Applied Math Colloquium - Spring 2020

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January 24

No Colloquium

January 31

Carlos Pérez, Pontificia Universidad Católica de Chile
Hosted by: Catalin Turc

Integral Equation Methods for Inverse Design of Metasurfaces

Metasurfaces are planar metamaterial slabs of subwavelength thickness composed by a large number of optical elements that can be engineered to manipulate the direction, amplitude, phase, and polarization of light. Inverse metasurface designs typically rely on the use of fast but low-order ray-optics (locally-periodic) approximations. Accurate full-wave metasurface simulations, in turn, pose significant challenges to well-established electromagnetic PDE solvers as they lead to intractably large computational domains and/or unreasonably long computing times.

In this talk, I will present two boundary integral equation approaches that partially overcome the aforementioned issues. I will first present a windowed Green function method based on a second-kind single-trace formulation for full-wave simulation of all-dielectric metasurfaces. Finally, I will present a novel integral equation framework that allows to compute corrections and assess the accuracy of a popular ray-optics approximation used in inverse metasurface design.

This is joint work with Raphäel Pestourie (Harvard), Steven G. Johnson (MIT).

February 7

Bryan Quaife, Florida State University
Hosted by: Yuan-Nan Young

Coupling Between Flow and Porous Structures​

In 1855, Henry Darcy experimentally established Darcy's law to relate the flow rate and pressure drop through the permeability. However, in many applications, the permeability is dynamically evolving with the flow.  For example, in groundwater flow, the porous structure is altered by sedimentation, deposition, and erosion. As another example, depending on the tension of an elastic membrane, it can be permeable to a solvent, but not the solute. I will describe recent work on both erosion in groundwater flow, and semi-permeability of elastic membranes. If time permits, I will also discuss a related clamped eigenvalue problem.

February 14

Roy Goodman, NJIT
Hosted by: David Shirokoff

Transfer Entropy for Network Reconstruction in a Simple Model

Dynamics on a network are ubiquitous in areas as diverse as genetics, meteorology, and the social sciences. By a network, we simply mean a system of interacting agents. A common problem is to reconstruct a network given the behavior of its agents, that is, given a set of time series describing the states of each node on the network, determine which other nodes are influencing that node. Information theory provides one set of tools for doing so in a model-independent manner. A popular tool in this game is called transfer entropy. Transfer entropy measures the reduction in uncertainty in predicting the state of one agent at a time (t+1) from its state at time t given the additional knowledge of the state of a second j at time t, compared with predicting it the knowledge of the first agent alone. This can be thought of as measuring the transfer of information from the second agent to the first. We analyze a simple model in which we derive asymptotic formulae relating the transfer entropy between two nodes to the strength of the connection between the nodes. Higher-order terms in this formula show the subtle effect of network topology on the transfer entropy. This allows for more accurate network reconstruction in this particular model and points to possible sources of errors when transfer entropy is used to reconstruct networks in a more general setting.

February 21

Alejandro Rodriguez, Princeton
Hosted by: David Shirokoff

Bringing to Light the Fundamental Limits of Optical Control at the Nanoscale

Nanophotonics deals with the myriad ways of manipulating light by nano-structuring materials at or below the scale of the electromagnetic wavelength. From sub-wavelength metallic resonators to periodic crystals, different combinations of material and geometry can be used to confine light to ultra small domains (thousandths of times smaller than its free space wavelength) and over long times (billions of times longer than the optical period), greatly enhancing its interaction with matter. Applications include new kinds of antennas, lasers, super-lenses, on-chip interconnects, and cloaking devices. In this talk, I will survey recent developments in the area of photonic computational (inverse) design methods, also known as large-scale optimization. Complementing the strengths of physically intuitive design paradigms, these computer-aided design techniques are beginning to probe the fundamental limits of the possible: new nonlinear micron-scale cavities that convert light from infrared to visible wavelengths, textured thin films that enhance single-photon extraction from electronic defects, and heated objects whose radiation greatly surpass (by hundreds even thousands of times) the predictions of the famous Stefan–Boltzmann blackbody limits. How does one know whether these designs are approaching optimal performance (as physically allowed by Maxwell's equations)? Concurrently, I will survey recently developed "dual optimization" methods whose main aim is to shed light on the question of optical bounds. Preliminary work suggests that while there is still plenty of room at the bottom for novel designs, inverse techniques are already producing devices that come within factors of unity of what is possible.

