• January 11, 2019
    Yusuke Asai
    Hokkaido University
    Location and Time: Parker Hall 328, 2pm-3pm
    Title: Development of hepatitis C virus dynamics model and numerical methods for random ordinary differential equations with time delay

    Abstract: Mathematical modeling by differential equations plays an important role to understand natural sciences. In particular, required data are frequently not obtained in biology and medicine and simulation using mathematical models helps us to analyze system behavior under various scenarios. Deterministic models have been long investigated and applied to natural phenomena, however, some factors might be ignored in model building process or we encounter random effect from environment in practice. To handle such uncertain effect, random ordinary differential equations (RODEs) can be an ideal tool because of its simplicity in model building as well as the regularity of the corresponding noise processes. Recently, several classes of numerical methods for RODEs have been developed and applied to real problems, yet time delay has been ignored in those applications. To capture and understand system behavior more accurately, we need to handle both of randomness and time delay simultaneously. In this talk, development of hepatitis C virus (HCV) dynamics models is briefly introduced and then numerical methods for RODEs with time delay are systematically constructed. The developed methods are applied to the introduced HCV model with target cells, infected cells and viruses compartments, and eclipse phase, the time elapsed between cell infection and virus production, and their behavior will be investigated.

  • January 25, 2019
    Junshan Lin
    Department of Mathematics and Statistics, Auburn University
    Location and Time: Parker Hall 328, 2pm-3pm
    Title: Embedded eigenvalues and Fano resonance for metallic structures with small holes

    Abstract: Fano resonance, which was initially discovered in quantum mechanics by Ugo Fano, has been extensively explored in photonics since the past decade due to its unique resonant feature of a sharp transition from total transmission to total reflection. Mathematically, Fano resonance is related to certain eigenvalues embedded in the continuum spectrum of the underlying differential operator. For photonic structures, the quantitative studies of embedded eigenvalues mostly rely on numerical approaches. In this talk, based on layer potential technique and asymptotic analysis, I will present quantitative analysis of embedded eigenvalues and their perturbation as resonances for a periodic array of subwavelength metallic structure. From a quantitative analysis of the wave field for the scattering problem, a rigorous proof of Fano resonance will be given. In addition, the field enhancement at Fano resonance frequencies will be discussed.

  • February 1, 2019
    Yanzhao Cao
    Department of Mathematics and Statistics, Auburn University
    Location and Time: Parker Hall 328, 2pm-3pm
    Title: Robust Robin-Robin Domain Decomposition Algorithms for Stokes Darcy Interface Problems

    Abstract: Many turbulent/porous flow problems can be modeled by Stokes-Darcy interface systems. In this talk we will discuss two efficient Robin-Robin domain decomposition algorithms to solve these systems. Both convergence analysis and numerical experiments will be presented.

  • February 8, 2019
    Shelvean Kapita
    Department of Mathematics and Statistics, University of Georgia
    Location and Time: Parker Hall 328, 2pm-3pm
    Title: Bivariate Spline Solutions to the Helmholtz Equation

    Abstract: lthough there are many computational methods for solving the Helmholtz equation, e.g. hp finite element methods, the numerical solution of the Helmholtz equation still poses challenges, particularly for large wavenumbers. We shall explain how to use bivariate splines to numerically solve the Helmholtz equation in both bounded and unbounded domains. In addition, we shall establish existence, uniqueness and stability of the weak solution of the Helmholtz equation, under the assumption that k^2, where k is the wavenumber, is not a Dirichlet eigenvalue of the associated Poisson equation. With this assumption, the standard assumption that the domain be strictly star-shaped is no longer needed. Finally, we will explain how to use bivariate splines to solve the exterior domain Helmholtz equation using a PML technique. We demonstrate the effectiveness of bivariate splines for the bounded domain and exterior Helmholtz equation with a variety of numerical examples.

  • February 11, 2019
    Jialin Hong
    Chinese Acaemey of Sciences
    Location and Time: Parker Hall 352, 4pm-5pm
    Title: Stochastic Symplecticity and Ergodicity of Numerical Methods for Stochastic Nonlinear Schroedinger Equation

