Meshfree Methods (MA 5324)

Meshfree methods are a comparatively new approach for the (numerical) interpolation or approximation of some unknown function. In contrast to more traditional techniques - like finite differences, finite elements, spectral methods or wavelets - they do not require a mesh of the underlying computational domain (where the unknown function lives) to be generated first. These methods are very flexible with respect to the geometry of the domain, work in higher space dimensions and their basic implementation is strikingly simple.

This lecture will cover the basics of meshfree methods with an emphasis on approximation by radial basis functions. The lecture will be intertwined with programming exercises in MATLAB.

News

No lecture on July 5. Replaced by the talk of Prof. Wendland on July 23, 17:15, MI 02.08.011

Lectures

Do 12-14, MI 02.08.011

• Chapter 1: Introduction
  • Problem 1.3: 1D interpolation with euclidean distance basis functions
  • Problem 1.4: error in dependence of the number of knots
  • Problem 1.5: 2D interpolation with euclidean distance basis functions
  • Problem 1.7: interpolation with Gaussian RBFs
• Chapter 2: Positive Definite Functions
  • Problem 2.11: interpolation with inverse multi quadrics RBFs
• Chapter 3: Completely monotone functions
• Chapter 4: Compactly supported RBFs
  • Problem 4.5: interpolation with compactly supported RBFs
  • Problem 4.6: comparison of full vs. sparse version
  • Problem 4.7: interpolation error in dependence of shape parameter
  • Problem 4.8: meshfree 2D interpolation with compactly supported RBFs
• Chapter 5: Native spaces
• Chapter 6: Native space error estimates
  • Problem 6.1: 2D cardinal basis functions
• Chapter 7: Moving least squares
  • Problem 7.1: Shepard's method

Software

• kdtree and sdist.m, MEX versions (faster)
  • How to compile the package:
    • at the MATLAB prompt, type "mex -setup" in order to set up the compiler ("doc mex" for options)
    • change to the kdtree directory, compile by, e.g. "mex kdtree.cc"
• kdtree_m and sdist.m, pure MATLAB versions
• points.m

Klausur

27.7., 10:15, MW 1350. Hilfsmittel: ein doppelseitig beschriebenes DIN A4-Blatt.

References

• Gregory E. Fasshauer: Meshfree approximation methods with MATLAB. World Scientific Publishing, 2007.
• Holger Wendland: Scattered data approximation. Cambridge University Press, 2005.