CSCI5636

Parallel Algorithms and Software for Numerical Solution of Partial Differential Equations

CSCI 5636

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:

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

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.

Reference books:

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.