The Radial Basis Function Toolbox

The 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 RBFT is under constant development as it is heavly used in my 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 I no longer use Matlab.  The Matlab versions will remain available for download and may see minor bug fixes and improvements but will not be developed further.  All future versions of the toolbox will be in the Python programming language.

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:

• rbfX - basic RBF method functionality
  
• 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
  

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

Future plans

• development of new classes for local RBF methods
• rbfLocal - local RBF method for derivative approximation
  
• rbfPU - partition of unity method for derivative approximation
  
• add Mercer's method to the GA class
  
• expansion of the rbfCenters class
• add hyper-viscosity methods for the stabilization of time-dependent PDE problems
• improve documentation
• expand the number and variety of example scripts
• parallel implementation of local methods via openCL
• add a multiquadric (MQ) RBF class
• agumentation of the basic RBF interpolant with polynomials
  

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

PRBFT Downloads - Python version

• version 1.2: (target release date, June 2019)
  

Change logs

References

The MRBFT is described in the manuscript

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

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

• 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 RBFT