Abstract: An emerging theme across many science and engineering domains is modeling materials that can change shape. With Morpho, domain scientists gain a powerful new simulation tool to tackle larger and more complex shape evolution problems than presently possible. One example involves modeling the evolution of nematic liquid crystals with free boundaries, known as nematic tactoids, that are in contact with an isotropic fluid. In this talk, we discuss results from applying a class of classical nonlinear numerical methods to this model and compare them with previously used gradient-descent methods. Moreover, by wrapping the algorithms in a multilevel nested iteration approach, we see significant improvements in the efficiency of the simulations with a variety of initial guesses.

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The development of public health policy is inextricably linked with governance structure. In our increasingly globalized world, human migration and infectious diseases often span multiple administrative jurisdictions that might have different systems of government and divergent management objectives. In this talk, I’ll introduce two examples in which spatial coordination may be critical for disease control. The first is in the management of rabies in vampire bats, where we demonstrate that spatial interactions likely play a key role in driving disease persistence. The second will describe a more general infectious disease in humans. We show that successful management may depend on both the actions of multiple managers and their desired objectives.

Abstract:  Data in high dimensional Euclidean space may often be described with fewer parameters than the ambient space, perhaps lying on or near a submanifold of lower intrinsic dimension. In part one of this talk, we will discuss a new method for estimating this intrinsic dimension from the data: a version of local PCA which is calibrated on quadratic embeddings which may better approximate a manifold at larger scales. The second part of the talk will focus on a technique of random embeddings to reduce ambient dimension. The topics of this talk are joint work with Anna Gilbert.

In this talk I discuss strong stability preserving (SSP) methods. After telling about the results of the last 30 years, I present a class of high order unconditionally strong stability preserving (SSP) implicit two-derivative Runge–Kutta schemes, and SSP implicit-explicit (IMEX) multi-derivative Runge–Kutta schemes where the time-step restriction is independent of the stiff term, as well as new SSP additive implicit Runge–Kutta schemes with downwinding that are unconditionally SSP. These methods depend on a backward-in-time assumption on the derivative of the operator. We show that this backward derivative condition is satisfied in many relevant cases where SSP IMEX schemes are desired. The unconditional SSP condition ensures that these methods are positivity preserving, and we present sufficient conditions under which such methods are also asymptotic preserving when applied to a range of problems, including a hyperbolic relaxation system, the Broadwell model, and the Bhatnagar-Gross-Krook (BGK) kinetic equation.

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Nematic liquid crystals are classified as partially-ordered materials due to the tendency of their molecules to align along preferred directions. This partial alignment, combined with their fluidic properties, render these materials useful in a range of applications, including their widespread use in optical displays. When confined to thin channels, the fluid flow extends the applications of nematics further to optofluidic devices and guided micro-cargo transport. We examine the intrinsic coupling between the fluid motion and the orientation of nematic molecules by considering steady unidirectional flows in thin channels. We work within a reduced Beris-Edwards framework which leads to a system of nonlinear and coupled differential equations. Our analytical and numerical results highlight the universality of order reconstruction solutions, where distinct sub-domains arise, separated by so-called domain walls.

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Recent years have seen promising developments in so-called certifiably optimal estimation, showing that many state estimation and planning problems in robotics, although highly non-convex, may be solved to global optimality. In particular, the quest for global optimality has led to 1) the development of solvers that are more likely to converge to the global optimum, 2) efficient certificates to verify if they did, and 3) a deeper understanding of the theory of when they do.

In this talk, I aim to introduce the audience to this exciting research field and to our recent efforts to make certifiably optimal solvers – for the field of state estimation in particular – not only faster, but also more accessible. Among those efforts, I will present efficient optimality certificates as a low-cost add-on to off-the-shelf local solvers, which apply to a variety of problems including range-only, stereo-camera and, more generally, matrix-weighted localization. Then, I will present our most recent solutions to automatically certify almost any state estimation problem, using a sampling-based method to automatically find a sufficient subset of so-called redundant constraints. I end with an outlook on promising research directions in state estimation and beyond.