RadialBasisFunctions.jl
Documentation for RadialBasisFunctions.
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 $n=1,3,5,7$ |
| Inverse Multiquadric | $1 / \sqrt{(r \varepsilon)^2+1}$ |
| Gaussian | $e^{-(r \varepsilon)^2}$ |
where $\varepsilon$ is a user-supplied shape parameter.
Installation
Simply install the latest stable release using Julia's package manager:
] add RadialBasisFunctionsCurrent Limitations
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.Interpolatoruses all points, but there are plans to support local collocation / subdomains like the operators use.