# Applied and Computational Mathematics

Seminar meets Fridays at 3:00 pm in Parker 362 unless otherwise noted.

Friday, November 8, 2013

Speaker: Hans-Werner van Wyk, Department of Scientific Computing, Florida State University

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, it 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.

Friday, October 25, 2013

Speaker: Feng Bao, AU
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.

Friday, October 18, 2013

Friday, October 11, 2013

Speaker: Zhongwei Shen﻿, AU
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.

Friday, September 27, 2013

Speaker: Hao-Min Zhou, School of Mathematics, Georgia Institute of Technology
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).

Friday, September 20, 2013

Speaker: Xiaoxia Xie
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.

Friday, September 6, 2013

Speaker: Junshan Lin
Subject: 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.

Last updated: 11/04/2013