Special Techniques for Kernel-Based Reconstruction of Functions from Meshless Data

Here are three short stories on meshless methods using kernel techniques: • Any well–posed linear problem in the native space NΦ of a symmetric (strictly) positive definite kernel Φ can be successfully solved by symmetric meshless collocation. This applies to a large variety of standard linear PDE problems. • Relaxing interpolation conditions by allowing some small absolute error can significantly reduce the complexity of meshless techniques, in particular in conjunction with greedy methods and learning algorithms. • The instability phenomena of badly scaled meshless techniques for smooth kernels can be overcome by an unexpected link to multivariate polynomial interpolation. In particular, there is a preconditioning technique that completely removes the instability in the limit and has a surprisingly simple form, separating the scale information from the geometric information. Since the readers can be assumed to be familiar with basic notions of meshless methods, and since detailed presentations are given in the references, it suffices to give a commented overview and suggestions for future research. 1 Short Story on Meshless Kernel Collocation We assert here that all reasonable analytic problems can in principle be solved numerically by meshless symmetric collocation using smooth positive definite kernels. Assume that a user has to find a function u that solves a very general analytic problem of the form Li(u) = fi on some set Ωi ⊂ IR, 1≤ i≤ K (1) where the linear operators Li may be of any type. Note that the Poisson problem and many, many others take this form for a mixed choice of operators within (parts of) domains and (parts of) boundaries. Assume further that there is a (strictly) positive definite [11, 13, 10] 1University of Göttingen, Germany ([email protected]).