Applied Mathematics 7400-006

Radial Basis Functions

Fall 2012

 

Instructor:

Office / Office hours:

Bengt Fornberg 

ECOT 214 /  Office hours: MWF 11-12. However, please feel free to stop by any time at your convenience. 

Class hours/ location: 

Prerequisites:

MWF  2.00-2.50 / ECEE  265
Recommended:   Numerical analysis (at least undergraduate version), linear algebra, complex variables.

Texts:

Main course text (manuscript in preparation):

Fornberg, B. and Flyer, N., Radial Basis Functions: Developments and Applications (final version to be published by SIAM)

Recommended further reading (on PS methods):

Fornberg, B., A Practical Guide to Pseudospectral methods (Cambridge Univ. Press, 1996) 

Recommended further reading (on RBF methods):

Fasshauer, G.E., Meshfree Approximation Methods with Matlab  (World Scientific Publishing Co., 2007)

Additional (optional) good reading:

Trefethen, L.N., Spectral Methods in Matlab (SIAM, 2000)

Boyd, J.P., Chebyshev and Fourier Spectral Methods (Dover, 2000).

Wendland, H., Scattered Data Approximation (Cambridge University Press, 2005)
Some additional
course materials:		 Slides for presentations:   Oxford 2010  , CU 2012     
Lecture notes - PS methods:  Periodic , Non-periodic, Numerical ODE solvers , Padé approximants, Generation of FD weights, Fourier series and transforms, Some PS notes, FFT algorithm .
Lecture notes - RBFs:  Non-singularity, Powell-Five Lectures on RBFs , RBF-GA algorithm, FD PS RBF RBF-FD slides   
Papers:  A split step approach for the 3-D Maxwell's equations , A hybrid RBF-PS method , RBF-GA algorithm.
Codes:  weights.m.

Assignments:

Some regular assignments will be given. However, the course will be largely project based, with studies of research papers and presentations, implementations of small research problems, etc. mainly replacing regular homeworks.

Assignment 1. Due 10/03/12.

Assignment 2. Due 10/22/12.

Assignment 3. Due 12/10/12.

Presentations:  Comparisons of transforms, Generalized Fourier transform, Lebesgue constant, Pade approximants, Runge phenomenon, Staggered grids, Wave equation at discontinuous interfaces . 

Exam:

Final exam will be held  Monday December 17, 4.30  - 7.00 pm. 

Final Grade:

Based on presentations and other assignments, possibly a midterm exam, and a final exam.

Course Description / Syllabus:

General course description:

This course is focused on two powerful numerical methodologies for solving PDEs to high accuracy in any number of dimensions: Pseudospectral (PS) methods and Radial Basis Functions (RBFs).  These methods apply to a wide range of PDEs, arising for example in fluid mechanics, wave motions, astro- and geo-sciences, etc. While this technically is an upper level graduate course on a numerical topic, one of its primary goals is to give students who are not specializing in numerical methods but who need to solve problems in these areas an up-to-date survey style overview of the methods. Our focus will be on how, when, and why the different approaches work, more by means of examples and heuristic explanations than by any rigorous theoretical arguments. The course assignments will be mostly take the form of small projects and presentations.

Four main course topics:

Both of the main techniques, PS and RBF methods, can be traced back to finite differences (FD). The present research frontier in the field appears to now be moving towards RBF-generated finite difference (RBF-FD) methods. The natural order for covering these areas follows quite closely how they developed historically.

Finite difference methods:   These were first proposed for solving PDEs in 1910, and they have remained a dominant methodology ever since. Generally, they are easy to implement, but more restrictive than for example finite elements in terms of geometric flexibility.

Pseudospectral methods:   For applications in very simple geometries (long intervals in 1-D, rectangular or circular domains in 2-D, periodic boxes in 3-D, spherical shells, etc.) it was noted in the early 1970's that the infinite order of accuracy limit of FD methods exists and that it can offer spectacular computational efficiencies. Another way to arrive at the same PS methods is via expansions in orthogonal functions (such as Fourier, Chebyshev, and spherical harmonics). These PS methods soon became prominent for solving PDEs in numerous areas, including fluid dynamics (such as direct numerical simulations of turbulent flows), weather forecasting, long time evolution of linear and nonlinear waves, and computational electromagnetics.

Radial basis functions:   Also this methodology has several origins. The theme we will follow is to generalize PS methods away from their geometric limitations and their dependence on very regular node layouts (which for PS methods makes it very complicated to carry out local refinements in critical solution areas). This can be done while preserving their spectral accuracy (beyond any power of the typical node spacing). It transpires that all PS methods can be seen as highly specialized (and typically not optimal) special cases of RBFs. Both the coding effort and the computational cost of RBFs are independent of how simple or complicated the geometry might be. In a recent large-scale 3-D geophysical flow application, an RBF-based code on a standard PCs competed very favorably against all previous methodologies, even when these were implemented on large supercomputer systems.

Radial basis function-generated FD methods:   It has very recently been discovered that using RBFs to create generalized FD methods might offer the best opportunity yet for combining the strengths of all the previous approaches. In particular, RBF-FD methods can often reach very high computational speed since they only rely on local approximations, and they also give rise to sparse rather than to full matrix problems.  Preliminary development work on RBF-FD methods is under way at several research centers, including a close collaboration between ourselves and NCAR.


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