Radial basis function (RBF) approximations for PDE problems
Members of the RBF research group documenting the view during the Dolomite Research Week on Approximation 2015. Photo: Alvise Sommariva.
The main focus of this project is to develop numerical techniques based on RBF methods that are stable, efficient and can be applied to real application problems. We are particularly interested in high-dimensional applications because of their extreme demands.
What is RBF approximation
The picture below is an example of how the RBFs can be visualized in a two-dimensional computational domain. Six weighted radial basis functions, drawn as red surfaces in the picture, are scattered over the computational domain. Their sum build up the interpolant, represented by the transparent surface with wireframe.
The main advantages of the RBF method are
- The method is meshfree, which means that it is flexible with respect to the geometry of the computational domain. It also means that the method is suitable for problems where data is only available at scattered points.
- The method is not more complicated for problems with many space dimensions, since the only geometrical property that is used is the pairwise distance between points.
- For smooth functions, approximations with smooth RBFs can give spectral convergence.
Master thesis projects
We regularly offer subjects for MSc thesis projects. Please consult our list of available projects. You may also contact us directly to discuss alternative topics.
Directions of research
Biomechanical simulation of the respiratory muscles
The focus is put to the simulation of the diaphragm, the main muscle of the respiratory system. The underlying model is based on the equations of nonlinear elasticity which are solved on a realistic 3-dimensional geometry using RBFs.
Numerical investigations and theory concerning the limit where the RBFs become flat. This limit is interesting because it can produce very accurate results for smooth functions and it reproduces multivariate polynomial interpolation.
RBFs for PDEs
General algorithms and methods for solving PDE problems using RBFs.
RBFs for high-dimensional PDEs
Algorithms and methods specifically designed for high-dimensional application fields such as financial mathematics and quantum dynamics.
RBFs for global climate simulation
RBF methods for linear and non-linear applications in geophysics.
Software
Various RBF codes, mostly in MATLAB are collected under the RBF software page.
Publications
Refereed
- Preconditioning for radial basis function partition of unity methods. In Journal of Scientific Computing, volume 67, pp 1089-1109, 2016. (DOI, fulltext:postprint).
- BENCHOP—The BENCHmarking project in Option Pricing. In International Journal of Computer Mathematics, volume 92, pp 2361-2379, 2015. (DOI, fulltext:postprint).
- A scalable RBF–FD method for atmospheric flow. In Journal of Computational Physics, volume 298, pp 406-422, 2015. (DOI, fulltext:postprint).
- A radial basis function partition of unity collocation method for convection–diffusion equations arising in financial applications. In Journal of Scientific Computing, volume 64, pp 341-367, 2015. (DOI, fulltext:postprint).
- A Galerkin radial basis function method for the Schrödinger equation. In SIAM Journal on Scientific Computing, volume 35, pp A2832-A2855, 2013. (DOI).
- Stable computation of differentiation matrices and scattered node stencils based on Gaussian radial basis functions. In SIAM Journal on Scientific Computing, volume 35, pp A2096-A2119, 2013. (DOI).
- Stable calculation of Gaussian-based RBF-FD stencils. In Computers and Mathematics with Applications, volume 65, pp 627-637, 2013. (DOI).
- A guide to RBF-generated finite differences for nonlinear transport: Shallow water simulations on a sphere. In Journal of Computational Physics, volume 231, pp 4078-4095, 2012. (DOI).
- Radial basis functions for the time-dependent Schrödinger equation. In Numerical Analysis and Applied Mathematics: ICNAAM 2011, volume 1389 of AIP Conference Proceedings, pp 1323-1326, American Institute of Physics (AIP), Melville, NY, 2011. (DOI).
- Stable computations with Gaussian radial basis functions. In SIAM Journal on Scientific Computing, volume 33, pp 869-892, 2011. (DOI).
- Stabilization of RBF-generated finite difference methods for convective PDEs. In Journal of Computational Physics, volume 230, pp 2270-2285, 2011. (DOI).
- Rotational transport on a sphere: Local node refinement with radial basis functions. In Journal of Computational Physics, volume 229, pp 1954-1969, 2010. (DOI).
- A note on radial basis function interpolant limits. In IMA Journal of Numerical Analysis, volume 30, pp 543-554, 2010. (DOI).
- Multi-dimensional option pricing using radial basis functions and the generalized Fourier transform. In Journal of Computational and Applied Mathematics, volume 222, pp 175-192, 2008. (DOI).
- Improved radial basis function methods for multi-dimensional option pricing. In Journal of Computational and Applied Mathematics, volume 222, pp 82-93, 2008. (DOI).
- A new class of oscillatory radial basis functions. In Computers and Mathematics with Applications, volume 51, pp 1209-1222, 2006. (DOI).
- Option pricing using radial basis functions. In Proc. ECCOMAS Thematic Conference on Meshless Methods, pp C24.1-6, Departamento de Matemática, Instituto Superior Técnico, Lisboa, Portugal, 2005.
- Theoretical and computational aspects of multivariate interpolation with increasingly flat radial basis functions. In Computers and Mathematics with Applications, volume 49, pp 103-130, 2005. (DOI).
- Some observations regarding interpolants in the limit of flat radial basis functions. In Computers and Mathematics with Applications, volume 47, pp 37-55, 2004. (DOI).
