The Department of Applied Mathematics is pleased to host this series of colloquium lectures, funded in part by a generous gift from the Boeing Company. This series will bring to campus prominent applied mathematicians from around the world.

The talks should be of general interest to researchers and students in the mathematical sciences and related fields. All are welcome to attend.

There will be three talks in this series each quarter, on Thursday afternoons at 4:00pm. Each talk will be followed by a reception. Boeing seminars are currently held in Smith Hall 205.

# Autumn Quarter 2014

## October 23, 2014.

Kevin Zumbrun, Indiana University

**Abstract: Modulation of spatially periodic patterns and behavior of thin film flows**

Periodic patterns and traveling waves arise quite generally in optics, biology, chemistry, and many other applications. A great success story over the past couple decades for the dynamical systems approach to PDE has been the rigorous treatment of modulation of periodic patterns in reaction diffusion systems. However, the techniques used were designed for modulations with a single degree of freedom. For systems possessing one or more conservation laws, hence two or more degrees of freedom in particular, the Kuramoto-Sivashinsky, Saint Venant, and other equations governing thin film flow- these methods do not apply. Here, we present an approach applying also to this more general situation, rigorously verifying an associated “Whitham system” formally governing slow modulations under suitable numerically verifiable stability assumptions on the spectra of the linearized operator about the background pattern. This verifies/explains a number of numerically observed phenomena in thin film flow, including `viscoelastic behavior” in cellular Kuramoto-Sivashinsky behavior, and the “homoclinic paradox” in inclined thin-film flow, the latter concerning the puzzling phenomenon that asymptotic behavior appears to consist of solitary waves, despite that solitary waves are readily seen to be exponentially unstable. We conclude by discussing verification of our spectral assumptions in weakly and strongly unstable (corresponding to small and large Froude number) regimes, giving simple power-law formulae describing the stability boundaries in each case. The latter, strongly unstable description, relevant in applications to hydraulic engineering/dam spillway construction, was unexpected and to our knowledge is completely new.

## October 30, 2014.

Charbel Farhat, Stanford University

**Abstract: **Energy-Conserving Sampling and Weighting for the Model Reduction of 2nd-Order Nonlinear Dynamical Systems

The computational efficiency of a typical projection-based nonlinear model reduction method hinges on the efficient approximation, for explicit computations, of the projection onto a subspace of a residual vector. For implicit computations, it also hinges on the additional efficient approximation of the projection of the Jacobian of this residual with respect to the solution. The Empirical Interpolation Method (EIM), its discrete counterpart DEIM, the Gauss-Newton with Approximated Tensors (GNAT) method, and the Gappy Proper Orthogonal Decomposition method (Gappy POD) are popular methods for performing such approximations. They differ in several aspects such as their applicability at the continuous, semi-discrete, or fully discrete level(s), their underlying left and right projectors, and their suitability for explicit and/or implicit schemes. However, they all share accuracy as the primary driver for their algorithmic design. They have also all demonstrated various forms of success for the reduction of nonlinear computational models emanating from elliptic, parabolic, and first-order hyperbolic partial differential equations. The first objective of this talk however is to show that they all lack robustness for second-order nonlinear dynamical systems because they do not necessarily preserve the numerical stability properties of the computational model they reduce. Consequently, the second objective of this talk is to present ECSW, an Energy-Conserving Sampling and Weighting method for the model reduction of second-order nonlinear dynamical systems such as those arising, for example, in structural dynamics, solid mechanics, wave propagation, and device analysis.

This proposed hyper reduction method is physics-based and natural for finite element semi-discretizations. It is applicable at both the semi-discrete and discrete levels. Unlike all aforementioned reduction methods, it preserves the Lagrangian structure associated with Hamilton’s principle and therefore is guaranteed to preserve the numerical stability properties of the nonlinear system it reduces. The error committed by ECSW during an online approximation is bounded by the error committed during the offline approximation of the training samples. Therefore, the online error can be estimated a priori and is controllable. The performance of ECSW will be first demonstrated for a set of academic but nevertheless challenging nonlinear dynamic response problems taken from the literature, and compared to that of DEIM and its unassembled variant recently introduced for finite element computations under the name UDEIM. Next, the potential of ECSW for complex second-order dynamical systems with strong nonlinearities will be highlighted with the realistic simulation of the transient response of a generic V-hull vehicle to an underbody blast event. For this hihgly nonlinear time-dependent problem, ECSW will be shown to deliver an excellent level of accuracy while enabling the reduction of CPU time by more than four orders of magnitude.

## November 20, 2014.

Erhan Bayraktar, Professor of Mathematics & Susan M. Smith Professor.

Department of Mathematics, University of Michigan

**Abstract: **Stochastic Perron’s method for Hamilton-Jacobi-Bellman equations

We show that the value function of a stochastic control problem is the unique solution of the associated Hamilton-Jacobi-Bellman (HJB) equation, completely avoiding the proof of the so-called dynamic programming principle (DPP). Using Stochastic Perron’s method we construct a super-solution lying below the value function and a sub-solution dominating it. A comparison argument easily closes the proof. The program has the precise meaning of verification for viscosity-solutions, obtaining the DPP as a conclusion. We will also discuss the effectiveness of this method in analyzing the robust feedback-type optimal control problems.

**Remaining schedule to be announced**