Encyclopedia of Applied and Computational Mathematics

Is there still a role for Encyclopedias in the age of Wikipedia and Google? This two-volume set from Springer is betting that the answer is yes. They have adopted a mixed strategy: like all Springer books today, the book exists both on paper and online, but the editors have also created a dynamic version online where authors and new contributors can update the papers. Once enough updates have been posted, a new edition will be prepared, and the process restarted. Judging from how my students interact with books, my guess is that the online version (either the stable or the dynamic one) will be where the action is; few libraries will buy the hard copy, and essentially no individuals. That, of course, is fine.

What the Encyclopedia tries to provide are up-to-date surveys of various parts of “Applied and Computational Mathematics.” As the editor notes in the preface,

Modern applied and computational mathematics is more applied, more computational, and more mathematical than ever. The exponential growth of computational power has allowed for much more complex mathematical models, and these new models are typically more realistic for applications.

There is an implied narrative here which is made more explicit later: traditional applied mathematics tended to use more elementary mathematical methods because those were the ones that allowed solution by hand; since we now have computers, we can use more sophisticated methods and solve them numerically:

Harmonic analysis is a typical example. It had its origin in applied and computational mathematics with the work of Fourier on head conduction. However, only very special cases can be studied in a quantitative way by hand. In the years after this beginning, there was substantial progress in pure directions of harmonic analysis. The emergence of powerful computers and the fast Fourier transform (FFT) algorithm drastically changed the scene. This resulted, for example, in wavelets, the inverse Radon transform, a variety of special techniques for PDEs, computational information, and sampling theory and compressed sensing.

Most of the articles are five to ten pages, with a few more general ones being longer. For example, “Classical Iterative Methods” fills twenty pages, while “Convergence Acceleration” gets ten. Many articles include “synonyms” for the topic in question (different communities often use different terms).

There is a list of entries, but not an index, which is a flaw in the printed version. (Online, one presumes that a text search will do the job of an index.) For example, I looked for an article on “multiscale methods” and did not find one. There are articles on “Multiscale Multi-Cloud Modeling and the Tropics” and on “Multiscale Numerical Methods in Atmospheric Science”; indeed, the term appears in several other article titles. Paging back, I found an article on “Multiresolution methods”, in which the “short definition” begins with

Multiresolution (or multiscale) methods decompose an object additively into terms on different scales or resolution levels…

But what if I hadn’t found that?

One impressive feature of the Encyclopedia is its sheer volume and variety, which certainly makes clear both the vitality and range of modern computational and applied mathematics. Students who want to know (in some detail) what is going on will find good information here. Use it online.

Fernando Q. Gouvêa admires applied mathematics but is better at number theory and history of mathematics. He is Carter Professor of Mathematics at Colby College in Waterville, ME.