Title:  Universal wave patterns

Abstract:  A feature of solutions of a (generally nonlinear) field
theory can be called "universal" if it is independent of side conditions like initial data. I will explain this phenomenon in some detail and then illustrate it in the context of the sine-Gordon equation, a fundamental relativistic nonlinear wave equation. In particular I will describe some recent results (joint work with R. Buckingham) concerning a universal wave pattern that appears for all initial data that crosses the separatrix in the phase portrait of the simple pendulum.  The pattern is fantastically complex and beautiful to look at but not hard to describe in terms of elementary solutions of the sine-Gordon equation and the collection of rational solutions of the famous inhomogeneous Painlev\'e-II equation.

Title:  Automating and Stabilizing the Discrete Empirical Interpolation Method for Nonlinear Model Reduction

Abstract:  The Discrete Empirical Interpolation Method (DEIM) is a technique for model reduction of nonlinear dynamical systems.  It is based upon a modification to proper orthogonal decomposition which is designed to reduce the computational complexity for evaluating reduced order nonlinear terms.  The DEIM approach is based upon an interpolatory projection and only requires evaluation of a few selected components of the original nonlinear term.  Thus, implementation of the reduced order nonlinear term requires a new code to be derived from the original code for evaluating the nonlinearity.  I will describe a methodology for automatically deriving a code for the reduced order nonlinearity directly from the original nonlinear code.  Although DEIM has been effective on some very difficult problems, it can under certain conditions introduce instabilities in the reduced model.  I will present a problem that has proved helpful in developing a method for stabilizing DEIM reduced models.