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public 01:24:58

Ju Sun : When Are Nonconvex Optimization Problems Not Scary?

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Many problems arising from scientific and engineering applications can be naturally formulated as optimization problems, most of which are nonconvex. For nonconvex problems, obtaining a local minimizer is computationally hard in theory, never mind the global minimizer. In practice, however, simple numerical methods often work surprisingly well in finding high-quality solutions for specific problems at hand.

In this talk, I will describe our recent effort in bridging the mysterious theory-practice gap for nonconvex optimization. I will highlight a family of nonconvex problems that can be solved to global optimality using simple numerical methods, independent of initialization. This family has the characteristic global structure that (1) all local minimizers are global, and (2) all saddle points have directional negative curvatures. Problems lying in this family cover various applications across machine learning, signal processing, scientific imaging, and more. I will focus on two examples we worked out: learning sparsifying bases for massive data and recovery of complex signals from phaseless measurements. In both examples, the benign global structure allows us to derive geometric insights and computational results that are inaccessible from previous methods. In contrast, alternative approaches to solving nonconvex problems often entail either expensive convex relaxation (e.g., solving large-scale semidefinite programs) or delicate problem-specific initializations.

Completing and enriching this framework is an active research endeavor that is being undertaken by several research communities. At the end of the talk, I will discuss open problems to be tackled to move forward.

public 01:34:51

Bruce Pitman : CANCELLED

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CANCELLED

public 01:14:36

Andrew J. Bernoff : Domain Relaxation in Langmuir Films

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We report on an experimental and theoretical study of a molecularly thin polymer Langmuir layers on the surface of a Stokesian subfluid. Langmuir layers can have multiple phases (fluid, gas, liquid crystal, isotropic or anisotropic solid); at phase boundaries a line tension force is observed. By comparing theory and experiment we can estimate this line tension. We first consider two co-existing fluid phases; specifically a localized phase embedded in an infinite secondary phase. When the localized phase is stretched (by a transient stagnation flow), it takes the form of a bola consisting of two roughly circular reservoirs connected by a thin tether. This shape relaxes to the minimum energy configuration of a circular domain. The tether is never observed to rupture, even when it is more than a hundred times as long as it is thin. We model these experiments by taking previous descriptions of the full hydrodynamics (primarily those of Stone & McConnell and Lubensky & Goldstein), identifying the dominant effects via dimensional analysis, and reducing the system to a more tractable form. The result is a free boundary problem where motion is driven by the line tension of the domain and damped by the viscosity of the subfluid. The problem has a boundary integral formulation which allows us to numerically simulate the tether relaxation; comparison with the experiments allows us to estimate the line tension in the system. We also report on incorporating dipolar repulsion into the force balance and simulating the formation of "labyrinth" patterns.

public 01:29:47

Elisabetta Matsumoto : Biomimetic 4D Printing

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The nascent technique of 4D printing has the potential to revolutionize manufacturing in fields ranging from organs-on-a-chip to architecture to soft robotics. By expanding the pallet of 3D printable materials to include the use stimuli responsive inks, 4D printing promises precise control over patterned shape transformations. With the goal of creating a new manufacturing technique, we have recently introduced a biomimetic printing platform that enables the direct control of local anisotropy into both the elastic moduli and the swelling response of the ink.

We have drawn inspiration from nastic plant movements to design a phytomimetic ink and printing process that enables patterned dynamic shape change upon exposure to water, and possibly other external stimuli. Our novel fiber-reinforced hydrogel ink enables local control over anisotropies not only in the elastic moduli, but more importantly in the swelling. Upon hydration, the hydrogel changes shape accord- ing the arbitrarily complex microstructure imparted during the printing process.

To use this process as a design tool, we must solve the inverse problem of prescribing the pattern of anisotropies required to generate a given curved target structure. We show how to do this by constructing a theory of anisotropic plates and shells that can respond to local metric changes induced by anisotropic swelling. A series of experiments corroborate our model by producing a range of target shapes inspired by the morphological diversity of flower petals.

public 01:34:55

Boyce E. Griffith : Multiphysics and multiscale modeling of cardiac dynamics

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The heart is a coupled electro-fluid-mechanical system. The contractions of the cardiac muscle are stimulated and coordinated by the electrophysiology of the heart; these contractions in turn affect the electrical function of the heart by altering the macroscopic conductivity of the tissue and by influencing stretch-activated transmembrane ion channels. In this talk, I will present mathematical models and adaptive numerical methods for describing cardiac mechanics, fluid dynamics, and electrophysiology, as well as applications of these models and methods to cardiac fluid-structure and electro-mechanical interaction. I will also describe novel models of cardiac electrophysiology that go beyond traditional macroscopic (tissue-scale) descriptions of cardiac electrical impulse propagation by explicitly incorporating details of the cellular microstructure into the model equations. Standard models of cardiac electrophysiology, such as the monodomain or bidomain equations, account for this cellular microstructure in only a homogenized or averaged sense, and we have demonstrated that such homogenized models yield incorrect results in certain pathophysiological parameter regimes. To obtain accurate model predictions in these parameter regimes without resorting to a fully cellular model, we have developed an adaptive multiscale model of cardiac conduction that uses detailed cellular models only where needed, while resorting to the more efficient macroscale equations where those equations suffice. Applications of these methods will be presented to simulating cardiac and cardiovascular dynamics in whole heart models, as well as in detailed models of cardiac valves and novel models of aortic dissection. Necessary physiological details will be introduced as needed.