Computing
As discussed in class, it is often necessary to conduct Local Fourier
Analysis computationally. Here is a sample Matlab script for LFA:
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Helpful MG Links
MGNET: Repository and info page for the multigrid research community.
MUDPACK: A multigrid software package (there are many of these out there).
Matlab: Much of the programming can be accomplished using Matlab. It is particularly useful for testing algorithms. Of course, you can do the assignments in C/C++, FORTRAN, or others as well.
MathSciNet: From Brown, you should have access to the AMS paper database.
Course Info
Text: A Multigrid Tutorial, Second Edition, by
William L. Briggs, Van Emden Henson, and Steve F. McCormick. Click here for the book's website, including errata and other information.
Other Texts:
- Multigrid by Trotenberg, Schuller, and Oosterlee.
- An Introduction to Multigrid Methods by Pieter Wesseling. Downloadable here
- Mathematics and Computational Techniques for Multilevel Adaptive Methods, by Ulrich Ruede
Note: The course will closely follow the first part of
A Multigrid Tutorial.
Course Description: Multigrid methods were first recognized in the 1970's for their ability to efficiently solve partial differential equations, however it has been only recently that the method has attained a wider appel. Early efforts in multigrid development were focused on elliptic-type problems and, in many cases, were shown to be optimal solvers. As research continues, the method is expanding into many areas of application, including aerodynamics, astrophysics, chemistry, elecromagnetics, hydrology, medical, applications, and quantum mechanics.
The purpose of this course is to introduce the fundamental concepts of the multigrid methodology. Key components of the multigrid process will be studied, including discretization of the PDE and iterative methods (smoothers). A geometric-based multigrid approach will be taken and the elements of the method will be considered in this context. Depending on time and interest, an algebraic-based multigrid method will also be examined.
Grading, Exams:The grade for the course will be based on (1) homework and (2) the final project with a weight of 50% each. There will be no exams, but homework will include programming and supplemented questions. The final project should be based on a culmination of topics covered in the course and is chosen based on the student's interests (with intructor's approval). The requirements for this are fairly broad and open and will be discussed in more detail as the course progresses.
Syllabus
Tentative outline of possible topics to be covered in the course:
- Numerical PDEs
- Finite differences
- Stencil notation
- Computational motivation
- Iterative Methods
- Smoothers: Gauss-Seidel, Jacobi
- Mode analysis
- Multigrid Components
- Interpolation, restriction
- Coarse-grid, correction
- V, W, and FMG cycles
- Implementation
- Convergence analysis
- Mode analysis, cont'd.
- Performance diagnostics
- Theoretical Observations
- Variational property
- Linear algebra
- Extensions
- Nonlinear PDEs
- Anisotropic PDEs
- Algebraic-based MG
- FAC
