Radial basis functions
It is necessary to estimate parameters by approximation and interpolation in many areas-from computer graphics to inverse methods to signal processing. Radial basis functions are modern, powerful tools which are being used more widely as the limitations of other methods become apparent. In the numerical analysis group we are interested in analyses of radial basic functions from the theoretical and practical implementation viewpoints. This includes in particular the ubiquitous multiquadric function and its generalisations. The particular advantages of radial basis functions lie in their availability in any space dimensions and in their versatility in applications such as solving partial differential equations, neural networks and deep learning. The need for approximations in high dimensions appear in particular in the current interest in big data analysis where multivariate approximation theory tools such as radial basis functions are state-of-the-art algorithms. For applications in PDEs, both collocation and Galerkin approaches are available methods.