[GTP] IMAGe Seminar- David Keyes-July 22 at 2:00pm

[Silvia Gentile]

David E. Keyes
Columbia University

Tuesday, July 22, 2008
Mesa Laboratory , Main Seminar Room (part of Theme- of-the-Year Summer 
School)
Lecture 2:00pm

Domain Decomposition Methods for Partial Differential Equations

Domain decomposition, a form of divide-and-conquer for mathematical 
problems posed over a physical domain is the most common paradigm for 
large-scale simulation on massively parallel, distributed, hierarchical 
memory computers. In domain decomposition, a large problem is reduced to 
a collection of smaller problems, each of which is easier to solve 
computationally than the undecomposed problem, and most or all of which 
can be solved independently and concurrently. Domain decomposition has 
proved to be an ideal paradigm not only for execution on advanced 
architecture computers, but also for the development of reusable, 
portable software. The most complex operation in a typical domain 
decomposition method -- the application of the preconditioner -- carries 
out in each subdomain steps nearly identical to those required to apply 
a conventional preconditioner to the undecomposed domain. Hence software 
developed for the global problem can readily be harvested for the local 
problems of a parallel implementation.
Finally, it should be noted that domain decomposition is often a natural 
paradigm for the modeling community. Physical systems are often 
decomposed into two or more contiguous subdomains based on 
phenomenological considerations, such as the importance or negligibility 
of viscosity or reactivity, or any other feature, and the subdomains are 
discretized accordingly, as independent tasks. This physically-based 
domain decomposition may be mirrored in the software engineering of the 
corresponding code, and leads to threads of execution that operate on 
contiguous subdomain blocks. This tutorial provides an overview of 
domain decomposition and focuses on the mathematical development of its 
two main paradigms: Schwarz (projection) and Schur (block elimination) 
preconditioning and their hybrids and nonlinear generalizations.