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• Nov. 8, 2013
Hans-Werner van Wyk
Department of Scientific Computing, Florida State University
Location and Time: Parker Hall 362, 3pm-4pm
Title: Uncertainty Quantification, Multilevel Sampling Methods and Parameter Identification

Abstract: As simulation plays an increasingly central role in modern science and engineering research, by supplementing experiments, aiding in the prototyping of engineering systems or informing decisions on safety and reliability, is has become critical to identify sources of model uncertainty as well as to quantify their effect on model outputs. For systems modeled by partial differential equations with random distributed parameters, statistical sampling methods such as Monte Carlo and stochastic collocation have proven both versatile and easy to implement. Multilevel sampling improves upon traditional sampling by dynamically incorporating the model's spatial discretization into the sampling procedure, thereby not only increasing efficiency but also allowing for a closer monitoring of overall convergence behavior. Originally developed for Monte Carlo sampling, these methods have since been extended to more general sampling methods, most notably stochastic collocation. We give a brief overview of the ideas underlying these methods and show how they can be used in the forward propagation of uncertainty and possibly to statistical inverse problems.

• Nov. 1, 2013, special applied math seminar
Xiaobing H. Feng
Department of Mathematics, The University of Tennessee
Location and Time: Parker Hall 250, 4:00pm-5:00pm
Title: Numerical Differential Calculus: A New Paradigm for Developing Numerical Methods for PDEs

Abstract: In this talk I shall first present a newly developed (discontinuous Galerkin finite element) numerical differential calculus theory for approximating weak (or distributional) derivatives of broken Sobolev functions. Various properties and calculus rules (such as product and chain rules, integration by parts formula and divergence theorem) for the proposed numerical derivatives will be discussed. I shall then discuss how this numerical differential calculus framework can be conveniently used to systematically construct numerical methods for various linear and nonlinear (including fully nonlinear) PDEs and how these new numerical methods are related to the existing numerical PDE methods. The materials of this talk is based on a joint work with Michael Neilan of University of Pittsburgh and Tom Lewis of the University of North Carolina at Greensboro.

• October 25, 2013
Feng Bao
Department of Mathematics and Statistics, Auburn University
Location and Time: Parker Hall 362, 3pm-4pm
Title: Numerical Algorithms for Nonlinear Filter Problems

Abstract: We consider the classical filtering problem where a signal process is modeled by a stochastic differential equation and the observation is perturbed by a white noise. The goal is to find the best estimation of the signal process based on the observation. Kalman filter, Particle filter and Zakai filter are some well known approaches to solve the optimal filter. In this talk, we shall show some new numerical algorithms to solve the nonlinear filtering problems. Both theoretical results and numerical experiments will be presented.

• Oct. 18, 2013, special applied math seminar
Thomas Caraballo
University of Sevilla
Location and Time: Parker Hall 249, 2:30pm-3:30pm
Title: Effects of Noise on the Asymptotic Behavior of Dynamical Systems

Abstract: The aim of this talk is to present some features concerning the effects of noise on the asymptotic behavior of dynamical systems. It is well-known now the stabilizing and destabilizing effects which the appearance of different kinds of noise (e.g. Ito or Stratonovich) may have on the stationary solutions (equilibria) of deterministic dynamical systems. Now we will report some results on the appearance of exponentially stable stationary (in the stochastic sense) solutions when some noise is added to the model, as well as, the analysis of the existence of random attractor when the deterministic model is not known to have (or does not have) a global attractor. These results will show some kind of stabilization on global attractors instead of only on equilibria.

• October 11, 2013
Zhongwei Shen
Department of Mathematics and Statistics, Auburn University
Location and Time: Parker Hall 362, 3pm-4pm
Title: Introduction to the Scattering Theory of Schrodinger Operators

Abstract: Scattering theory studies the large time behavior of quantum-mechanical systems. Besides the dynamics, it has important spectral implications, say, in particular, the stability of absolutely continuous spectrum. In this talk, I will first give an introduction to the spectral theory and scattering theory of self-adjoint operators. Then, I will present some classical results about Schrodinger operators with decaying potentials as well as some open problems about Schrodinger operators with long-range potentials. Classical and recent results about Schrodinger operators with sparse potentials will also be presented.

