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, as far as they are related to kernel techniques. It addresses non-expert readers and focuses on practical guidelines for using kernels in applications.

Meshfree methods are the topic of recent research in many areas of computational science and approximation theory. These methods come in various flavors, most of which can be explained either by what is known in the literature as radial basis functions (RBFs), or in terms of the moving least squares (MLS) method. Over the past several years meshfree approximation methods have found their way into many different application areas ranging from artificial intelligence, computer graphics, image processing and optimization to the numerical solution of all kinds of (partial) differential equations problems. Applications in computational nanotechnology are still somewhat rare, but do exist in the literature. In this chapter we will focus on the mathematical foundation of meshfree methods, and the discussion of various computational techniques presently available for a successful implementation of meshfree methods. At the end of this review we mention some initial applications of meshfree methods to problems in computational nanotechnology, and hope that this introduction will serve as a motivation for others to apply meshfree methods to many other challenging problems in computational nanotechnology.

A. Local rotational and Galilean translational transformations can be obtainedto reduce the conservation equations into steady state forms for the inviscid Euler equations or Navier-Stokes equations. B. The entire set of PDEs are transformed into the method of lines approachyielding a set of coupled ordinary differential equations whose homogeneous solution is exact in time. C. The spatial components are approximated by expansions of meshless RBFs;each individual RBF is volumetrically integrated at one of the sampling knots xi, yielding a collocation formulation of the method of lines structure of theODEs. D. Because the volume integrated RBFs increase more rapidly away from thedata center than the commonly used RBFs, we use a higher order preconditioner to counter-act the ill-conditioning problem. Domain decomposition isused over each piece-wise continuous subdomain.

During the last years, there has been an increased interest in developing efficient radial basis function (RBF) algorithms to solve partial differential problems of great scale. In this article, we are interested in solving large PDEs problems, whose solution presents rapid variations. Our main objective is to introduce a RBF dynamical domain decomposition algorithm which simultaneously performs a node adaptive strategy. This algorithm is based on the RBFs unsymmetric collocation setting. Numerical experiments performed with the multiquadric kernel function, for two stationary problems in two dimensions are presented.

Originally, the motivation for the basic meshfree approximation methods (radial basis functions and moving least squares methods) came from applications in geodesy, geophysics, mapping, or meteorology. Later, applications were found in many areas such