# Updates

## Friday, March 17, 10:00-11:00AM, Conference room

## Monday, January 23rd, 3:30-4:30 PM, Conference room

Valery Romanovski, Center for Applied Mathematics and Theoretical Physics |

Title: Some algebraic tools for investigation of systems of ODEs |

Abstract: We give an introduction to algorithms of the elimination theory and methods for solving polynomial systems and show how they can be used for the qualitative investigation of autonomous systems of ordinary differential equations. We then apply them to study the May-Leonard system which models some ecological and chemical processes. |

## Thursday, January 5th, 11:00AM in Conference room

Duk-Soon Oh, Department of Mathematics, Rutgers University |

Title: Domain Decomposition Methods |

Abstract: |

## Friday, January 13rd, 10:00AM in Conference room

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. |

## Friday, November 4th, 2:00PM in Conference room

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. |

## Friday, December 2nd, Conference room, 2:00PM

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. |

## Friday, November 18th, 2:00PM in Conference room

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 thenpresent 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. |

## Tuesday, November 8th, 2:00PM in Conference room

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. |

## Friday, October 28th, at 2:00PM Fretwell 410

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. |