Department of Mathematical Sciences

Colloquium Series

Purpose: To discuss mathematical research and interdisciplinary problems in all areas of mathematics and applied mathematics. Talks are welcome from faculty, graduate students, and outside speakers from academia and industry. Talks should be at a level accessible to graduate students. Students and faculty at all levels and from all departments are welcome to attend.

Date/Time Speaker Title/Abstract
Friday, January 23

3-4pm Crawford 210

Dr. Daniel Onofrei

Department of Mathematics

University of Houston

Title: Active control of electromagnetic fields

Abstract: In this talk we will discuss the problem of control of electromagnetic fields by using active sources (antennas), i.e., characterization of surface currents needed on the active sources so that their radiated fields will approximate desired patterns in several given disjoint external regions. Mentioning that any realistic design will need to consider a series of important feasibility constraints, this problem can be placed at the intersection of several exciting research areas: inverse source problems, optimal control of PDE's, antenna synthesis and optimization theory. In the first part of the talk, after a brief introduction of the subject, we will discuss the problem of controlling transverse normal modes in a wave guide. We will present our analytical approach, discuss the feasibility of the approach and conclude with several relevant numerical results. Our analysis indicates, among other things, that the proposed control strategy seems to be feasible only in the near field region of the defending antenna. In the second part of the talk we will present the extension of our results to the case of free space electromagnetics, discuss the feasibility of the approach in this general context, and highlight several future research goals and the challenges we anticipate for this project.

Friday, January 30

3-4pm Crawford 210

Dr. Michael V. Klibanov

Department of Mathematics and Statistics

University of North Carolina at Charlotte

Title: Global convergence for inverse problems and phaseless inverse scattering

Abstract:We will present two types of results on Inverse Problems. The first type is about the globally convergent numerical methods. Both the theory and the numerical results will be presented. Numerical studies were done for experimental data collected in our campus. The second type is the solution of a long standing (since 1977) problem. Namely, we show that the coefficient of the Schrodinger equation in 3d can be reconstructed via the classical Radon transform from the scattering data, which do not contain the phase information: only the modulus of the complex valued wave field is measured, whereas the phase is not measured.

Monday, Feb 9

12-1pm Skurla 103

Dr. Ariel Barton

Department of Mathematics

University of Missouri

Title: Potential methods for higher-order boundary-value problems

Abstract: The theory of boundary-value problems for the Laplacian in Lipschitz domains is by now very well developed. Furthermore, many of the existing tools and known results for the Laplacian have been extended to the case of second-order linear equations of the form $\nabla\cdot A\nabla u=0$, where $A$ is a matrix of variable coefficients. However, at present there are many open questions in the theory of higher-order elliptic differential equations. In this talk I will describe a generalization of layer potentials to the case of higher-order operators $\nabla^m\cdot A\nabla^m$; layer potentials are a common and very useful tool in the theory of second-order equations. I will then describe some applications of layer potentials to the theory of boundary-value problems. This is joint work with Steve Hofmann and Svitlana Mayboroda.

Wednesday, Feb 11

11am-12 Skurla 116

Dr. Francesco Cellarosi

Department of Mathematics

University of Illinois

Title: Quadratic Weyl sums, Automorphic Functions, and Invariance Principles

Abstract: In 1914, Hardy and Littlewood published their celebrated approximate functional equation for quadratic Weyl sums (theta sums). Their result provides, by iterative application, a powerful tool for the asymptotic analysis of such sums. The classical Jacobi theta function, on the other hand, satisfies an exact functional equation, and extends to an automorphic function on the Jacobi group. We construct a related, almost everywhere non-differentiable automorphic function, which approximates quadratic Weyl sums up to an error of order one, uniformly in the summation range. This not only implies the approximate functional equation, but allows us to replace Hardy and Littlewood's renormalization approach by the dynamics of a certain homogeneous flow. The great advantage of this construction is that the approximation is global, i.e., there is no need to keep track of the error terms accumulating in an iterative procedure. Our main application is a new functional limit theorem, or invariance principle, for theta sums. The interesting observation is that the paths of the limiting process share a number of key features with Brownian motion (scale invariance, invariance under time inversion, non-differentiability), although time increments are not independent, the value distribution at each fixed time is distinctly different from a normal distribution. Joint work with Jens Marklof.

