Radial basis function (RBF) approximations for PDE problems

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

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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.

rbf.png

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

The flat RBF limit

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.

Flexible software for RBFs

Implementation and parallelization aspects of RBF approximation.

Publications

Refereed

  1. A guide to RBF-generated finite differences for nonlinear transport: Shallow water simulations on a sphere. Natasha Flyer, Erik Lehto, Sébastien Blaise, Grady B. Wright, and Amik St-Cyr. In Journal of Computational Physics, volume 231, number 11, pp 4078-4095, 2012. (DOI).
  2. Radial basis functions for the time-dependent Schrödinger equation. Katharina Kormann and Elisabeth Larsson. 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).
  3. Stable computations with Gaussian radial basis functions. Bengt Fornberg, Elisabeth Larsson, and Natasha Flyer. In SIAM Journal on Scientific Computing, volume 33, pp 869-892, 2011. (DOI).
  4. Stabilization of RBF-generated finite difference methods for convective PDEs. Bengt Fornberg and Erik Lehto. In Journal of Computational Physics, volume 230, pp 2270-2285, 2011. (DOI).
  5. Rotational transport on a sphere: Local node refinement with radial basis functions. Natasha Flyer and Erik Lehto. In Journal of Computational Physics, volume 229, pp 1954-1969, 2010. (DOI).
  6. A note on radial basis function interpolant limits. Martin D. Buhmann, Slawomir Dinew, and Elisabeth Larsson. In IMA Journal of Numerical Analysis, volume 30, pp 543-554, 2010. (DOI).
  7. Multi-dimensional option pricing using radial basis functions and the generalized Fourier transform. Elisabeth Larsson, Krister Åhlander, and Andreas Hall. In Journal of Computational and Applied Mathematics, volume 222, pp 175-192, 2008. (DOI).
  8. Improved radial basis function methods for multi-dimensional option pricing. Ulrika Pettersson, Elisabeth Larsson, Gunnar Marcusson, and Jonas Persson. In Journal of Computational and Applied Mathematics, volume 222, pp 82-93, 2008. (DOI).
  9. A new class of oscillatory radial basis functions. Bengt Fornberg, Elisabeth Larsson, and Grady Wright. In Computers and Mathematics with Applications, volume 51, pp 1209-1222, 2006. (DOI).
  10. Option pricing using radial basis functions. Ulrika Pettersson, Elisabeth Larsson, Gunnar Marcusson, and Jonas Persson. In Proc. ECCOMAS Thematic Conference on Meshless Methods, pp C24.1-6, Departamento de Matemática, Instituto Superior Técnico, Lisboa, Portugal, 2005.
  11. Theoretical and computational aspects of multivariate interpolation with increasingly flat radial basis functions. Elisabeth Larsson and Bengt Fornberg. In Computers and Mathematics with Applications, volume 49, pp 103-130, 2005. (DOI).
  12. Some observations regarding interpolants in the limit of flat radial basis functions. Bengt Fornberg, Grady Wright, and Elisabeth Larsson. In Computers and Mathematics with Applications, volume 47, pp 37-55, 2004. (DOI).
  13. A numerical study of some radial basis function based solution methods for elliptic PDEs. Elisabeth Larsson and Bengt Fornberg. In Computers and Mathematics with Applications, volume 46, pp 891-902, 2003. (DOI).

Reports

  1. Stable computations with Gaussian radial basis functions in 2-D. Bengt Fornberg, Elisabeth Larsson, and Natasha Flyer. Technical report / Department of Information Technology, Uppsala University nr 2009-020, 2009. (External link).

Supervised M.Sc. theses

  • 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 project participants

  • Elisabeth Larsson, Ph.D., Docent, Dept. of IT, Scientific Computing, Uppsala University.
  • Alfa Heryudono, Ph.D., Dept. of Mathematics, University of Massachusetts, Dartmouth, MA, USA (visiting researcher jun 2010-aug 2011).
  • Katharina Kormann, M.Sc., Ph.D. student, Dept. of IT, Scientific Computing, Uppsala University.
  • Erik Lehto, M.Sc., Ph.D. student, Dept. of IT, Scientific Computing, Uppsala University.
  • Ali Safdari-Vaighani, M.Sc., Ph.D. student, Dept. of Applied Mathematics, Iran University of Science and Technology, Tehran, Iran (visiting Ph.D. student 2011).
  • Ulrika Sundin, M.Sc., Ph.D. student, Dept. of IT, Scientific Computing, Uppsala University (on maternity leave).
  • Martin Tillenius, M.Sc., Ph.D. student, Dept of IT, Scientific Comptuing, Uppsala University.

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Some recent and current 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.
  • Sverker Holmgren, Dept. of Information Technology, Scientific Computing, Uppsala University.
  • Simon Hubbert, Ph.D., Dept. of Economics Mathematics and Statistics, Birckbeck college, University of London, London, UK.
  • Amir Malekpour, M.Sc., Ph.D. student, Dept. of Water Sciences Engineering,Tabriz University, Tabriz, Iran.
  • Axel Målqvist, Dept. of Information Technology, Scientific Computing, Uppsala University.
  • 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.
  • Lina von Sydow, Dept. of Information Technology, Scientific Computing, Uppsala University.
  • Grady Wright, Ph.D., Dept. of Mathematics, University of Utah, Salt Lake City, UT, USA.

Former collaborators

  • Jonas Persson, Ph.D., SunGard Front Arena.
  • Krister Åhlander, Ph.D., Docent, TietoEnator.

Contact

For more information contact Elisabeth Larsson.