Radial Basis Function Methods

One of my research interests is in radial basis function (RBF) collocation methods for pde.

Publications and Reports:

[1] Driscoll and Heryudono
Adaptive Residual Subsampling Methods for Radial Basis Function Interpolation and Collocation Problems
Computers & Mathematics with Applications, Volume 53, Issue 6, March 2007, Pages 927-939

[2] Heryudono
Adaptive radial basis function methods for the numerical solution of partial differential equations, with application to the simulation of the human tear film.
Thesis (Ph.D.)-University of Delaware. 2008.

[3] Neves, Heryudono, Driscoll, Ferreira, Soares
Adaptive methods for analysis of composite beams and plates with radial basis functions.
Proceedings of the ninth international conference on computational structures technology, Civil-Comp Press, Stirlingshire, UK, Paper 8, 2008.

[4] Heryudono and Driscoll
Radial Basis Function Interpolation on Irregular Domain Through Conformal Transplantation
Journal of Scientific Computing, Volume 44, Number 3, 2010.

[5] Neves, Driscoll, Heryudono, Ferreira, Soares, Jorge
Adaptive Methods for Analysis of Composite Plates with Radial Basis Functions
Mechanics of Advanced Materials and Structures, Volume 18, Issue 6, Sept 2011, Pages 420-430.

[6] Sarra, Heryudono, Wang
A numerical study of a technique for shifting eigenvalues of radial basis function differentiation matrices
Technical report, Department of Mathematics, Marshall University, 2011.

[7] Heryudono and Larsson
FEM-RBF: A geometrically flexible, efficient numerical solution technique for partial differential equations with mixed regularity
Marie Curie FP7 Technical Report, August 16 2012.

[8] Larsson, Lehto, Heryudono, Fornberg
Stable Computation of Differentiation Matrices and Scattered Node Stencils Based on Gaussian Radial Basis Functions
SIAM Journal on Scientific Computing, Volume 35, Issue 4, 2013.

[9] Larsson, Gomes, Heryudono, Safdari-Vaighani
Radial Basis Function Methods in Computational Finance
Proceedings of the 13th International Conference on Computational and Mathematical Methods in Science and Engineering, CMMSE 2013.

[10] Safdari-Vaighani, Heryudono, Larsson
A Radial Basis Function Partition of Unity Collocation Method for Convection-Diffusion Equations Arising in Financial Applications
Journal of Scientific Computing, Volume 64, Issue 2, 2015. Matlab code to generate 3D results in the paper: rbfpu_convdiff3D.zip

[11] Chen, Gottlieb, Heryudono, Narayan
A Reduced Radial Basis Function Method for Partial Differential Equations on Irregular Domains
Journal of Scientific Computing, Volume 66, Issue 1, 2016.

[12] Heryudono, Larsson, Ramage, von Sydow
Preconditioning for Radial Basis Function Partition of Unity Methods
Journal of Scientific Computing, Volume 67, Issue 3, 2016.

[13] Kowalewski, Larsson, Heryudono
An Adaptive Interpolation Scheme for Molecular Potential Energy Surfaces
The Journal of Chemical Physics, Volume 145, Issue 8, 2016.

Matlab 7 Codes for Adaptive Radial Basis Functions Methods

I like the idea of sharing MATLAB codes to support reproducible research in RBF. Feel free to modify, use, and tweak the codes to solve other PDE problems. I welcome any bugs report, updated codes, feedbacks, and advices for improvement.

Adaptive 1-D interpolation problem
coarserefine.m contains a short matlab codes that implement the adaptive residual subsampling method for radial basis function 1-D interpolation problem. The default test function used is Runge function but users can provide their own 1-D function.

Adaptive 1-D initial-boundary value problem
adaptburgers_mol.zip contains matlab codes that solve time-dependent Burgers' Equation using method of lines with adaptive residual subsampling for RBFs in space. This example has been used many times to check many adaptive methods. Unzip the file, and run adaptburgers_mol.m.

Adaptive 2-D interpolation problem (updated Nov 30 2006 to include boundary adaptation)
coarserefine2d.zip contains matlab codes that provide basic template for adaptive residual subsampling method for radial basis function 2-D interpolation problem. The default test function used is Franke function but users can provide their own 2-D function. Unzip the file, and run coarserefine2d.m.


Adaptive 2-D initial-boundary value problem



adaptburgers2d_mol.zip contains matlab codes that solve time-dependent 2D Burgers' Equation using method of lines with adaptive residual subsampling method for RBFs in space. The program is not fast but it is usable. I do adaptation at every time step but users can tweak it to readapt at every two or other steps. Unzip the file, and run adaptburgers2d_mol.m. I included this demo in my 8 minute presentation