Parallel Algorithms and Software for Numerical
Solution of Partial Differential Equations
The mathematical models of most scientific and engineering problems
take the form of partial differential equations. With the rapid
development of high speed computers over the past decades, the
possibilities of efficiently utilizing these models have dramatically
increased. With the new possibilities, intensive researches in
computer simulation of large scale engineering problems are bringing
revolutionary changes in science and engineering. The focus of this
course is on efficient numerical methods and software for solving partial
differential equations, with emphasis on the newly developed parallel
algorithms, such as domain decomposition and multigrid methods.
Among the topics are:
Students will be expected to implement some of the algorithms
on either a parallel machine.
A knowledge of a programming language, such as Fortran, C or C++
is essential. Parallel programming experience is not required.
Grade will be based on class projects.
- Introduction to partial differential equations
- Graph partitioning, sparse matrices, direct methods
- Algebraic iterative methods for linear systems, Conjugate Gradient, GMRES, QMR, ...
- Multigrid methods
- Domain decomposition methods
- Scalability of iterative methods
- Using MPI (the Message-Passing Interface) on cluster of workstations
- Algorithms implementation on sequential and parallel computers
- Performance measurement
- C. Johnson, Numerical Solution of Partial Differential Equations
by the Finite Element Method, Cambridge University Press, 1987.
- S. Brenner and L. Scott, The Mathematical Theory of Finite
Element Methods, Springer, 1994.
- J. Strikwerda, Finite Difference Schemes and Partial Differential
Equations, Wadsworth and Brook/Cole Mathematics Series, 1989.
- G. Golub and J. Ortega, Scientific Computing:
An Introduction with Parallel Computing, Academic Press, 1992.
- W. Hackbusch, Iterative Solution of Large Sparse Systems
of Equations, Springer--Verlag, Berlin, 1993.
- B. Smith, P. Bjorstad and W. Gropp, Domain Decomposition: Parallel
Multilevel Methods for Elliptic Partial Differential Equations,
Cambridge University Press, 1996.
- B. Briggs, V. Henson, and S. McCormick, A Multigrid
Tutorial (Second Edition), SIAM, 2000.
- W. Gropp, E. Lusk and A. Skjellum, Using MPI: Portable Parallel
Programming with the Message-Passing Interface, The MIT Press, 1994.
- Y. Saad, Iterative Methods for Sparse Linear Systems,
PWS Kent, 1995.