Some Brief Mathematical Notes

Prof. Steven G. Johnson, Dept. of Mathematics, MIT

Overview

Below are some brief notes I've written up for my students and for use in my courses, especially 18.369 (Mathematical Methods in Nanophotonics) and 18.336 (Numerical Methods for PDEs).

For the most part, they are reviews and introductions to topics that appear in the literature, but may not have a compact pedagogical presentation that includes all of the information that I would like my students to be aware of. The references in the notes are by no means exhaustive (these are not formal review papers), but I try to give at least a pointer or two for more information.

The Notes

• Notes on adjoint methods for 18.336 — adjoint methods provide ways to evaluate gradients of complicated functions quickly, and are very important for optimization and sensitivity analysis. Matlab code: schrodinger_fd_adj.m, schrodinger_fd_opt.m. You will also need the nonlinear conjugate-gradient code by H. B. Nielsen: conj_grad.m, linesearch.m.
  • Notes on adjoint methods and sensitivity analysis for recurrence relations — an extension of the above notes to gradients of the solution of recurrence relations, since this example does not seem to be explicitly treated in any of the references I found (although I'm sure someone has done it).
• Notes on the algebraic structure of wave equations — a few notes pointing out how all of the common wave equations can be written in a particular anti-Hermitian algebraic form, from which their properties can be derived in a unified way. (In some sense, this is the definition of a wave equation.)
• Notes on Perfectly Matched Layers (PML) — discussion of various viewpoints on PML absorbing layers, and their properties and limitation, focusing on the fundamentals rather than on the detailed implementation in, for example, FDTD.
• Notes on coordinate transformation and invariance in electromagnetism — a concise derivation of a beautiful result by Ward & Pendry, showing that any coordinate transformation of Maxwell's equations can be absorbed into a change of materials ε and μ.
• A brief survey of computational photonics, excerpted from a draft manuscript of the upcoming Photonic Crystals: Molding the Flow of Light, second edition (to be published spring 2008).
• Saddle-point integration of C_∞ "bump" functions — discussion of asymptotic Fourier analysis of the C_∞ (infinitely differentiable) "bump" functions that often appear in analysis, using saddle-point methods (other common asymptotic methods fail here because of the essential singularity in the integrand).
• Notes on a discontinuous f(x) satisfying f(x+y) = f(x)⋅f(y) — a relatively elementary discussion of how one can construct (though not explicitly) a non-exponential discontinuous function satisfying f(x+y) = f(x)⋅f(y), a simple real-analysis question that is surprisingly non-trivial to answer
• Notes on the convergence of trapezoidal-rule quadrature: the error rate of trapezoidal-rule quadrature, and by extension Clenshaw-Curtis quadrature, can be linked in an elementary way to the convergence rate of a Fourier series and hence to the smoothness of a function.