Mathematical Sciences

Mathematical Sciences

Math Colloquia

Date/TimeSpeakerTitle/Abstract

Friday, September 14

3:00 - 4:00pm

The MAC 

Dr. Ugur G. Abdulla

Department of Mathematical Sciences

Florida Institute of Technology 

Title: Breast Cancer Detection through Electrical Impedance Tomography and Optimal Control of Elliptic PDEs -- Invitation to Research

Abstract: In this talk I am going to discuss the inverse Electrical Impedance Tomography (EIT) problem or Calderon problem on recovering electrical conductivity tensor and potential in the body based on the measurement of the boundary voltages on the electrodes for a given electrode current. The inverse EIT problem presents an effective mathematical model of breast cancer detection based on the experimental fact that the electrical conductivity of malignant tumors of the breast is significantly different from conductivity of the normal tissue. I am going to introduce a mathematical model of the inverse EIT problem as a PDE constrained optimal control problem in a Sobolev-Besov spaces framework, where the electrical conductivity tensor and boundary voltages are control parameters, and the cost functional is the norm declinations of the boundary electrode current from the given current pattern and boundary electrode voltages from the measurements. The state vector is a solution of the second order elliptic PDE in divergence form with bounded measurable coefficients under mixed Neumann/Robin type boundary condition. Some recent results on the existence of the optimal control, Frechet differentiability in the Besov space setting, derivation of the he formula for the Frechet gradient, optimality condition, and extensive numerical analysis in the 2D case through implementation of the gradient method in Banach spaces will be presented. Talk will end with the formulation of some major open problems and the perspectives of future advance. 

Friday, October 5

3:00 - 4:00pm

The MAC

Dr. Yalchin Efendiev

Department of Mathematics

Texas A&M University 

Title: Data Integration in Multiscale Simulations

Abstract: In this talk, I will discuss several data integration techniques in multiscale simulations. I will give a brief overview of multiscale simulation concepts that will be used. These multiscale techniques are designed for problems when the coarse grid does not resolve scales and contrast. I will describe the relation between multiscale and upscaling methods. I will describe three data integration techniques. The first one, Bayesian multiscale modeling, will sample basis functions and incorporate available data. In the second approach, we will use deep learning techniques to design and modify existing multiscale methods in the presence of data and nonlinearities.

 Friday, October 12

3:00 - 4:00pm

The MAC

 Dr. Alexandr Tamasan

Department of Mathematics

University of Central Florida

Title: Current Density based Impedance Imaging (CDII)

Abstract: In this talk I will present the inverse hybrid problem of CDII, where the electrical conductivity of body is to be recovered from of the magnitude of one current density field. Physically we use a connection between the current density field generated from the boundary, with interior measurements of the magnetization in a MRI machine. The basic idea is to first recover the voltage potential, which solves a generalized 1-Laplacian. Mathematical methods involved in solving this problem combines ideas from Riemannian geometry with geometric measure theory, and touch on some algebraic topology. 

 Friday, February 22

3:00 - 4:00pm

The MAC

 Dr. Benjamin F. Akers

 

Department of Mathematics & Statistics

Air Force Institute of Technology

Title:  Asymptotics and Numerics for Modulational Instabilities of Traveling Waves.

Abstract:  The spectral stability problem for periodic traveling waves for water wave models is considered. The structure of the spectrum is discussed from the perspective of resonant interaction theory. Modulational asymptotic expansions are used to predict the location of instabilities in frequency-amplitude space. These predictions explain numerical results in [1]. Asymptotics results are presented in the potential ow equations [2] as well as weakly nonlinear models [3]. The asymptotic predictions are compared to the results of a direct numerical simulation of the modulational spectrum.

References

 [1]  Nicholls, David P., Spectral data for travelling water waves: singularities and stability, Journal of Fluid Mechanics, 624 (2009), 339-360.

[2] Akers, Benjamin F., Modulational Instabilities of periodic traveling waves in deep water, Physica D: Nonlinear Phenomena, 300 (2015), 26-33.

[3] Akers, Benjamin F. and Milewski, Paul A., A Model Equation for Wavepacket Solitary Waves Arising from Capillary-Gravity Flows, Studies in Applied Mathematics, 122 (2009), 249-274.

 Friday, April 19

3:00 - 4:00pm

The MAC

 Dr. Robert Talbert

Department of Mathematics

Grand Valley State University

 Title: Flipped learning in theory and practice for mathematics

Abstract: Flipped learning is a pedagogical model in which learners get first contact with new ideas through guided and structured self-learning activities prior to group meetings, allowing significantly more attention to be paid during class meetings to more advanced topics through challenging active learning tasks done collaboratively. Flipped learning combines the best traditions of tutorial and case study methods with modern technologies and evidence-based teaching practices, to help create highly effective learning environments for all learners. In this talk, we will trace the origins of flipped learning, present a framework for flipped learning based in self-determination theory, and illustrate real-life examples from applications to mathematics courses.