CS 791/891 Spring 2006 
Combinatorial Scientific Computing:
Algorithms in the Real World: 
T, Th 3:00 -- 4:15 P.M. (tentative) ECS 2120 

Alex Pothen 

How does Google search for web pages when you type in a query? 
How do bioinformatics companies mine large biological networks 
to identify a  protein as a drug target? 
How do  semiconductor researchers  create computational simulations 
of nanomaterials for the future processors? 
What common computational tools are used by scientists and engineers 
working in such diverse fields as  fusion energy, automobile  design, 
aircraft design and financial optimization?

A common ingredient in modeling these problems on a computer is scientific
computing algorithms.  
In this graduate course, we will discuss some of the combinatorial and 
algebraic algorithms employed by these computational applications.  
A knowledge of these algorithms should give students the
background needed to work on these emerging applications  
at a DOE national laboratory or in industry. 

Students who wish to take this course will find knowledge of 
algorithms and linear algebra at the undergraduate level essential. 
They should also be comfortable with programming in a modern 
programming language or an environment such as Matlab.
Students will also work with  a state of the art software package,  
OBLIO, designed and implemented 
by Florin Dobrian, a member of our research team at Old Dominion University. 

Florin Dobrian will give some of the lectures in the course.
Our plan is to write a text book on combinatorial scientific computing
based  on these lectures. 
Currently there is no book discussing  these modern algorithms
in combinatorial scientific computing, 
which  have been developed in the last twenty years. 
Some lectures will also be given by Assefaw Gebremedhin. 

Grades will be based on regular homework assignments,
which will include  analytical and computational problems.
Students will also have the opportunity to complete a  
computational project implementing some of the algorithms 
discussed in class. 

A tentative  syllabus for the course follows: 

I: Overview
   Modern scientific computing algorithms 
   Matrix factorizations

II: Simple graph models for scientific computing
     Matrix-vector multiplication, triangular solution
     Sparse matrix computations in Matlab 

III: Solution of Positive Definite Systems: A first look

    Sparse Cholesky factorization
    Graph models, chordal graphs 
    Data structures: elimination tree, symbolic factors
    Factorization Algorithms: left-looking, and right-looking algorithms 
    The design of OBLIO

IV: Solution of Positive Definite Systems: A second  look

    High performance algorithms via blocking: 
    Supernodes and supernodal algorithms
    The multifrontal algorithm
    Ordering algorithms: minimum degree and nested dissection
    Quotient graphs 
    Graph partitioning algorithms 

V: Solution of Symmetric Indefinite Systems 
    The need for numerical pivoting and  2 by 2 pivots
    Delaying columns in numerical factorization
    Modifications for an indefinite factorization 
        
VI: Solution of Unsymmetric Systems of Equations
    Directed graph model
    Directed Acyclic Graphs and graph reductions 
    Factorization algorithms
    
VII: Matchings 
     Weighted matchings  for ``heavy''  diagonals
     Fast approximation algorithms for  weighted matchings
     Block triangular form of a sparse matrix 

VIII: Graph Coloring 
     Computing derivative matrices  and coloring 
     Jacobian computation and distance-2 coloring 
     Hessian computation and star coloring
     Automatic differentiation 

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