|Duk-Soon Oh, Department of Mathematics, Rutgers University|
|Title: Domain Decomposition Methods|
|Luan Hoang, Department of Mathematics, Texas Tech University|
|Title: Non-Darcy flows in heterogeneous porous media|
|Abstract: The most common equation to describe fluid flows in porous media is the Darcy law. However, this linear equation is not valid in many situations, particularly, when the Reynolds number is large or very small.
In the first part of this talk, we survey the Forchheimer models and their generalizations for compressible fluids in heterogeneous porous media. The Forchheimer coefficients in this case are functions of the spatial variables. We derive a parabolic equation for the pressure which is both singular/degenerate in the spatial variables, and degenerate in the pressure’s gradient.
In the second part, we model different flow regimes, namely, pre-Darcy, Darcy and post-Darcy, which may be present in different portions of a porous medium. To study these complex flows, we use a single equation of motion to unify all three regimes. Several scenarios and models are then considered for slightly compressible fluids. A nonlinear parabolic equation for the pressure is derived, which is degenerate when the pressure’s gradient is either small or large.
We estimate the pressure and its gradient for all time in terms of the initial and boundary data. We also obtain their particular bounds for large time which depend on the asymptotic behavior of the boundary data but not on the initial one. Moreover, the continuous dependence of the solutions on the initial and boundary data, and the structural stability for the equations are established.
|Todd Wittman, Department of Mathematics, The Citadel|
|Title: Enhancing Satellite Imagery using the Calculus of Variations|
|Abstract: Image processing is an interdisciplinary field that draws on various branches of mathematics including optimization, differential equations, and numerical analysis. I will discuss a mathematical approach to enhancing satellite imagery based on the calculus of variations. Satellite spectral images give more information about the objects in the scene, but this comes at the cost of reduced spatial resolution. To address this issue, we can fuse the spectral image with a high-resolution panchromatic image. This process is called pan-sharpening. Traditional pan-sharpening methods work well for low-dimensional multispectral datasets (4-6 bands), but do not extend to high-dimensional hyperspectral datasets (100-200 bands). We present a variational method that incorporates wavelets and Total Variation to sharpen hyperspectral images. Time permitting, we will discuss applications to density estimation. This is a joint work with Michael Moeller, Andrea Bertozzi, and Martin Burger.|
|Brigita Fercec, Center for Applied Mathematics and Theoretical Physics, University of Maribor|
|Title: Integrability in planar polynomial systems of ODE’s|
|Abstract: The integrability problem consists in the determination of local or global first integrals and is one of the main open problems in the qualitative theory of differential systems. An essential part of the theory of integrability of ODE’s is devoted to studying local integrability of two dimensional analytic systems of differential equations (two dimensional analytic vector fields) in a neighborhood of a singular point of center or focus type. In this talk we describe an approach for studying integrability in two dimensional polynomial systems. Then, we discuss new criteria for existence of a first integral of the certain form.|
|Zhongqiang Zhang, Mathematical Sciences, WPI|
|Title:Structure-preserving numerical methods fo highly nonlinear stochastic differential equations (SDEs)|
|Abstract:Numerical methods are discussed for SDEs with local Lipschitz coefficients growing at most polynomially at infinity. We first review numerical methods for such nonlinear SDEs and then
present our recent work on stability-preserving implicit schemes and explicit numerical schemes including modified forward Euler schemes and modified Milstein schemes.
We also discuss some positivity-preserving schemes for SDEs with both local Lipschitz coefficients and Holder coefficients. Numerical comparison among various schemes for nonlinear SDEs is presented.
|Linh Nguyen, University of Idaho|
|Title: Mathematics of Photoacoustic Tomography|
Abstract:Photoacoustic tomography (PAT) is a hybrid method of imaging. It combines the high contrast of optical imaging and high resolution of ultrasound imaging. A short pulse of laser light is scanned through the biological object of interest. The photoelastic effect produces an ultrasound pressure propagating throughout the space, which is measured by transducers located on an observation surface. The goal of PAT is to find the initial pressure inside the object, since it contains helpful information of the object.
