One of my research interests is in radial basis function (RBF) collocation methods for PDEs.
Publications and Reports:
[18] Tominec, Larsson, Heryudono
A Least Squares Radial Basis Function Finite Difference Method with Improved Stability Properties
SIAM Journal on Scientific Computing, volume 3, issue 2, 2021.
[17] Mollapourasl, Haghi, Heryudono
Numerical Simulation and Applications of the Convection-Diffusion-Reaction Equation with
the Radial Basis Function in a Finite-Difference Mode
Journal of Computational Finance, volume 23, number 5, 2020.
[16] Safdari-Vaighani, Larsson, Heryudono
Radial Basis Function Methods for the Rosenau Equation and Other Higher Order PDEs
Journal of Scientific Computing, 2017.
[15] Bernal, Heryudono, Larsson
Numerical Solution of the Viscous Flow Past a Cylinder with a Non-global Yet Spectrally
Convergent Meshless Collocation Method
Lecture Notes in Computational Science and Engineering, Volume 119, Pages 495-507, 2017.
[14] Larsson, Shcherbakov, Heryudono
A Least Squares Radial Basis Function Partition of Unity Method for Solving PDEs
SIAM Journal on Scientific Computing, Volume 39, Issue 6, 2017.
[13] Kowalewski, Larsson, Heryudono
An Adaptive Interpolation Scheme for Molecular Potential Energy Surfaces
The Journal of Chemical Physics, Volume 145, Issue 8, 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.
[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.
[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
[09] 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.
[08] 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.
[07] 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.
[06] 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.
[05] 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.
[04] Heryudono and Driscoll
Radial Basis Function Interpolation on Irregular Domain Through
Conformal Transplantation
Journal of Scientific Computing, Volume 44, Number 3, 2010.
[03] 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.
[02] 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.
[01] 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
I like 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, feedback, and advice for improvement.
Adaptive 1-D interpolation problem
coarserefine.m
contains short MATLAB codes that implement the adaptive residual
subsampling method for radial basis function 1-D interpolation problem.
The default test function used is the 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 a basic template for adaptive residual
subsampling method for radial basis function 2-D interpolation problem.
The default test function used is the 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 the 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