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Mathematics Department

Applied Math Seminar

Fall  2017

All talks are from 12:00-1:00 p.m. in the Seminar Room CH351, unless otherwise specified.

  • Dec
    08
  • Mathematical Aspects of Distributed Control for Compressible Fluids
    Stefan Doboszczak
    Dept. of Mathematics Air Force Institute of Technology

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    The compressible Navier-Stokes equations comprise a system of PDE describing the evolution of a linearly viscous compressible fluid. We consider the general problem of driving the fluid to a given state over a fixed time, under the influence of a distributed control in the form of a body force. An optimal control is sought in order to minimize a certain cost functional. We first obtain the existence of optimal controls for the nonlinear system. Our result relies on the weak-strong uniqueness property of the compressible Navier-Stokes equations to ensure the existence of unique states. Next we obtain the first order necessary conditions for a linearized version of the compressible system in the form of a Pontryagin maximum principle. Time permitting we will also discuss the problem of optimally accelerating an object in compressible fluid to a desired speed.
  • Dec
    01
  • TBA
    Brice Nguelifack
    United States Naval Academy

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    Talk Cancelled
  • Nov
    13
  • Lyme Disease in New England: Multilevel population interactions as affected by human driven development
    Joe Skufca
    Clarkson University
    Location: CH 351
    Time: 03:45 AM

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    The prevalence of Lyme Disease continues to grow in North America, not only in geographic extent but also with respect to number of cases, disease burden and other economic impacts. While typical studies of the spatial spread of the disease focus on the dynamics at low population levels, we seek to understand the potential implications of endemic levels of the disease. In particular, we want to understand what mechanism might be in play that control the tick population levels — the key driver with respect to morbidity rates.
  • Nov
    02
  • The Hyperspectral Diffuse Optical Tomography Forward and Inverse Problems
    Rachel Grotheer
    Goucher College

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    Medical imaging devices play an increasingly significant role in disease diagnosis and detection. Diffuse optical tomography (DOT), which uses a low-energy light source in the visible to near infrared range, has emerged as a potential alternative to traditional imaging techniques. DOT uses the diffusion approximation of the radiative transport equation, an elliptic PDE, to model the diffusion of photons in the tissue during the imaging process. The DOT inverse problem is to create an image by reconstructing a spatial map of the optical parameters of the tissue being imaged given a known source and boundary measurements. In recent years, researchers have sought to apply hyperspectral imaging, the use of hundreds of optical wavelengths in the imaging process, to DOT in order to improve the resolution of the image by adding new information. In hyperspectral DOT (hyDOT), the optical parameters have both spatial and spectral dependence, adding an extra dimension to both the forward and inverse problems. The implications of this have not yet been studied thoroughly in a mathematical framework. We present an initial examination of how the spectral dependence of the optical parameters affects both the forward problem and the image reconstruction problem of hyDOT.
  • Oct
    27
  • Control Problems with Vanishing Lie Bracket Arising from Complete Odd Circulant Evolutionary Games
    Chris Griffin
    Dept. of Mathematics USNA

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    We study an optimal control problem arising from a generalization of rock-paper-scissors in which the number of strategies may be selected from any positive odd number greater than 1 and in which the payoff to the winner is controlled by a control variable $\gamma$. Using the replicator dynamics as the equations of motion, we show that a quasi-linearization of the problem admits a special optimal control form in which explicit dynamics for the controller can be identified. We show that all optimal controls must satisfy a specific second order differential equation parameterized by the number of strategies in the game. We show that as the number of strategies increases, a limiting case admits a closed form for the open-loop optimal control. In performing our analysis we show necessary conditions on an optimal control problem that allow this analytic approach to function.
  • Oct
    20
  • Computing the quasi-potential for the quantification of rare events in stochastic systems
    Maria Cameron
    University of Maryland College Park

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    A number of processes in nature such as population dynamics and genetic switches are modeled using non-gradient Stochastic Differential Equations with small while noise terms. The function that allows one to quantify the escape process from any neighborhood of an attractor of the system is called the quasi-potential. Once the quasi-potential is found, one can readily find the maximum likelihood escape paths, and estimate the escape rates and the invariant probability density near the attractor. Unfortunately, the quasi-potential is the solution of an optimal control problem that can be solved analytically only in special cases. In this talk, I will introduce a family of Dijkstra-like numerical methods called the Ordered Line Integral Methods for computing the quasi-potential in whole regions surrounding the attractors.
  • Oct
    06
  • Influence Maximization Problems on Social Networks
    Raghu Raghavan
    University of Maryland College Park

