Adaptive Gaussian radial basis function methods for initial value problems: construction and comparison with adaptive multiquadric radial basis function methods (Q2195928)

Radial basis functions with parameters are a particularly important class of kernel functions for univariate and multivariable approximation. Their success is based on their ability to generate highly accurate approximations in any dimension to gridded or scattered data. These approximants are usually formed by quasi-interpolation (semi-discrete convolutions with data vectors) or interpolation (so-called collocation in the language of ODE or PDE solvers). For the latter the so-called multiquadric kernel is especially well-known and tried and tested with outstanding results, but for the former quasi-interpolation it works excellently as well. Using the kernels with parameters has the distinct advantage that the parameters can be used to improve stability and/or approximation orders dependent or independent of spacings of data points as an additional feature of the approximation scheme (i.e., we keep the same good kernel, but do have a variability through the parameters). The authors of these papers use these ideas to approximate the solution of initial value problems and form new numerical schemes for these situations. They use Euler's method and describe collocation methods with radial basis functions. Many numerical examples are presented to underline the theoretical results.