RadialBasisFunctions.jl

This package intends to provide tools for all things regarding Radial Basis Functions (RBF).

Feature	Status
Interpolation	✅
Regridding	✅
Partial derivative ($\partial f$)	✅
Laplacian ($\nabla^2 f$, $\Delta f$)	✅
Gradient ($\nabla f$)	✅
Directional Derivative ($\nabla f \cdot v$)	✅
Custom / user supplied ($\mathcal{L} f$)	✅
divergence ($\textrm{div} \mathbf{F}$ or $\nabla \cdot \mathbf{F}$)	❌
curl ($\nabla \times \mathbf{F}$)	❌
Reduced Order Models (i.e. POD)	❌

Currently, we support the following types of RBFs (all have polynomial augmentation by default, but is optional)

Type	Function, $\phi(r)$
Polyharmonic Spline	$r^n$ where $n=1,3,5, \text{ or } 7$
Inverse Multiquadric	$1 / \sqrt{(r \varepsilon)^2+1}$
Gaussian	$e^{-(r \varepsilon)^2}$

where $r = \lvert \mathbf{x}-\mathbf{x}_{c} \rvert$ is the Euclidean distance between two points and $\varepsilon$ is a user-supplied shape parameter.

Installation

Simply install the latest stable release using Julia's package manager:

] add RadialBasisFunctions

Current Limitations

1. A critical dependency of this package is NearestNeighbors.jl which requires that the dimension of each data point is inferrable. To quote from NearestNeighbors.jl:

   The data, i.e., the points to build up the tree from. It can either be

   • a matrix of size nd × np with the points to insert in the tree where nd is the dimensionality of the points and np is the number of points
   • a vector of vectors with fixed dimensionality, nd, which must be part of the type. Specifically, data should be a Vector{V}, where V is itself a subtype of an AbstractVector and such that eltype(V) and length(V) are defined. (For example, with 3D points, V = SVector{3, Float64} works because eltype(V) = Float64 and length(V) = 3 are defined in V.)

   That said, we currently only support the second option here (Vector{AbstractVector}), but plan to support matrix inputs in the future.

2. Interpolator uses all points, but there are plans to support local collocation / subdomains like the operators use.