• September 27, 2013
Hao-Min Zhou
School of Mathematics, Georgia Institute of Technology
Location and Time: Parker Hall 362, 3pm-4pm
Title: Fokker-Planck equations, Free Energy, and Markov Processes on Graphs

Abstract: The classical Fokker-Planck equation is a linear parabolic equation which describes the time evolution of probability distribution of a stochastic process defined on an Euclidean space. Corresponding to a stochastic process, there often exists a free energy functional which is defined on the space of probability distributions and is a linear combination of a potential and entropy. In recent years, it has been shown that the Fokker-Planck equation is the gradient flow of the free energy functional defined on the Riemannian manifold of probability distributions whose inner product is generated by a 2-Wasserstein distance. In this talk, we present results on similar matters for a free energy functional or Markov process defined on a graph with a finite number of vertices and edges. If $N\ge 2$ is the number of vertices of the graph, we show that the corresponding Fokker-Planck equation is a system of $N$ {\it nonlinear} ordinary differential equations defined on a Riemannian manifold of probability distributions. However, in contrast to stochastic processes defined on Euclidean spaces, the situation is more subtle for discrete spaces. We have different choices for inner products on the space of probability distributions resulting in different Fokker-Planck equations for the same process. It is shown that there is a strong connection but also substantial discrepancies between the systems of ordinary differential equations and the classical Fokker-Planck equation on Euclidean spaces. Furthermore, both systems of ordinary differential equations are gradient flows for the same free energy functional defined on the Riemannian manifolds of probability distributions with different metrics. Some examples will also be discussed. The work is jointly with Wen Huang (USTC) and Yao Li (Georgia Tech).

• September 20, 2013
Xiaoxia Xie
Department of Mathematics and Statistics, Auburn University
Location and Time: Parker Hall 362, 3pm-4pm
Title: Approximations of Random Dispersal Operators/Equations by Nonlocal Dispersal Operators/Equations

Abstract: Both random dispersal evolution equations (or reaction diffusion equations) and nonlocal dispersal evolution equations (or differential integral equations) are widely used to model diffusive systems in applied science and have been extensively studied. It has been shown that the random dispersal operator and the nonlocal dispersal operator share many similar properties and they are also essentially different in some way. But there are not many studies on the how they are related.

In this talk, I will present the approximations of random dispersal operators/equations by nonlocal dispersal operators/equations. In particular, we show that (1). the solutions of properly rescaled nonlocal dispersal initial-boundary value problems converge to the solutions of the corresponding random dispersal initial-boundary value problems; (2). the principal spectrum points of nonlocal dispersal operators with properly rescaled kernels converge to the principal eigenvalues of the corresponding random dispersal operators; (3). the unique positive stationary solutions of nonlocal dispersal KPP equations with properly rescaled kernels converge to the unique positive stationary solutions of the corresponding random dispersal KPP equations.The results obtained have potential applications on the population dynamics and Turing patterns.

• September 6, 2013
Junshan Lin
Department of Mathematics and Statistics, Auburn University
Location and Time: Parker Hall 362, 3pm-4pm
Title: Scattering Resonances for Photonic Structures and Schrodinger Operators

Abstract: Resonances are important in the study of transient phenomena associated with the wave equation, especially in understanding the large time behavior of the solution to the wave equation when radiation losses are small. In this talk, I will present recent studies on the scattering resonances for photonic structures and Schrodinger operators. In particular, for a finite one dimensional symmetric photonic crystal with a defect, it is shown that the near bound-state resonances converge to the point spectrum of the infinite structure with an exponential rate when the number of periods increases. Such an exponential decay rate also holds for the Schrodinger operator with a potential function that is a low-energy well surrounded by a thick barrier. We introduce a general method that is suitable for the Schrodinger operator in both low and high dimensions, and has the potential to be extended to the photonic case.