Suggested Project Topics

Symbolic computing packages, such as Maple, are wonderful tools for exploration and experiment, but are typically too slow for large-scale applications. Numerical computing tools, such as Matlab, can be highly efficient (if used properly) but lack the facility for symbolic manipulation. However, a new Matlab-based system called Chebfun is trying to change that.

A student who undertakes this project would investigate the mathematics that underpins the system, while developing new programmes and examples to solve problems with chebfuns. A particularly interesting example might be the (numerical) solution of certain PDEs.

You should consider this project if you like numerical analysis and programming.

A different example is Schwarz Methods for linear boundary value problems. These have existed for over 100 years, but are being ``rediscovered'' because they are so suitable for parallel computers

A project on this topic would mainly involve programming and algorithm design. Code could be tested on some Linux computers running software to emulate parallel and distributed systems.

The student who does this project should like programming. It would be useful to have studied numerical analysis.

Given a function, it is not hard to construct a numerical scheme to estimate its derivatives at certain points. However, it is easy to show that such schemes are highly susceptible to numerical error.

Automatic differentiation is an approach based on repeated use of the chain rule and which can be applied to arbitrary functions and should be accurate to machine precision. It can be particularly useful in problems where one needs to find derivatives of vector-valued functions (Jacobians...).

This project will begin with reading a recent article: Introduction to Automatic Differentiation and MATLAB Object-Oriented Programming, Richard D. Neidinger. SIAM Rev. 52, pp. 545-563. The ideas and tools will be developed and applied to (I hope!) some interesting but non-trivial problems.

Numerical Analysis is the area of mathematics that is concerned with the design and analysis of methods and algorithms for obtaining useful solutions to mathematical problems. Two of the main sources of problems are differential equations and linear algebra. The applications are far too numerous to mention.

The interested student should browse a few books in the library. For example Afternotes on Numerical Analysis and Afternotes goes to Graduate School (G.W. Stewart), Introduction of Numerical Analysis (Stoer and Bulirish),

Depending on the students' interests, projects may have little or no computing aspects, or may be strongly focused on computational problems, or something in between.

Why not come up with your own idea? A good place to start would be the Education articles in SIAM Review.

Or read some mathematical blogs. The most famous is Terry' Tao's.

David Gleich's blog features some neat ideas and demos.

Or you could go to the library and have a look at such titles as

• Amy N. Langville and Carl D. Meyer Google's PageRank and beyond : the science of search engine rankings
• Michael W. Berry and Murray Browne, Understanding search engines : mathematical modeling and text retrieval
• John Fauvel, Music and mathematics: from Pythagoras to fractals
• Gardner, Martin, The night is large : collected essays, 1938-1995