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RadialBasisFunctions.jl

Build Status Stable Dev License File DOI

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.