Radial Basis Functions
Fall 2008
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Instructor: Office / Office hours: |
ECOT 214 / M 3-4, WF 2-3. |
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Class hours/ location: Prerequisites: |
MWF 12.00-12.50 / ECST 1B21 |
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Texts: |
Course text (on PS methods): Fornberg, B., A Practical Guide to Pseudospectral methods (Cambridge Univ. Press, 1996)
Additional (optional) good reading: PS methods: Trefethen, L.N., Spectral Methods in Matlab (SIAM, 2000) Boyd, J.P., Chebyshev and Fourier Spectral Methods (Dover, 2000). RBF methods: Fasshauer, G.E., Meshfree Approximation Methods with Matlab (World Scientific Publishing Co., 2007) Wendland, H., Scattered Data Approximation (Cambridge University Press, 2005) |
| Some additional course materials: |
Slides for presentations: Uppsala,
May 2008, Göttingen,
Sept 2008. Lecture notes - Background and PS methods: Numerical ODE solvers , Padé approximants, Generation of FD weights, Fourier series and transforms, Some PS notes Lecture notes - RBFs: Non-singularity, Powell-Five Lectures on RBFs . Papers: Observations on the behavior of RBF approximations near boundaries . Preliminary manuscript Modeling in Applied Mathematics; See in particular sections of Fourier transform/series (pages 146-163) and FFT (pages 371-376). Codes: weights.m. |
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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 Sept. 22 Assignment 2. Due Oct. 20. Solution (by Erik) Assignment 3. Due Oct. 29 |
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Exam: |
Final exam will be held Wednesday Dec 17, 1.30 - 4.00 pm. |
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Final Grade: |
Based on presentations and other assignments, a midterm and a final exam. |
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Course Description / Syllabus: |
This course will provide a foundation for two of the most powerful methodologies that are known for solving PDEs to high accuracy both on both regular and highly irregular domains, in any number of dimensions. Pseudospectral Methods: On a geometrically simple domain with smooth data, pseudospectral (PS) methods usually form the most powerful known approach. For the same amount of coding and computational cost, these methods often achieve 10-15 digits of accuracy where a finite difference or finite element method would give two or three. At lower accuracies, they typically require less memory than the alternatives. Shortly after their discovery some 30 years ago, PS methods quickly became recognized as often successful and, at times, far superior alternatives to methodologies such as finite differences and finite elements in several application areas, such as fluid dynamics (e.g. wave motion and weather forecasting), computational electromagnetics, etc. In this course, we will explain how, when, and why the PS approach works, more by means of examples and heuristic explanations than by rigorous theoretical arguments. A key theme will be the close connection that exists between PS and finite difference methods. This approach to PS methods not only gives many insights, but also provides numerous algorithmic variations which do not appear naturally from more spectral-based approaches to the subject (such as representing functions in terms of expansions of orthogonal bases functions, like Fourier or Chebyshev). The main limitations of PS methods are (i) PS methods are very restricted geometrically - domain shapes have to be extremely simple (like rectangles or circles in 2-D, etc.), and (ii) It is not possible to employ local mesh refinement (unless using 'spectral elements', a domain decomposition approach with problematic aspects). Radial basis functions (RBF) can be seen as a generalization of PS methods, again offering spectral accuracy but without the main limitations of the PS approach. Domains can be irregular and local node refinement can be highly effective. The coding effort for RBF is very small (a fraction of what is needed for, say, finite elements), and the computational cost is not only independent of the number of dimensions, but also independent of the complexity of the geometry. It is therefore not surprising that RBF have recently found extensive use in many applications, such as solving PDEs from a variety of applications. In this course, we will first discuss a number of properties of RBF approximations, including their particularly high accuracy in a limit that until very recently was thought to be impractical for computing. Their application to numerical solution of PDEs is at the forefront of present research in high order methods. A particularly interesting application area is to computations in spherical domains, which often arise in geophysical and astrophysical applications. The coverage of the first part of the course
(PS methods) will mainly follow my book "A Practical Guide to
Pseudospectral
Methods" (Cambridge University Press, 1996). The supplementary book
"Spectral Methods in Matlab" by L.N.
Trefethen ( The main topics that will be discussed
are listed in the course
outline. |
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Some General Information about CU Courses:
Students with Disabilities: Students with disabilities who qualify for academic accommodations must provide a letter from Disability Services (DS) and discuss specific needs with the professor, preferably during the first two weeks of class. DS determines accommodations based on documented disabilities (303-492-8671, Willard 322, www.colorado.edu/sacs/disabilityservices)
Religious Conflicts: Students who have conflicts between religious observance dates and course examinations or assignments please contact me before September 6. The campus policy can be viewed at (http://www.colorado.edu/policies/index.html).
Student Honor Code: The
Student Honor
Code may be viewed at (http://www.colorado.edu/academics/honorcode/)