Friday, Feb 13

3-4pm Crawford 210

Dr. Thanh Nguyen

Department of Mathematics

Iowa State University

Title: Reconstruction Methods for Inverse Scattering Problems and Applications

Abstract: Inverse scattering theory aims at determining geometrical and/or physical properties of an object or medium based on measurements of waves which are generated by the scattering of incident waves by the object/medium. These problems can be stated as coefficient inverse problems or shape reconstruction problems for partial differential equations of hyperbolic or elliptic types. Applications of inverse scattering theory can be found in different fields such as nondestructive evaluations, medical imaging, geophysics and remote sensing. In this talk I will give a short introduction to the three main questions in inverse scattering theory: (i) uniqueness, (ii) stability, and (iii) reconstruction methods. In particular, it is well-known that most numerical methods for solving inverse problems require a priori information about the unknown object/medium, which is usually not available in many practical situations. The central part of this talk will be devoted to numerical methods for solving the inverse scattering problems which do not require a priori information about the unknown object/medium. The application of our methods to imaging of buried objects will also be discussed.

Friday, Feb 27

3-4pm Crawford 210

Dr. Olympia Hadjiliadis

Department of Mathematics

City University of New York

Title: Quickest detection and sequential identification in systems with correlated noise

Abstract: We address the problems of sequential classification and quickest detection in Brownian channels with correlated noise. In the former we demonstrate that the maximum of two SPRTs is asymptotically optimal in identifying the channel with signal as the error probabilities decrease to zero, while in the latter, asymptotic optimality of the minimum of two CUSUM stopping rules in detecting the first instance of a signal as the frequency of false alarms decreases to zero.

Friday, Mar 6

3-4pm Crawford 210

Dr. Monica Gentili

Department of Mathematics

University of Salerno, Italy

&

Institute for People and Technology

Georgia Institute of Technology

Title: Optimal Location of Sensors on Traffic Networks

Abstract: The problem of optimally locating sensors on a traffic network to measure flows has been object of growing interest in the past few years, due to its relevance in transportation systems. Different locations of sensors on the network can allow, indeed, the collection of data whose usage can be useful for traffic management and control purposes. Many different models have been proposed in the literature as well as corresponding solution approaches. The proposed existing models differ according to different criteria: (i) sensor types to be located on the network (e.g., counting sensors, image sensors, Automatic Vehicle Identification (AVI) readers), (ii) available a-priori information, (iii) flows of interest (e.g., OD flows, route flows, link flows). The purpose of this talk is to give a unifying picture of these models by categorizing them into two main problems: the Sensor Location Observability Problem and the Sensor Location Estimation Problem. Both the problems will be described and the computational complexity will be discussed. A polynomial case for a special class of problems will also be shown.

Friday, April 3

3-4pm Crawford 210

Dr. Vladislav Bukshtynov

Department of Energy Resources Engineering

Stanford University

SIAM Speaker

Title: Gradient-based Parameter Estimation for Industrial Applications

Abstract: Parameter estimation problems are often solved by employing numerical optimization techniques. These require objective function evaluations that are very expensive in industrial applications due to large model sizes and multiphysics effects. The solution is often obtained by performing PDE-based optimizations characterized by a large number of design variables along with objectives and constraints that are complex in structure. Despite its known drawbacks, e.g. searching locally, gradient-based strategies may be viable if supplied with additional features boosting their overall performance. An example taken from the petroleum engineering field will demonstrate how the problem of reservoir characterization is solved within the so-called "discretize-then-optimize" framework with the help of automatic differentiation and effective parameterization. The second part of the talk will present a new computational approach applicable to a broad range of problems arising in nonequilibrium thermodynamics and continuum mechanics where, in contrast to "classical" parameter reconstruction problems, the control variable is a function of the states, i.e. the dependent variables. The main novelty is that the nonlinear constitutive relation is determined in a very general form with no prior knowledge except some assumptions on smoothness.

Friday, April 17

3-4pm Crawford 210

Dr. Hasanjan Sayit

Department of Mathematical Sciences

Durham University

Title: No Arbitrage Conditions in Markets with Friction

Abstract: In the classical arbitrage theory all the continuous time trading strategies with lower bounded wealth processes are admissible. The no-arbitrage condition in this class of trading strategies restrict asset prices to models that admit equivalent martingale measures. Since fractional Brownian motion is not a semi-martingale it admits free lunches with continuous time trading strategies. In this talk, we discuss no-arbitrage conditions in smaller class of trading strategies. More specifically, we restrict the class of trading strategies to simple integrands and show that fractional Brownian motion does not admit arbitrage with simple strategies. We also discuss no arbitrage conditions in financial markets where proportional transaction costs are present.

Friday, April 24

3-4pm Crawford 210

Dr. Charles Fulton

Department of Mathematical Sciences

Florida Institute of Technology

Title: New Formulas and Algorithms for Spectral Density Function Computations

Abstract: Some new formulas for the Spectral Density Functions associated with the Sturm-Liouville equation when absolutely continuous spectrum occurs will be presented. These formulas derive from the ``Value Distribution'' theory of D. Pearson and S. Breimesser, and have the feature of being applicable on any interval where absolutely continuous spectrum in known to occcur. A prototype algorithm implementing some of these formulas will be discussed.

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