Srivastava
8. Stéphane P.A. Bordas

Citations in the corpus

• Control Functionals for Quasi-Monte Carlo Integration (2015, background): “Our results below apply also to Sobolev spaces with noninteger k, but this construction is more technical and we refer the reader to [27] for details.”
• Scalable Log Determinants for Gaussian Process Kernel Learning (2017, neutral): “[37] Robert Schaback and Holger Wendland.”
• Limit problems for interpolation by analytic radial basis functions (2008, background): “But modern large-scale methods like domain decomposition or partition-of-unity techniques [17] will localize large problems anyway, and thus get away with reasonably small values of N .”
• Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks (2021, background): “Schaback and Wendland [47] discuss the ties of kernel methods (for example, radial basis functions) to machine learning and meshfree methods.”
• Approximation of eigenfunctions in kernel-based spaces (2014, neutral): “Details can be retrieved from the monographs [2, 7, 27] and the surveys [1, 25].”
• Solving PDEs with radial basis functions * (2015, background): “Acta Numerica featured RBF articles in 2000 [8] and in 2006 [70].”
• On the choice of source points in the method of fundamental solutions (2009, neutral): “[33,34]).”