Department of Mathematics

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Past talks for Fall 2011

September 12 2011
Applied Math Gathering: Short talks highlighting Applied Mathematics research in our department, followed by snacks and refreshments. Speakers include Peter Alfeld, Alexander Balk, Samit Bhattacharyya, Alla Borisyuk, David Dobson, Yekaterina Epshteyn, Ken Golden, Fernando Guevara Vasquez, and Graeme Milton.

September 19 2011 (student talk)
Speaker: Paul Nuñez, Physics Dept., University of Utah
Title: High angular resolution imaging with Intensity Interferometry
Abstract: In order to image stars with sub-milliarcsecond resolution, several interferometric techniques have been developed. However, most stars are still merely detected as point sources at optical frequencies, and not as extended objects. Intensity interferometry measures the correlation of intensity fluctuations between neighboring detectors, which in turn provides information about the Fourier transform of the stellar object. Planned air Cherenkov telescope arrays will be ideal for the implementation of optical intensity interferometry. These arrays will contain nearly 100 telescopes separated by up to 1km, and will provide an unprecedented coverage in the Fourier plane. This will allow the possibility of high angular resolution imaging. The subject of recovering images from (simulated) intensity interferometric data will be discussed. Additionally, current experimental efforts towards sub-milliarcsecond imaging with intensity interferometry will be discussed.

September 26 2011
Speaker: Simon Foucart, Drexel University, Department of Mathematics
Title: Compressive Sensing and the Hard Thresholding Pursuit algorithm
Abstract: This talk will provide an overview of the field of Compressive Sensing, where one aims at reconstructing sparse vectors from a seemingly incomplete set of linear measurements. The focus will be put on the reconstruction process rather than the measurement process. First, we will review the classical $\ell_1$-minimization scheme. We will then discuss alternative iterative algorithms, in particular an algorithm which we called Hard Thresholding Pursuit. We will highlight its advantages in terms of simplicity, speed, and theoretical performances.

October 21 2011 (FRIDAY LCB 225 4pm, joint with Math BIO)
Speaker: Jennifer J. Young, Rice University, CAAM Dept.
Title: A poroelastic model of intestinal edema
Abstract: Intestinal edema is a medical condition referring to the accumulation of excess fluid in the interstitium of the intestinal wall. To study this phenomenon, we developed a computational, poroelastic model of edema formation in the intestinal wall. The intestinal wall is a multi-layered material, whose individual layers have distinct mechanical properties and structure. Partial differential equations are used to describe the deformation, fluid volume changes, and pressure changes in the intestinal wall. The problem is solved using a discontinuous Galerkin finite element method. To validate the model, simulation results are compared to results from four experimental scenarios.

October 24 2011
Speaker: Rajesh Menon, Electrical and Computer Engineering Dept., University of Utah
Title: Multi-wavelength diffractive optics for imaging, nanopatterning and non-imaging applications
Abstract: Diffractive optics offer significant advantages over conventional refractive optics due to their lightweight, planar geometries. In addition, they provide degrees of freedom that are readily accessible via modern microfabrication technologies. However, they suffer from strong chromatic aberrations and their broadband optical performance is typically poor. Here, we apply an extension of the conventional direct binary search algorithm to design diffractive optics that has the potential to operate over multiple wavelengths as well as continuous large spectral bands at high optical efficiencies. I will describe the use of such optics in novel super-resolution imaging, nanoscale patterning and non-imaging applications.

November 7 2011
Speaker: Andrej Cherkaev, Math Dept., University of Utah
Title: Optimal Three-Material Wheel Assemblage of Conducting and Elastic Composites
Abstract: We describe a new type of three material microstructures which we call wheel assemblage, that correspond to extremal isotropic conductivity and extremal bulk modulus for a composite made of two materials and an ideal material. The exact lower bound for effective conductivity and matching laminates were found in (Cherkaev, 2009) and for anisotropic composites, in (Cherkaev, Zhang, 2011). Here, we show different optimal structures that generalize of the classical Hashin-Shtrikman coated spheres (circles). They consists of circular inclusions which contain a solid central circle (hub) and radial spikes in a surrounding annulus, and (for larger volume fractions of the best material) an annulus filled with it. The same wheel assemblages are optimal for the couple of dual problems of minimal conductivity (resistivity) of a composite made from two materials and an ideal conductor (insulator), in the problem of maximal effective bulk modulus of elastic composites made from two linearly elastic material and void, and the dual one.