- A numerical study of some radial basis function based solution methods for elliptic PDEs. In Computers and Mathematics with Applications, volume 46, pp 891-902, 2003. (DOI).
Reports
Supervised M.Sc. theses
- Evaluation of a least-squares radial basis function approximation method for solving the Black-Scholes equation for option pricing. Student thesis, supervisor: Elisabeth Larsson, examiner: Lina von Sydow, Jarmo Rantakokko, IT nr 12 051, 2012. (fulltext).
- Adapting a Radial Basis Functions Framework for Large-Scale Computing. Student thesis, supervisor: Elisabeth Larsson, examiner: Jarmo Rantakokko, IT nr 12 050, 2012. (fulltext).
- Parallelizing a Software Framework for Radial Basis Function Methods. Student thesis, supervisor: Martin Tillenius, Elisabeth Larsson, examiner: Michael Thuné, Anders Jansson, IT nr 11 084, 2011. (fulltext).
- Designing a Flexible Software Tool for RBF Approximations Applied to PDEs. Student thesis (Master Programme in Computational Science), supervisor: Elisabeth Larsson, Martin Tillenius, examiner: Michael Thuné, Anders Jansson, IT nr 10 058, 2010. (fulltext).
- Björn Rodhe, A discontinuous Galerkin method with local radial basis function interpolation, UPTEC Report F 07 066, School of Engineering, Uppsala University, 2007. (Advisors: E. Larsson and S.-E. Ekström)
- Andreas Hall, Pricing financial derivatives using radial basis functions and the generalized Fourier transform, UPTEC Report IT 05 036, School of Engineering, Uppsala University, 2005. (Advisors: E. Larsson and K. Åhlander)
- Gunnar Marcusson, Option pricing using radial basis functions, UPTEC Report F 04 078, School of Engineering, Uppsala University, 2004. (Advisors: E. Larsson and L. von Sydow)
- Ulrika Pettersson, Radial basis function approximations for the Helmholtz equation, UPTEC Report F 03 082, School of Engineering, Uppsala University, 2003. (Advisor: E. Larsson)
Current RBF research group members
- Elisabeth Larsson, Ph.D., Docent, Dept. of IT, Scientific Computing, Uppsala University.
- Lina von Sydow, Ph.D., Docent, Dept. of IT, Scientific Computing, Uppsala University.
- Slobodan Milovanovic, M.Sc., Ph.D. student, Dept. of IT, Scientific Computing, Uppsala University.
- Victor Shcherbakov, M.Sc., Ph.D. student, Dept. of IT, Scientific Computing, Uppsala University.
- Ulrika Sundin, M.Sc., Ph.D. student, Dept. of IT, Scientific Computing, Uppsala University (on maternity leave).
- Igor Tominec, M.Sc., Ph.D. student, Dept. of IT, Scientific Computing, Uppsala University.
Research group at Katalin, Feb 2011 | In the Dolomites, Sep 2015 |
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Former RBF research group members
Still collaborators, but at a longer distance.
- Alfa Heryudono, Ph.D., Dept. of Mathematics, University of Massachusetts, Dartmouth, MA, USA (visiting researcher jun 2010-aug 2011).
- Ali Safdari-Vaighani, Ph.D., Allameh Tabatabai University, Tehran, Iran (visiting Ph.D. student 2011).
- Cecile Piret, Ph.D., Applied Mechanics and Mathematics (MEMA), Université Catholique de Louvain (UCL), Belgium (visiting researcher jul 2012-sep 2012).
- Erik Lehto, Ph.D., Numerical Analysis, KTH Royal Institute of Technology, Stockholm (PhD from Uppsala University 2012).
- Katharina Kormann, Ph.D., Max-Planck-Institut für Plasmaphysik, Munich, Germany (PhD from Uppsala University in 2012).
- Martin Tillenius, Ph.D., COMSOL, Stockholm (PhD from Uppsala University in 2014).
- Ahmad Saeidi, Iran University of Science and Technology, Tehran, Iran (visiting PhD student 2015).
- Jamal Amani Rad, Ph.D., Shahid Beheshti University, Tehran, Iran (visiting PhD student 2015).
Some international collaborators
- Martin Buhmann, Prof., Mathematical Institute, Justus-Liebig-Universität Giessen, Germany.
- Natasha Flyer, Ph.D., Div. of Scientific Computing, The National Center for Atmospheric Research (NCAR), Boulder, CO, USA.
- Bengt Fornberg, Prof., Dept. of Applied Mathematics, University of Colorado, Boulder, CO, USA.
- Sônia Gomes, Prof., Dept. of Applied Mathematics, University of Campinas, Brazil.
- Simon Hubbert, Ph.D., Dept. of Economics Mathematics and Statistics, Birckbeck college, University of London, London, UK.
- Amir Malekpour, Ph.D., Hydraulic Structure Engineering, University of Guilan, Iran.
- Alison Ramage, Reader, Dept. of Mathematics and Statistics, University of Strathclyde, Scotland.
- Robert Schaback, Prof., Institute for Numerical and Applied Mathematics, Georg-August-University Göttingen, Germany.
- Grady Wright, Ph.D., Dept. of Mathematics, University of Utah, Salt Lake City, UT, USA.
Contact
For more information contact Elisabeth Larsson.