Multigrid Methods and Parallel Computation

Mathematics 234

Developed by Professor Murli Gupta, Department of Mathematics

Bulletin Copy:

Math 234 Multigrid Methods and Parallel Computation (3) Gupta

Multigrid and multilevel techniques for solving partial differential equations; techniques and practice of parallel computation. Numerical solution of partial differential equations; Basic iterative techniques; Multigrid methods; Multilevel preconditioners; Parallel Computation; Domain decomposition methods; Algorithm development; Software packages. Prerequisites: Math 124, Math 153 or equivalent, or permission of instructor; knowledge of a programming language.

Course Description:

The main purpose of this course is to introduce the study of multigrid and multilevel techniques for solving partial differential equations. Multigrid techniques are among the most efficient, advanced methods for solving large scale problems arising in scientific and engineering computation.

The course includes both theory and practice. The theory and analysis of basic multigrid techniques is presented to set the direction of the course. This is done through the study of standard iterative techniques that often require substantial computational resources. Various multigrid and multilevel techniques are introduced and the performance of these methods is analyzed theoretically, where possible, and demonstrated numerically. Students are exposed to standard software available in the field and are required to write their own software components.

An added feature of this course is the introduction of parallel computational techniques for solving partial differential equations. Parallel computers are increasingly becoming available for scientific computation and, for optimal usage, require special methods for planning and coding the algorithms. The course brings in the parallel computation in conjunction with the standard iterative techniques as well as multigrid methods.

Prerequisites: Students should have a background in matrix theory (Math 124) and numerical analysis (Math 153) at advanced undergraduate level. Most of the course work will require the use of a computer programming language (e.g., Fortran, C, Pascal). Students may also be able to use the computational packages such as Matlab, Mathematica, Maple and spreadsheets such as Microsoft Excel and Corel Quattro Pro.

Grading:  Students will be graded on the basis of take home assignments. The assignments will often involve computer programming to obtain solutions. A course project, developed in consultation with the individual students, may also be required. Students may be expected to make class presentations on their course projects. Time will be allowed for extensive consultations and discussions about the solution and programming techniques.

Target Audience: Math 234 was originally designed as a Mathematics Specialty Track course for the Computational Sciences Program. It is intended to be accessible to a wide audience, including engineers and scientists, and it should appeal to graduate students pursuing a variety of degrees (both MS and PhD) in mathematics, physics and engineering.

Text:

William L. Briggs, Van Emden Hansen and Steve F. McCormick, A Multigrid Tutorial, 2nd Edition, SIAM Publications, 2000.

Recommended Text:

Gene H. Golub and Jim M. Ortega, Scientific Computing. An Introduction with Parallel Computing, Academic Press, 1993.

 Topics Introduction to numerical solution of partial differential equations in scientific and engineering computation (4 lectures) Basic iterative techniques: Jacobi, Gauss-Seidel, SOR methods (4 lectures) Multigrid methods; Multilevel preconditioners (8 lectures) Parallel Computation; Domain decomposition methods (8 lectures) Algorithm development; Software packages; Applications (4 lectures)

Instructor
Mathematics Department (Funger 428W)
2201 G Street, NW
Washington, DC 20052
Office Hours for Spring 2003 semester: Mondays and Wednesdays 10:30 a.m.- 12:00 noon,  and by appointment.
Telephone: 202-994-4857
FAX: 202-994-6760
Email: mmg@gwu.edu