The goal of this course is to give you an increased understanding of and practice in using computational techniques to answer scientific questions. We will study how to set up problems, what techniques and software are available to solve these problems, what the possible pitfalls are, and learn some physics and a bit of other science along the way. As part of learning how to run simulations, we will create virtual "demonstrations" of physical phenomena using languages such as Java to visualize the simulations.

The relationship between science and computers as developed over this century is quite deep. To a large degree, this is a reflection of the inseparability of mathematics and science; the computer is a tool that allows us to perform very complex calculations. The computer is also a physical object, of course, and its abilities are governed by technology and are limited by physical laws. Computations performed by computers have allowed scientists and engineers to design products from space shuttles to video games to faster computers. Besides the design of complex experiments and collecting data from these experiments, they have also been crucial in the theoretical exploration of subjects from chaos to particle physics to the evolution of galaxies.

The computational tools that will be used include Mathematica, C/C++, Java, and possibly PERL.

This course is under development, so the topics may shift some. Currently, the plan is to cover the following:

General principles in computational science

How does one set up and solve a problem on the computer? Does one treat the problem as a "continuum" problem or a "discrete" problem. Does one take a "Monte Carlo" approach (based on random sampling) or a deterministic approach? Is one interested in a numeric answer or a symbolic one? What types of approximations are used in the model and in the algorithms? How accurate is the answer? What resources are needed?

Random walks

A recurring theme in modelling is that of random walks. This is one of the simplest "Monte Carlo" methods. Here, we will be initially concerned with what a "Monte Carlo" method is and how to generate "random" numbers. We will then study the motion of small particles, the behavior of light in clouds, the motion of bacteria, and the price of stock options.

Chaos and Fractals

In the study of dynamics, the characterization of "chaos" is used for systems which have very unpredictable behavior. This type of behavior is quite different from a single planet orbiting a star or a falling object. The unpredictabliity does not come from ignorance, but from the "sensitivity" of the motion to small effects. Often, the behavior of chaos is reflected in the fractal nature of the path the system follows (in "phase space"). A fractal shape is one that does not have a simple dimension like 1 or 2, but is a non-integral number.

Waves

Partial differential equations. Wave propagation, reaction-diffusion equations (leopard spots).

Other topics

Models of evolution. Least squares fitting. Optimization.