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

October 26, 2016 by Duan Chen
Categories: Spring 2022
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

October 26, 2016 by Duan Chen
Categories: Spring 2022
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.

 

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

October 26, 2016 by Duan Chen
Categories: Spring 2022
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

October 26, 2016 by Duan Chen
Categories: Spring 2022
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.

 

Friday, October 14th, at 2:00PM Conference room

October 14, 2016 by Duan Chen
Categories: Spring 2022
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)

 

Friday, September 16th, at 11:00AM Conference room

September 14, 2016 by Duan Chen
Categories: Spring 2022
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.

 

Thursday September 8th, at 11:00AM Fretwell 315

September 07, 2016 by Duan Chen
Categories: Spring 2022
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.

 

Wednesday August 23rd, at 11:00AM Math Conference room

August 20, 2016 by Duan Chen
Categories: Spring 2022
Yang Yang, Department of Mathematical Sciences, Michigan Tech University
Title:Bound-preserving discontinuous Galerkin method for compressible miscible displacement problem in porous media
Abstract: In this talk, I will talk about the bound-preserving discontinuous Galerkin (DG) methods for the coupled system of compressible miscible displacement problems. We consider the problem with two components and the (volumetric) concentration of the ith component of the fluid mixture, c_i, should be between 0 and 1. However, c_i does not satisfy the maximum-principle due to the existence of the source terms. Therefore, the numerical techniques introduced in (X. Zhang and C.-W. Shu, Journal of Computational Physics, 229 (2010), 3091-3120) cannot be applied directly. The main idea is to apply the positivity-preserving techniques to both c_1 and c_2, respectively and enforce c_1+c_2=1 simultaneously to obtain physically relevant approximations. By doing so, we have to treat the time derivative of the pressure dp/dt as a source in the concentration equation. Moreover, c_i’s are not the conservative variables, as a result, the classical bound-preserving limiter in (X. Zhang and C.-W. Shu, Journal of Computational Physics, 229 (2010), 3091-3120) cannot be applied directly. Therefore, another limiter will be introduced. Numerical experiments will be given to demonstrate the good performance of the numerical technique.

Wednesday May 4th, at 2:00PM Math Conference room

May 03, 2016 by Duan Chen
Categories: Spring 2022
Thomas Hudson, CERMICS – École des Ponts ParisTech
Title: Stability and motion of screw dislocations in a lattice model
Abstract:Dislocations are defects found in crystalline solids, and their motion provides a mechanism through which such materials may plastically deform. In this talk, I will present a series of results concerning a lattice model in which configurations containing screw dislocations may be identified as local minima of a certain energy. Based upon this description, a stochastic model is then proposed for thermally-driven dislocation motion. In the limit of low temperature, a Large Deviations Principle for the evolution can be found, showing that the most probable trajectory of the system is along solutions of a generalized gradient flow. These results provide a first rigorous identification of a regime in which Discrete Dislocation Dynamics models, commonly simulated by Materials Scientists in practice, is valid.

Friday April 29th, at 11:00AM Math Conference room

April 25, 2016 by Duan Chen
Categories: Spring 2022
Kelin Xia, Michigan State University
Title: Topological modeling and analysis of complex data in biomolecules
Abstract: Proteins are the most important biomolecules for living organisms. The understanding of protein structure, function, dynamics, and transport is one of the most challenging tasks in biological science. We have introduced persistent homology for extracting molecular topological fingerprints (MTFs) based on the persistence of molecular topological invariants. MTFs are utilized for protein characterization, identification, and classification. The multidimensional persistent homology is proposed and further used to quantitatively predict the stability of protein folding configurations generated by steered molecular dynamics. An excellent consistence between my persistent homology prediction and molecular dynamics simulation is found. Further, we introduce multiresolution persistent homology to handle complex biomolecular data. The essential idea is to match the resolution with the scale of interest so as to represent large scale datasets with appropriate resolution. By appropriately tuning the resolution of a density function, we are able to focus the topological lens on the scale of interest. The proposed multiresolution topological method has potential applications in arbitrary data sets, such as social networks, biological networks and graphs.