February 28

Rongjie Lai, RPI
Hosted by: Brittany Hamfeldt

Understanding Manifold-Structure Data via Geometric Modeling and Learning

Analyzing and inferring the underlying global intrinsic structures of data from its local information is critical in many fields. Different from image and signal processing which handle functions on flat domains with well-developed tools for processing and learning, manifold-structured data sets are even more challenging due to their complicated geometry. For example, a geometric object can have very different coordinate representations due to various embeddings, transformations or representations (imagine the same human body shape can have different poses as its nearly isometric embedding ambiguities). In this talk, I will first discuss geometric modeling based methods for 3D shape analysis. This approach uses geometric PDEs to adapt the intrinsic manifolds structure of data. It extracts various invariant descriptors to characterize and understand data through solutions of differential equations on manifolds. Inspired by recent developments of deep learning, I will also discuss our recent work of a new way of defining convolution on manifolds. This geometric way of defining parallel transport convolution (PTC) provides a natural combination of modeling and learning on manifolds. PTC allows for the construction of compactly supported filters and is also robust to manifold deformations. It enables further applications of comparing, classifying and understanding manifold-structured data by combining with recent advances in deep learning.

March 6

Misha Tsodyks, Institute for Advanced Study
Hosted by: Victor Matveev

Mathematical Models of Human Memory

Human memory is a multi-staged phenomenon of extreme complexity, which results in highly unpredictable behavior in real-life situations. Psychologists developed classical paradigms for studying memory in the lab, which produce easily quantifiable measures of performance at the cost of using artificial content, such as lists of randomly assembled words. I will introduce a set of simple mathematical models describing how information is maintained and recalled in these experiments. Surprisingly, they provide a very good description of experimental data obtained with internet-based memory experiments on large number of human subjects. Moreover, more detailed mathematical analysis of the models leads to some interesting ideas for future experiments with potentially very surprising results.

March 13

Canceled

March 20

No Colloquium - Spring Break

March 27

Canceled

April 3

Canceled

April 10

No Colloquium - Good Friday

April 17

Canceled

April 24

Shang-Huan Chiu, NJIT
Hosted by: David Shirokoff

Title: Erosion and Binary Encounters of bodies in Stokes Flows

Abstract:  Complex particulate fluid suspensions appear in chemical, biological, petroleum, and environmental applications. People are interested in different fluids with a suspension of bodies that are being sedimented, transported, or eroded. In this talk, I will first present a boundary integral equation formulation to simulate two-dimensional erosion in porous media. One numerical challenge is accurately resolving the interactions between nearly touching eroding bodies at low porosity. A quadrature method has been used to resolve these interactions and compared with the standard trapezoid rule. To reduce the computational time, we use a hybrid method that combines this quadrature rule and the trapezoid rule. We compute the velocity, vorticity, and tracer trajectories in the geometries that include dense packings of 20, 50, and 100 eroding bodies. Then, having computed tracer trajectories, we characterize the transport inside of eroding geometries by computing and analyzing the tortuosity and anomalous diffusion rates. For bodies transported, I will present a Distributed Lagrange Multiplier/ Fictitious Domain Method (DLM/FD) to simulate binary encounters of spheres in bounded shear flow of Newtonian and non-Newtonian fluids. It is well-known that, up to the initial sphere displacement, binary encounters of spheres in bounded shear flow of a Newtonian fluid can have either swapping or non-swapping trajectories under creeping flow conditions. The motion of dilute sphere suspensions in bounded shear flow of Oldroyd-B fluids at zero Reynolds number has been studied. The pass and return trajectories of the two ball mass centers in a two wall driven shear flow are similar to those in a Newtonian fluid; but they lose the symmetry due to the effect of elastic force arising from viscoelastic fluids. A tumbling chain of two balls (a dipole) may occur, depending on the value of the Weissenberg number and the initial vertical displacement of the ball mass center to the middle plane between two walls.

*This seminar will be held via Webex. An invitation will be sent out in the near future.

May 1

Mohammad Farazmand, North Carolina State University
Hosted by: David Shirokoff

Extreme Events: Dynamics, Prediction and Mitigation​

A wide range of natural and engineering systems exhibit extreme events, i.e., spontaneous intermittent behavior manifested through sporadic bursts in the time series of their observables. Examples include ocean rogue waves, intermittency in turbulence, extreme weather patterns and epileptic seizure. Because of their undesirable impact on the system or the surrounding environment, the real-time prediction and mitigation of extreme events is of great interest. In this talk, I will discuss three aspects of extreme events. First, I introduce a variational method that unravels the mechanisms underpinning the formation of extreme events. Next, I show how this framework enables the data-driven, real-time prediction of extreme events. I demonstrate the application of this method with several examples, including the prediction of ocean rogue waves and the intermittent energy dissipation bursts in turbulent fluid flows. Finally, I will discuss a closed-loop adaptive control and a delay feedback control for mitigating extreme events. 

Updated: May 5, 2020