
Advanced Numerical Linear Algebra  AMSC/CMSC 763, Fall 2017

Basic Information



Overview


The course will survey topics in numerical linear algebra, with emphasis
on solution algorithms for sparse linear systems of equations and numerical
methods for solving eigenvalue problems.
Both theoretical and computational issues will be studied.



Outline of Topics Covered

 Iterative Methods
 Krylov subspace methods:
the conjugate gradient, MINRES, GMRES methods and generalizations
 Preconditioning methods: uses and derivation
 Multigrid methods for partial differential equations
Algebraic multigrid
 Communicationavoiding methods
 Sparse Direct Methods
 Reordering schemes and general sparse elimination
 Hierarchical semiseparable matrices
 Mathematical software
 Computational Methods for Eigenvalue Problems
 Lanczos and Arnoldi methods
 The implicitly restarted Arnoldi method
 Subspace iteration
 The JacobiDavidson method
 Solution Methods for Structured Problems
 Saddlepoint problems
 Constraint preconditioners
 Linear algebra and control
 Linear algebra and uncertainty quantification


Homework

 
References

There will be no assigned text.
Much of the material can be found in the references listed below.
These are largely affordable books and it will be useful to purchase
one or two of them.

T. A. Davis,
Direct Methods for Sparse Linear Systems,
SIAM Publications, Philadelphia, 2006.

J. W. Demmel,
Applied Numerical Linear Algebra,
SIAM Publications, Philadelphia, 1997.

H. C. Elman, D. J. Silvester and A. J. Wathen,
Fast Solvers and Finite Elements,
Oxford University Press, Oxford, 2005.

A. George and J. W.H. Liu,
Computer Solution of Large Positive Definite Systems,
PrenticeHall, New Jersey, 1981.

G. H. Golub and C. Van Loan,
Matrix Computations,
Third Edition, Johns Hopkins University Press, 1996.

A. Greenbaum,
Iterative Methods for Solving Linear Systems,
SIAM Publications, Philadelphia, 1997.

Y. Saad,
Iterative Methods for Sparse Linear Systems,
Second Edition, SIAM Publications, Philadelphia, 2003.

Y. Saad,
Numerical Methods for Large Eigenvalue Problems,
Manchester University Press, Manchester, 1992.
Available at
http://wwwusers.cs.umn.edu/~saad/books.html.

G. W. Stewart,
Matrix Algorithms Volume II: Eigensystems,
SIAM Publications, Philadelphia, 2001.

D. S. Watkins,
The Matrix Eigenvalue Problem: GR and Krylov Subspace Methods
SIAM Publications, Philadelphia, 2007.


Grading

Grades will be determined as follows:
 46 homework assignments: 40%
 Inclass midterm examination: 25%
 Course Project: 35%
The homework will consist of both analysis and computational testing.
Computations can be done using any convenient programming tools although
Matlab will be encouraged.
Plagiarism:
You are welcome to discuss assignments in a general way among yourselves,
but you may not use other students' written work or programs.
Use of external references for your work should be cited.
Clear similarities between your work and others will result in a grade
reduction for all parties.
Flagrant violations will be referred to appropriate university authorities.

 
