# Numerical Analysis: Radial basis function methods for partial differential equations

In this project radial basis functions are used to discretize and solve PDEs.

Radial basis functions (RBFs) were introduced in the late 1960s for interpolation of scattered data, with the specific purpose of reconstruction of topography from field measurements. The original idea is very simple; place a radially symmetric basis function at each data location and form a linear combination of the basis functions that fits the data at every data location. With a judicious choice of basis function (e.g., a Gaussian), a unique interpolant always exists. It also exhibits some very desirable properties, such as geometric flexibility, spectral accuracy and algorithmic simplicity.

Discretizing partial differential equations (PDEs) using RBFs is equally simple; merely replace the unknown solution by an RBF interpolant, and substitute derivatives of the solution with exact derivatives of the interpolant. While this approach has been very successful in a wide variety of applications (see e.g., [1]), it is hampered by numerical ill-conditioning and poor scaling of the computational cost.

A solution to these issues is studied in this project. By considering separate local interpolants in the neighborhood of each data location instead of one global interpolant, scaling can be drastically improved while still retaining most of the advantages of the original formulation. The resulting method can be viewed as an extension of traditional finite difference methods to scattered data points, hence the acronym RBF-FD.

The figure shows the solution to the Galewsky test case of the shallow-water eqautions on the sphere, obtained using the RBF-FD model described in [2]. This set of non-linear hyperbolic equations is an important test case in the atmospheric modeling community. Weather and climate modeling is one application area where these methods are gaining traction as a potential discretization method in the next generation of large-scale computational models.

[1] Wright, G. B., Flyer, N., & Yuen, D. A. (2010). A hybrid radial basis function pseudospectral method for thermal convection in a 3D spherical shell. Geochemistry, Geophysics, Geosystems, 11(7).
[2] Flyer, N., Lehto, E., Blaise, S., Wright, G. B., & St-Cyr, A. (2012). A guide to RBF-generated finite differences for nonlinear transport: Shallow water simulations on a sphere. Journal of Computational Physics, 231(11), 4078-4095.