Numerical methods blog posts

I recently got a review copy of Scientific Computing: A Historical Perspective by Bertil Gustafsson. I thought that thumbing through the book might give me ideas for new topics to blog about. It still may, but mostly it made me think of numerical methods I've already blogged about.

In historical order, or at least in the order of Gustafsson's book:

• Fixed point iteration
• Newton's method for root-finding
• Runge phenomenon
• Gauss quadrature
• Taylor series
• Fourier series
• Gibbs phenomenon
• Fourier transform
• Runge-Kutta
• Linear programming
• Nonlinear optimization
• Iterative linear solvers
• Natural cubic spline interpolation
• Wavelets
• Fast Fourier Transform (FFT)
• Singular value decomposition
• Monte Carlo methods
• MCMC

The links above include numerical methods I have written about. What about the methods I have not written about?

I like to write fairly short, self-contained blog posts, and I've written about algorithms compatible with that. The methods in Gustafsson's book that I haven't written about are mostly methods in partial differential equations. I don't see how to write short, self-contained posts about numerical methods for PDEs. In any case, my impression is that not many readers would be interested. If you are one of the few readers who would like to see more about PDEs, you may enjoy the following somewhat personal rambling about PDEs.

I studied PDEs in grad school-mostly abstract theory, but also numerical methods-and expected PDEs to be a big part of my career. I've done some professional work with differential equations, but the demand for other areas, particularly probability and statistics, has been far greater. In college I had the impression that applied math was practically synonymous with differential equations. I think that was closer to being true a generation or two ago than it is now.

My impression of the market demand for various kinds of math is no doubt colored by my experience. When I was at MD Anderson we had one person in biomathematics working in PDEs, and then none when she left. There may have been people at MDACC doing research into PDEs for modeling cancer, but they weren't in the biostatistics or biomathematics departments. I know that people are working with differential equations in cancer research, but not at the world's largest cancer center, and so I expect there aren't many doing research in that area.

Since leaving MDACC to work as a consultant I've seen little demand for differential equations, more in Kalman filters than in differential equations per se. A lot of companies hire people to numerically solve PDEs, but there don't seem to be many who want to use a consultant for such work. I imagine most companies with an interest in PDEs are large and have a staff of engineers and applied mathematicians working together. There's more demand for statistical consulting because companies are likely to have an occasional need for statistics. The companies that need PDEs, say for making finite element models of oil reservoirs or airplanes, have an ongoing need and hire accordingly.