Seminar meets Fridays at 2:00 pm in Parker 328 unless otherwise noted.

2014 - 2015

November 21

Speaker: **Guannan Zhang**, Oak Ridge National Lab and Auburn University

Title: High-Order Numerical Methods for Forward-Backward Stochastic Differential Equations with Jumps and Applications in Nonlocal Diffusion Problems

Abstract: We propose a new numerical scheme for decoupled forward backward stochastic differential equation (FBSDE) with jumps, where the jumps are characterized by Poisson random measures. A variant of Crank-Nicolson scheme is proposed for time discretization. We proved that our scheme can achieve second-order convergence as long as a second-order scheme, e.g. order-2.0 weak Taylor scheme, is used to discretize the forward SDE. A high-order fully discrete scheme is also proposed in the case of Poisson random measures with finite activities. Compared to existing methods, the development of the high-order time-space discretization schemes is the main contribution of this paper. On the other hand, our approach is also an effective tool for nonlocal diffusion problems, where the governing equation is a class of semi-linear partial-integral differential equations.

November 14

Speaker: **Michael Neilan**, Department of Mathematics, University of Pittsburgh

Title: Finite Element Methods for Elliptic Problems in Non-divergence Form

Abstract: The finite element method is a powerful and ubiquitous tool in numerical analysis and scientific computing to compute approximate solutions to partial differential equations (PDEs). A contributing factor of the method's success is that it naturally fits into the functional analysis framework of variational models. In this talk I will discuss finite element methods for PDEs problems that do not conform to the usual variational framework, namely, elliptic PDEs in non--divergence form. I will first present the derivation of the scheme and give a brief outline of the convergence analysis. Finally, I will present several challenging numerical examples showing the robustness of the method as well as verifying the theoretical results.

November 7

Speaker: **Richard Zalik**

Title: On the Nonlinear Jeffcott Equations

Abstract: The Jeffcott equations are a system of coupled, nonlinear, ordinary differential equations. The primary application of their study is directed towards understanding the reasons for the excessive vibrations recorded in the cryogenic pumps of the second stage main engine of the Space Shuttle, during hot firing ground testing. In this talk we shall examine some properties of the solutions of the Jeffcott equations. In particular, we show how bounds for the solutions of these equations can be obtained from bounds of the solutions of the linearized equations. By studying the behavior of the Fourier transforms of the solutions, we are also able to predict the onset of destructive vibrations. These conclusions are verified by means of numerical solutions of the equations, and of power spectrum density plots.

This work shows how numerical simulations can be used to obtain an insight into the correct solution to a problem. Once this correct answer is known, it then becomes possible to give a rigorous proof.

October 10

Speaker: **Anthony Skjellum**, Department of Computer Science and Software Engineering, Director of the Auburn Cyber Research Center, Auburn University

Title: Toward Fault Tolerant Parallel Computing with MPI

Abstract: In this talk, I present issues with the MPI parallel programming model, and the emerging issues of fault tolerance at scale. Over the past fifteen years, various efforts have been made to make MPI and parallel programs in general more tolerant to faults. We describe two current models - FA-MPI and ULFM, which are current models for making MPI fault tolerant, or at least enabling fault tolerance with MPI applications. We also describe basic reasons for faults, how emerging architectures are leading to higher fault rates, and where other opportunities coming from mathematical analysis and algorithm theory can help address problems in such scenarios such as Newton-Krylov solvers inside simulation applications.

October 3

Speaker: **Zhongwei Shen**, Department of Mathematics and Statistics, Auburn University

Title: Front Propagation in Reaction-Diffusion Equations with Ignition Nonlinearities

Abstract: In this talk, I will present the developments of front propagation in diffusive media of ignition type, with the focus on traveling waves and their generalizations. I will first present some classical results in the homogeneous media. Then, I will move to the recent developments in space heterogeneous media. Finally, I will present my recent work with W. Shen in time heterogeneous media.

September 26

Speaker: **Jiayin Jin**, Department of Mathematics, Michigan Sate University

Title: Global Dynamics of Boundary Droplets for the 2-d Mass-conserving Allen-Cahn Equation

Abstract: In this talk I will present how to establish the existence of a invariant manifold of bubble states for the mass-conserving Allen-Cahn equation in two space dimensions, and give the dynamics for the center of the bubble.

September 19

Speaker: **Catalin Turc**, Department of Mathematical Sciences, New Jersey Institute of Technology

Title: Well-conditioned boundary integral equation formulations for the solution of high-frequency scattering problems

Abstract: We present several versions of Regularized Combined Field Integral Equation (CFIER) formulations for the solution of two and three dimensional frequency domain scattering problems with various kinds of boundary conditions. These formulations are based on suitable approximations to Dirichlet-to-Neuman operators and can be shown to be well posed, under certain assumptions on the regularity of the scatterers. For a wide variety of scatterers, solvers based on these formulations outperform solvers based on the classical Combined Field Integral Equations.

September 12

Speaker: **Shan Zhao**, Department of Mathematics, University of Alabama

Title: New Developments of Alternating Direction Implicit (ADI) Algorithms for Biomolecular Solvation Analysis

Abstract: In this talk, I will first present some tailored alternating direction implicit (ADI) algorithms for solving nonlinear PDEs in biomolecular solvation analysis. Based on the variational formulation, we have previously proposed a pseudo-transient continuation model to couple a nonlinear Poisson-Boltzmann (NPB) equation for the electrostatic potential with a geometric flow equation defining the biomolecular surface. To speed up the simulation, we have reformulated the geometric flow equation so that an unconditionally stable ADI algorithm can be realized for molecular surface generation. Meanwhile, to overcome the stability issue associated with the strong nonlinearity, we have introduced an operator splitting ADI method for solving the NPB equation. Motivated by our biological applications, we have also recently carried out some studies on the algorithm development for solving the parabolic interface problem. A novel matched ADI method has been developed to solve a 2D diffusion equation with material interfaces involving complex geometries. For the first time in the literature, the ADI finite difference method is able to deliver a second order of accuracy in space for arbitrarily shaped interfaces and spatial-temporal dependent interface conditions.

August 29

Speaker: **Junshan Lin**

Title: Electromagnetic Field Enhancement for Metallic Nano-gaps

Abstract: There has been increasing interest in electromagnetic field enhancement and extraordinary optical transmission effect through subwavelength apertures in recent years, due to its significant potential applications in biological and chemical sensing, spectroscopy, terahertz semiconductor devices, etc. In this talk, I will present a quantitative analysis for the field enhancement when an electromagnetic wave passes through small metallic gaps. We focus on a particular configuration when there is extreme scale difference between the wavelength of the incident wave, the thickness of metal films, and the size of gap aperture. Based upon a rigorous study of the perfect electrical conductor model, we show that enormous electric field enhancement occurs inside the gap. Furthermore, the enhancement strength is proportional to ratio between the wavelength of the incident wave and the thickness of the metal film, which could exceed 10000 due to the scale difference between the two. On other hand, there is no significant magnetic field enhancement inside the gap. The ongoing work along this research direction will also be discussed.