November 21 2011 (student talk)
Speaker: Christian Sampson, Math Dept., University of Utah
Title: Tipping Points and the Thinning of Arctic Sea Ice
Abstract: Sea ice in the Arctic Ocean is essential to the survival of many arctic animal species. Many communities along the coast of the Arctic Ocean depend on the sea ice for supply transport and use as a platform to hunt for food. The most recent observations conducted in the Arctic Ocean show rapid declines of sea ice extent and thickness over recent years. Many believe that with in the next century we may have an ice free or seasonally ice free Arctic Ocean. This raises the question; How will this transition occur and perhaps more importantly is it reversible? In this talk I will present a model for sea ice thickness and extent which depends on not only on the internal properties of the ice itself but on external parameters. We will talk about some of these parameters and the effect changing them has on the over all health of the arctic ice cover. Current work using bifurcation analysis with these models suggest that abrupt changes in ice cover can occur given the right conditions, so called "tipping points", and show hysteresis leading to irreversible changes in the ice cover of the Arctic Ocean. Yet others suggest a more mild transition. We will discuss these observations and their controlling factors and what they imply for the future of the Arctic Ocean.

November 22 2011 (TUESDAY 4pm LCB 323)
Speaker: Yuliya Babenko, Kennesaw State University
Title: Adaptive approximation on box partitions: polynomial and harmonic splines.
Abstract: Approximation by various types of splines is one of the standard procedure in many applications. In all these applications, there is a standard distinction between uniform and adaptive methods. In the uniform case, the domain of interest is decomposed into a partition where elements do not vary much. Adaptive partitions, on the other hand, take into consideration local variations in the function behavior and therefore provide more accurate approximations. However, adaptive methods are highly nonlinear and no polynomial time algorithm exists to provide an optimal approximant for each given function. Therefore, the next natural question would be to construct asymptotically optimal sequences of partitions and approximants on them.
First we shall briefly present a general scheme for obtaining the asymptotic estimates for the error of interpolation and approximation by splines in various settings (bivariate as well as multivariate). Then we shall introduce our recent results on sharp asymtotics of the interpolation error by polynomial splines on box partitions. However, polynomial splines are not always the optimal choice and (poly)harmonic splines can be a more natural interpolation tool. We will discuss the advantages of using harmonic splines and present the error analysis in the case of approximation by interpolating harmonic splines.

November 28 2011 (student talk)
Speaker: Ben Murphy, Math Dept., University of Utah
Title: Critical Theory of Two-Phase Random Media
Abstract: Composite media arise naturally throughout the physical and biological sciences, and are employed in a broad range of engineering and technological applications. The behavior of those media exhibiting a critical transition as system parameters are varied is particularly challenging to describe physically, and to predict mathematically.
I will demonstrate how techniques of statistical mechanics, random matrix theory, and percolation theory may be used to characterize critical (phase) transitions in two-phase random media. In particular, I will apply these techniques to characterize critical transitions exhibited by electrorheological fluids, sea ice, bone, and percolation models of two-phase random media.

December 5 2011
Speaker: Linh Nguyen, University of Idaho, Math Dept.
Title: Mathematics of thermoacoustic tomography
Abstract: Thermoacoustic tomography (TAT) is an emerging hybrid modality of biomedical imaging. It combines the good contrast of microwave tomography with good resolution of ultrasound tomography. A brief electromagnetic pulse at radio frequency is scanned through to slightly heat up the biological object of interest. The elastic expansion, due to the thermoelastic effect, produces an ultrasound pressure propagating throughout the space. The pressure is recorded by detectors located on an observation surface. The goal of TAT is to reconstruct the initial ultrasound pressure, which contains useful information for cancer detection.
In this talk, we will discuss the following issues of TAT:

  1. Reconstruction procedures: we will discuss several known reconstruction procedures. They include inversion formulas, time-reversal technique, and series solutions. We will also discuss effects of limited data.
  2. Range description: Range descriptions play an important role in tomography. They help to detect and correct measurement errors. We will discuss this topic for TAT.
  3. Speed recovery: Most of the current work in TAT assumes that the ultrasound speed is known. However, in applications, it is unknown. Moreover, ultrasound speed also contains useful information for cancer detection. We will discuss the problem of speed recovery in TAT.

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