Experimental mathematics: the role of computation in nonlinear science

Reviewer: Herbert Maisel

For centuries, mathematicians and scientists studying nature have advanced their studies by abstracting and simplifying the real problem, modeling the simplified version, and analyzing the model. Greatest progress has been made where the models were mathematical and linear. In this case, it was frequently possible to obtain comprehensive solutions analytically. However, nature is complex; it is nonlinear. Analytic solutions can be found for only a small subset of these nonlinear problems. In about 40 years, the electronic digital computer has both increased in speed by a factor of a billion and decreased in cost per computation by a factor of ten million. This has permitted the study of nonlinear phenomena by replacing the space-time continuum with a discrete lattice containing as much as a billion points and by performing calculations that move through the lattice generating billions of results. In recent years it has become possible to obtain output from a computer using color coding and other techniques in such a way that billions of results can be reduced to a series of tens or hundreds of meaningful graphic images. As a result, complex, nonlinear phenomena can be successfully studied by combining intuition and analytic methodology with numerical experimentation. Zabusky [1] calls this “computational synergetics.” This approach to the study of complex phenomena was anticipated by John von Neumann; Birkoff describes von Neumann's foresight in [2]. This well-written, interesting paper describes the way this is done and offers some predictions (perhaps “suggestions” would be a better word) for the computational resources that will prove to be most effective in this kind of numerical experimentation. The paper is well worth reading. Those of you who might like to read about an application of these techniques (to the study of how gas spirals from black holes) should also read Smarr's paper [3].