    Abstract: In this talk we present a review on stochastic symplecticity (multi-symplecticity) and ergodicity of numerical methods for stochastic nonlinear Schroedinger (NLS) equation. The equation considered is charge conservative and has the multi-symplectic conservation law. Based a stochastic version of variational principle, we show that the phase flow of the equation, considered as an evolution equation, preserves the symplectic structure of the phase space. We give some symplectic integrators and multi-symplectic methods for the equation. By constructing control system and invariant control set, it is proved that the symplectic integrator, based on the central difference scheme, possesses a unique invariant measure on the unit sphere. Furthermore, by using the midpoint scheme, we get a full discretization which possesses the discrete charge conservation law and the discrete multi-symplectic conservation law. Utilizing the Poisson equation corresponding to the finite dimensional approximation, the convergence error between the temporal average of the full discretization and the ergodic limit of the symplectic method is derived (In collaboration with Dr. Chuchu Chen, Dr. Xu Wang and Dr. Liying Zhang)

  • February 22, 2019
    Erkan Nane
    Department of Mathematics and Statistics, Auburn University
    Location and Time: Parker Hall 328, 2pm-3pm
    Title: Blow-up results for space--time fractional dynamics

    Abstract: TBA

  • March 29, 2019
    Matthias Chung
    Department of Mathematics, Virginia Tech
    Location and Time: Parker Hall 328, 2pm-3pm
    Title: Computational Challenges of Inverse Problems

    Abstract: Inverse problems are omnipresent in many scientific fields such as systems biology, engineering, medical imaging, and geophysics. The main challenges toward obtaining meaningful real-time solutions to large, data-intensive inverse problems are ill-posedness of the problem, large parameter dimensions, and/or complex model constraints. For instance, we consider iterative methods based on sampling for computing solutions to massive inverse problems where the entire dataset cannot be accessed or is not available all-at-once. Oftentimes, the selection of a proper regularization parameter is the most critical and computationally intensive task and may hinder real-time computations of the solution. For the linear problem, we describe a limited-memory sampled Tikhonov method, and for the nonlinear problem, we describe an approach to integrate the limited-memory sampled Tikhonov method within a nonlinear optimization framework. The proposed method is computationally efficient in that it only uses available data at any iteration to update both sets of parameters. Numerical experiments applied to massive super-resolution image reconstruction problems show the power of these methods.

  • April 5, 2019
    Ratnasingham Shivaji
    Department of Mathematics and Statisitcs, University of North Carolina at Greensboro
    Location and Time: Parker Hall 328, 2pm-3pm
    Title: Existence and multiplicity of positive radial solutions for singular superlinear elliptic systems in the exterior of a ball

    Abstract: TBA

  • April 12, 2019
    Guihong Fan
    Department of Mathematics, Columbus State University
    Location and Time: Parker Hall 328, 2pm-3pm
    Title: Periodic phenomena and driving mechanisms in the transmission of West Nile virus

    Abstract: West Nile virus is a type of mosquito-borne disease which can cause severe illness in humans. Mosquitoes play a critical role in the transmission and spread of the diseases. We propose a system of equations to model the transmission of West Nile virus between mosquitoes and avian hosts. The system compromises five equations in which maturation delay in mosquitoes is incorpo- rated. Analytical analysis shows that mosquitoes can force the system to oscillate under the impact of maturation delay while the species interaction enhances the oscillation. Our analysis indicates the existence of global Hopf bifurcation, period doubling bifurcation, and fold bifurcation of period solutions as well as the existence of a bi-stability in the form of a boundary periodic solution and a positive periodic solution. This is a collaborated work with Prof. Chunhua Shan (The University of Toledo) and Prof. Huaiping Zhu (York University).

  • April 19, 2019
    Braxton Osting
    Department of Mathematics, University of Utah
    Location and Time: Parker Hall 328, 2pm-3pm (Cancelled)
    Title: TBA

    Abstract: TBA

  • April 26, 2019
    Dinh Liem Nguyen
    Department of Mathematics, Kansas State University
    Location and Time: Parker Hall 328, 2pm-3pm
    Title: Solving nonlinear inverse scattering problems with little a priori information

    Abstract: The goal of inverse scattering problems is, broadly speaking, to determine information about an object from measurements of the field scattered by that object. These inverse problems arise in a wide range of applications, including non-destructive evaluations, detection of explosives, medical imaging, radar imaging, and geophysical prospecting. Inverse scattering problems are in general highly nonlinear and ill-posed problems, causing substantial challenges in studying their numerical solution. Optimization-based methods are the most widely studied approach among the numerical methods for solving these inverse problems. However, these methods suffer from the fact that they may converge to a local minimum of the cost function, which is not the true solution to the inverse problem. In addition, they require strong a priori information about the true solution which are not always available in some practical applications. In this talk, after a short introduction to inverse scattering problems, we will discuss our recent results in developing two different numerical methods for solving these problems, without using any detailed a priori information of the solution. The first method is to solve a coefficient inverse problem for the Helmholtz equation and the second one is to solve a shape inverse problem for Maxwell's equations.