

January 11, 2019
Yusuke Asai
Hokkaido University
Location and Time: Parker Hall 328, 2pm3pm
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, 2pm3pm
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, 2pm3pm
Title: Robust RobinRobin Domain Decomposition Algorithms for Stokes Darcy Interface Problems
Abstract: Many turbulent/porous flow problems can be modeled by StokesDarcy interface systems. In this talk we will discuss two efficient RobinRobin 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, 2pm3pm
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 starshaped 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, 4pm5pm
Title: Stochastic Symplecticity and Ergodicity of Numerical Methods for Stochastic Nonlinear Schroedinger Equation
Abstract: In this talk we present a review on stochastic symplecticity (multisymplecticity) and ergodicity of numerical methods for stochastic nonlinear Schroedinger (NLS) equation. The equation considered is charge conservative and has the multisymplectic 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 multisymplectic 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 multisymplectic 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, 2pm3pm
Title: Blowup results for spacetime fractional dynamics
Abstract: TBA

March 29, 2019
Matthias Chung
Department of Mathematics, Virginia Tech
Location and Time: Parker Hall 328, 2pm3pm
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 realtime solutions to large, dataintensive inverse problems are illposedness 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 allatonce. Oftentimes, the selection of a proper regularization parameter is the most critical and computationally intensive task and may hinder realtime computations of the solution. For the linear problem, we describe a limitedmemory sampled Tikhonov method, and for the nonlinear problem, we describe an approach to integrate the limitedmemory 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 superresolution 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, 2pm3pm
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, 2pm3pm
Title: Periodic phenomena and driving mechanisms in the transmission of West Nile virus
Abstract: West Nile virus is a type of mosquitoborne 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 bistability 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, 2pm3pm (Cancelled)
Title: TBA
Abstract: TBA

April 26, 2019
Dinh Liem Nguyen
Department of Mathematics, Kansas State University
Location and Time: Parker Hall 328, 2pm3pm
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 nondestructive evaluations, detection of explosives, medical imaging, radar imaging, and geophysical prospecting. Inverse scattering problems are in general highly nonlinear and illposed problems, causing substantial challenges in studying their numerical solution. Optimizationbased 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.
