The Python Radial Basis Function Toolbox

The Python Radial Basis Function Toolbox (RBFT) is software for implementing RBF interpolation methods and RBF methods for the numerical solution of PDEs on scattered centers located in complexly shaped domains. The PRBFT is under constant development as it is heavily used in RBF research projects.

The two earlier versions of the toolbox, versions 1.0 and 1.1, were programmed in Matlab. For a multitude of reasons Matlab is no longer used. The Matlab versions will remain available for download and may see minor bug fixes and improvements but will not be developed further. All future development will be in the Python version.

Comments, questions, bug reports, code requests, etc. can be sent to sarra@marshall.edu

Functionality

The functionality of the toolbox is organized via object oriented programming into several classes.

Classes and modules:

rbfX - basic RBF method functionality and defines methods that all RBF classes must implement

gax - Gaussian RBF
iqx - Inverse Quadratic RBF
imqx - inverse multiquadric RBF class (new in version 1.1)

rbfCenters - center locations
rbfCentro - reduced flop count and storage algorithms for RBF methods in symmetric domains.
Functions - test functions and derivatives.
rbfRational - (new in version 1.1) a rational RBF method
rbfMisc - (new in version 1.1) miscellaneous functions used in examples such as a 4th order Runge-Kutta method, eigenvalues of differentiation matrices, etc.
rbfLocal - (new in version 1.2) - local RBF method
rbfPU - (new in version 1.2) - two versions of partition of unity RBF methods
AnimatePlot - module for animating the solutions of time-dependent PDEs
rbfc - (new in version 1.2) parent class for compactly supported RBFs

W32 - Wendland functions that are SPD in up to 3 space dimensions and in C4
W33 - Wendland functions that are SPD in up to 3 space dimensions and in C6

Scripts: The toolbox comes with a collection of scripts that demonstrate its usage, benchmark its performance, and verify that its algorithms produce the correct results. The scripts are located in the following folders.

\examples
\tests
\benchmarks
\papers

Future plans

add Mercer's method to the GA class
expansion of the rbfCenters class
improve documentation
expand the number and variety of example scripts
parallel implementation of local methods via openCL
add option to augment the RBF interpolant with polynomials
add extended precision to the Python version

PRBFT Downloads - Python version

version 1.2: coming in August 2021

new in version 1.2:

artifical viscosity methods for the stabilization of time-dependent PDE problems
local RBF method
partition of unity (PU) RBF method
a new partition of unity RBF method which is easier to implement than is the standard PU method
new shape parameter selection method based on the RBF system matrix being numerical symmetric positive definite
added a new quasi-random center algorithm which has a lower-discrepancy than methods such as Halton and Hammersly points that are currently commonly used
added a module to assist in the animation of the solution of time-dependent PDEs

Older Matlab version

MRBFT Downloads - Matlab version

The Matlab version uses the Multiprecision Computing Toolbox to seamlessly implement extended precision floating point arithmetic in all RBFT routines.

version 1.0: (May 27, 2016), can be downloaded from GitHub or MRBFTv1p0.zip
version 1.1: (April 20, 2017), MRBFTv1p1.zip

MRBFT References

The MRBFT is described in the manuscript

The Matlab Radial Basis Function Toolbox. Journal of Open Research Software, vol. 5, 2017.

All the numerical results in the following manuscript were produced with the MRBFT (in particular the rbfCentro class was used)

Radial Basis Function Methods - the case of symmetric domains. Numerical Methods for Partial Differential Equations, 2018. DOI:10.1002/num.22272

The double and extended precision "standard basis" RBF results in the following used the MRBFT

An examination of evaluation algorithms for the RBF methods. Engineering Analysis with Boundary Elements, volume 75, p. 36-45, February 2017. DOI: 10.1016/j.enganabound.2016.11.006.

The rbfRational class that is new in version 1.1 was used in

A rational radial basis function method for accurately resolving discontinuities and steep gradients (with Y. Bai). Applied Numerical Mathematics 130, p. 131-142, 2018. DOI: 10.1016/j.apnum.2018.04.001

Citing the MRBFT

The academic reference for the MRBFT is this paper. The BibTex entry for the paper is

@article{Sarra2017,
doi = {10.5334/jors.131},
year = {2017},
month = {March},
volume = {5},
author = {S. A. Sarra},
title = {The {M}atlab Radial Basis Function Toolbox},
journal = {Journal of Open Research Software}
}

or in plain text:

S. A. Sarra. The Matlab Radial Basis Function Toolbox. Journal of Open Research Software, vol. 5, 2017.