By David S. Watkins
A challenge for readers of this issue will be to decipher the pretty pictures in our featured review by Elias Wegert. The book under consideration is Riemann--Hilbert Problems, Their Numerical Solution, and the Computation of Nonlinear Special Functions, by Thomas Trogdon and Sheehan Olver. The reviewer illustrates some of the issues by making some nice colorful plots involving the Airy function. Can you decipher them? They make sense to me because I have read Wegert's book Visual Complex Functions. You may have to do the same, and I suspect this is a ploy on the wily reviewer's part to get us all to read his book. By the way, I first learned of Wegert's book from a review by Nick Trefethen in this journal (Vol. 55, Issue 4, December 2013). I bought a copy, I read it, and I recommend it. And what of the book under review? Wegert's view is generally quite favorable—I was impressed by the wealth of material presented, the elaborate techniques, the involved formulas, the sophisticated diagrams, and the high accuracy of numerical computations łdots''—but he also has a few criticisms.
I enjoyed Elisabeth Larsson's review of A Primer of Radial Basis Functions with Applications to the Geosciences, by Bengt Fornberg and Natasha Flyer. Larsson gives a quick overview of the development of radial basis function (RBF) methods over the past thirty years or so, including RBF-FD (finite difference) and RBF-PUM (partition of unity) methods. The book under review develops the RBF-FD method. Larsson tells us that the main message is that RBF methods consistently outperform classical approaches for large-scale geophysical flow problems, and it is time that they are given a place alongside finite element and finite difference methods.
In addition we have several other substantial and informative reviews on a variety of topics, including inequalities for graph eigenvalues, an introduction to scientific computing, reduced basis methods for partial differential equations, variational methods for nonlinear elliptic problems, and two books on mesh generation.
David Watkins is the section editor of SIAM Review.