Scientific Computation II

General Information

This is the second half of a two semester introduction to graduate level numerical analysis and scientific computing. The main goal is to build the necessary foundation for numerically solving problems that arise in the sciences. The procedures by which ordinary and partial differential equations as well as integral equations are discretized are described. Numerical linear algebra is needed in all these problems and forms a substantial part of the course. The course is intended for a general audience of graduate students from the sciences. While basic theoretical concepts are carefully investigated, substantial interest is placed on applications.

Syllabus

Numerical Linear Algebra

• Orthogonal vectors and matrices
• Singular value decomposition
• QR factorization
• Least squares
• Conditioning and stability
• Linear systems - direct methods
• Eigenvalues
• Linear systems - iterative methods

Linear systems from discretized PDEs

• PDE classification
• Diffusion equation
• Poisson, Helmholtz equations
• Finite difference, finite volume, finite element approaches

N-body problems and integral equations

• Fredholm problems of first and second kind
• Green's function
• Fast methods

Grading Policy

Coursework

• Homework - 6 assignments x 6 points = 36 points (3 theory, 3 application questions per homework, 2-week cycle)
• Computational project = 14 points (projects given after Spring Break, due May 1st)
• Midterm - 4 problems x 5 points = 20 points
• Final - 6 problems x 5 points = 30 points
• Extra credit - A number of topics will be proposed during the course. You can choose to draft a small report on two of these topics. Each report can receive 4 course points.

The midterm examination can be retaken during the Finals.

Mapping of point score to grades

Course Texts, Notes, Supplementary Material

Numerical Linear Algebra, Lloyd N. Treffethen and David Bau, SIAM, 1997 ISBN-13: 978-0-898713-61-9, $57.50 from SIAM

Here are some additional references which are useful to complement the necessarily limited view of the problems of mathematical physics problems presented in the lecture notes.

• An Introduction to Partial Differential Equations, Michael Renardy, Robert C. Rogers, Springer 2004.
• Methods of Mathematical Physics, Harold Jeffreys, Bertha Swirles Jeffreys, Cambridge, 1956
• Boundary Value Problems of Mathematical Physics, Ivar Stakgold, SIAM

• Original Hestenes & Stiefel paper on conjugate gradient method: hestenes-stiefel-cg.pdf

• Historical note on conjugate gradient methods: CGhistory.pdf

Homework, computational projects

CourseProject.pdf (due May 1st)

Bonus point projects

Here's a list of the various topics that have been proposed during the lectures as additional work that will be awarded bonus points.

• Revisit SVD and change from 2-norm to 1-norm and inf-norm. Redo all of Lect. 4, 5 (4 points)
• Present an analytical Gram-Schmidt procedure using a symbolic computation package (1 point)

• Use ATLAS in course project (4 points)

Due dates:

• March 12 for extra credit from first half of semester
• April 27 for extra credit from second half of semester

Examinations

• midterm.pdf (March 6. regular class hour)

• Upload new attachment "final.pdf" (Monday, May 4 at noon, 3 hours)

Course Bulletin Boards

Please choose the appropriate section for your question. Questions are answered daily, typically late at night.