Department of Information Technology

Radial basis function (RBF) approximations for PDE problems

RBFphotogroup.jpg
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

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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

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.

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.

Software

Various RBF codes, mostly in MATLAB are collected under the RBF software page.

Publications

Refereed

  1. Preconditioning for radial basis function partition of unity methods. Alfa Heryudono, Elisabeth Larsson, Alison Ramage, and Lina von Sydow. In Journal of Scientific Computing, volume 67, pp 1089-1109, 2016. (DOI, fulltext).
  2. BENCHOP—The BENCHmarking project in Option Pricing. Lina von Sydow, Lars Josef Höök, Elisabeth Larsson, Erik Lindström, Slobodan Milovanovic, Jonas Persson, Victor Shcherbakov, Yuri Shpolyanskiy, Samuel Sirén, Jari Toivanen, Johan Waldén, Magnus Wiktorsson, Jeremy Levesley, Juxi Li, Cornelis W. Oosterlee, Maria J. Ruijter, Alexander Toropov, and Yangzhang Zhao. In International Journal of Computer Mathematics, volume 92, pp 2361-2379, 2015. (DOI, fulltext).
  3. A scalable RBF–FD method for atmospheric flow. Martin Tillenius, Elisabeth Larsson, Erik Lehto, and Natasha Flyer. In Journal of Computational Physics, volume 298, pp 406-422, 2015. (DOI, fulltext).
  4. A radial basis function partition of unity collocation method for convection–diffusion equations arising in financial applications. Ali Safdari-Vaighani, Alfa Heryudono, and Elisabeth Larsson. In Journal of Scientific Computing, volume 64, pp 341-367, 2015. (DOI, fulltext).
  5. A Galerkin radial basis function method for the Schrödinger equation. Katharina Kormann and Elisabeth Larsson. In SIAM Journal on Scientific Computing, volume 35, pp A2832-A2855, 2013. (DOI).
  6. Stable computation of differentiation matrices and scattered node stencils based on Gaussian radial basis functions. Elisabeth Larsson, Erik Lehto, Alfa Heryudono, and Bengt Fornberg. In SIAM Journal on Scientific Computing, volume 35, pp A2096-A2119, 2013. (DOI).
  7. Stable calculation of Gaussian-based RBF-FD stencils. Bengt Fornberg, Erik Lehto, and Collin Powell. In Computers and Mathematics with Applications, volume 65, pp 627-637, 2013. (DOI).
  8. 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, pp 4078-4095, 2012. (DOI).
  9. 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).
  10. 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).
  11. 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).
  12. 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).
  13. 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).
  14. 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).
  15. 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).
  16. 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).
  17. 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.
  18. 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).
  19. 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).
  20. 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

Radial basis function partition of unity methods for pricing vanilla basket options. Victor Shcherbakov and Elisabeth Larsson. Technical report / Department of Information Technology, Uppsala University nr 2015-001, 2015. (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 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.
1270862.jpg Research group at Katalin, Feb 2011 DolomitesSep15.jpg In the Dolomites, Sep 2015

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.
Updated  2017-09-01 14:40:29 by Igor Tominec.