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    The study of viral marketing strategies on social networks has become an area of significant research interest. Roughly, these problems seek to identify which individuals in a social network to target so that a given proportion of the target population is influenced at minimum cost. Motivated by the desire to develop a better understanding of fundamental problems in social network analytics, we study two problems in this domain and seek to develop optimization techniques to solve them exactly. We first consider the weighted target set selection (WTSS) problem. Here the weights (on the nodes) have a natural interpretation as the cost or effort associated with influencing the node (or individual). Next, we consider the least cost influence problem (LCIP). The difference between the LCIP and the WTSS problem lies in the fact that nodes selected for targeting in the LCIP may be provided partial incentives (i.e., payments) in the viral marketing context; while in the WTSS problem no partial payments are permitted. Both problems are NP-hard on general graphs. Motivated by the desire to develop mathematical programming approaches to solve these two problems, we focus on developing strong formulations for these problems on trees. In particular, we are interested in formulations whose linear programming relaxations are integral. In radically different ways we develop tight and compact formulations for the WTSS and LCIP problem on trees. Based on the observation that the influence propagation network is a directed acyclic graph, these integral formulations can be embedded into a larger exponential sized integer programming formulation for these two problems on general graphs. Using this idea we design and implement a branch-and-cut approach to solve the WTSS and LCIP problems on general graphs and share our computational experience on networks with up to 10,000 nodes. Extensions of this work to related problems and variants will be discussed.
  • Sep
    29
  • Using Optimization to Balance Fairness and Efficiency in Barter Markets
    John Dickerson
    University of Maryland College Park

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    The exchange of indivisible goods without money addresses a variety of constrained economic settings where a medium of exchange - such as money - is considered inappropriate. Participants are either matched directly with another participant or, in more complex domains, in barter cycles and chains with other participants before exchanging their endowed goods. We show that techniques from computer science and operations research, combined with the recent availability of massive data and inexpensive computing, can guide the design of such matching markets and enable the markets by running them in the real world. A key application domain for our work is kidney exchange, an organized market where patients with end-stage renal failure swap willing but incompatible donors. We present new models that address three fundamental dimensions of kidney exchange: (i) uncertainty over the existence of possible trades, (ii) balancing efficiency and fairness, and (iii) inherent dynamism. Next, we combine these dimensions, along with high-level human-provided guidance, into a unified framework for learning to match in a general dynamic setting. This framework, which we coin FutureMatch, takes as input a high-level objective (e.g., "maximize graft survival of transplants over time") decided on by experts, then automatically learns based on data how to make this objective concrete and learns the "means" to accomplish this goal - a task that, in our experience, humans handle poorly. Throughout, we draw on insights from our work with the United Network for Organ Sharing (UNOS) US-wide exchange and experiments on data from the National Health Service UK-wide exchange. Bio: John P. Dickerson is an Assistant Professor of Computer Science at the University of Maryland, and a recent CS PhD graduate of Carnegie Mellon University. His research centers on solving practical economic problems using techniques from computer science, stochastic optimization, and machine learning. He has worked extensively on theoretical and empirical approaches to kidney exchange, where his work has set policy at the UNOS nationwide exchange; game-theoretic approaches to counter-terrorism and negotiation, where his models have been deployed; and computational advertising through Optimized Markets, a CMU spin-off company. He created FutureMatch, a general framework for learning to match subject to human value judgments; that framework won a 2014 HPCWire Supercomputing Award. Prior to his Ph.D., he worked at IBM and in R&D at a defense agency. He is an NDSEG Fellow, Facebook Fellow, and Siebel Scholar.
  • Sep
    15
  • Hyperspherical coordinates in few-body quantum mechanics
    Seth Rittenhouse
    United States Naval Academy, Physics Department

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    Few-body quantum mechanics is really the study of a specific partial differential equation, the Schroedinger equation. Because the Schroedinger equation is a linear PDE, on the surface this seems to be a rather straightforward endeavour. However, as more particles are added to the system, the high dimensionality of the problem quickly becomes intractable. In this talk I will discuss one approach that has seen a great deal of success in recent year in combating this issue, the adiabatic hyperspherical method. In this method the system is parameterized in the high dimensional analog of spherical coordinates, hyperspherical coordinates. I will discuss how these coordinates are generated, and how they can be used to solve the schrodinger equation. Finally, I show how this approach can be used to solve the quantum mechanical 3-body problem with short-range inter-particle interactions.
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