The mathematical model for PAT is an inverse source problem for the wave equation. In this talk, we will discuss several methods for solving this inverse problem. They include inversion formulas, time reversal techniques, and iterative methods.
|Teng Zhang: Mechanical and Aerospace Engineering, Syracuse University|
|Title: Mathematical Models for Topological Defects in Graphene|
Abstract:Topological defects such as disclination, dislocation and grain boundary are ubiquitous in large-scale fabricated graphene. Due to its atomic scale thickness, the deformation energy in a free standing graphene sheet can be easily released through out-of- plane wrinkles which, if controllable, may be used to tune the electrical and mechanical properties of graphene.
In this talk, I will first demonstrate that a generalized von Karman equation for a flexible solid membrane can be used to describe graphene wrinkling in the presence of topological defects. In this framework, a given distribution of topological defects in a graphene sheet is represented as an eigenstrain field which is determined from a Poisson equation and can be conveniently implemented in finite element (FEM) simulations. Comparison with atomistic simulations indicates that the proposed continuum model is capable of accurately predicting the atomic scale wrinkles near disclination/dislocation cores while also capturing the large scale graphene configurations under specific defect distributions such as those leading to a sinusoidal surface ruga or a catenoid funnel. A great challenge in designing arbitrarily curved graphene with topological defects is that the defect distribution for a specific 3D shape of graphene membrane is usually unknown, which is actually an inverse problem involving highly nonlinear deformation. In the second part of my talk, I will show how to apply the phase field crystal (PFC) method to search for a triangular lattice pattern with the lowest energy on a given curved surface, which then serves as a good approximation of the graphene lattice structure conforming to that surface.
|Jianfeng Lu, Department of Mathematics, Duke University|
|Title: Green’s function in electronic structure calculations|
|Abstract: The Green’s function offers a useful description to the electronic structure models, alternative to using eigenfunctions of the Hamiltonian operator. In this talk, we will demonstrate the usefulness of the Green’s function perspective by two recent results: A mathematical analysis of the divide-and-conquer method and a new Green’s function embedding approach PEXSI-\Sigma.
(based on joint work with Jingrun Chen, Xiantao Li, and Lin Lin)
|Xiaochuan Tian, Department of Mathematics, Columbia University|
|Title: Asymptotically Compatible Schemes for Robust Discretization of Nonlocal Models|
|Abstract:Many problems in nature, being characterized by a parameter, are of interests both with a fixed parameter value and with the parameter approaching an asymptotic limit. Numerical schemes that are convergent in both regimes offer robust discretizations which can be highly desirable in practice. The asymptotically compatible(AC) schemes discussed here meet such objectives for a class of parametrized problems. An abstract mathematical framework is given here together with applications to the numerical solutions of some nonlocal models featured with a horizon parameter which characterizes the nonlocal interaction length. In particular, we will discuss AC schemes for robust discretization of nonlocal diffusion with horizon parameter going to zero or infinity, where the two cases approximates classical diffusion and fractional diffusion respectively. Moreover, a nonconforming DG scheme is proposed for nonlocal models with its convergence established by the theory of AC schemes.|
|Zhennan Zhou, Department of Mathematics, Duke University|
|Title: Asymptotically Compatible Schemes for Robust Discretization of Nonlocal Models|
|Abstract: We develop a surface hopping algorithm based on frozen Gaussian approximation for semiclassical matrix Schr\”odinger equations. The algorithm is asymptotically derived from the Schr\”odinger equation with rigorous approximation error analysis. The resulting algorithm can be viewed as a path integral stochastic representation of the semiclassical matrix Schr\”odinger equations. Our results provide mathematical understanding to and shed new light on the important class of surface hopping methods in theoretical and computational chemistry. Also, I would like to report our recent progress on the improved surface hopping algorithm with various numerical tests.|