# Applied and Computational Mathematics

Seminar meets Fridays at 2:00 pm in Parker 356 unless otherwise noted.

Friday, February 28, 2014

Speaker: Paul Schmidt

Title: Oscillatory Entire Solutions of Polyharmonic Equations with Power Nonlinearities

Abstract: There is a vast amount of literature on positive entire solutions (or "ground states") of semilinear elliptic equations with superlinear growth, with numerous results concerning the existence, uniqueness or multiplicity, and asymptotic behavior of such solutions. Typically, positive entire solutions do not exist if the growth of the nonlinearity is subcritical in a certain sense. A natural question, then, is whether there are sign-changing entire solutions in such cases. I will present recent and ongoing work with Monica Lazzo (University of Bari, Italy) on the existence, uniqueness up to scaling and symmetry, and asymptotic behavior of oscillatory entire radial solutions for a subcritical biharmonic equation with power nonlinearity. Time allowing, I will also discuss possible generalizations to the polyharmonic case.

Thursday, February 20, 2014  4:00--5:00 Parker Hall 356

Speaker: Yuesheng Xu
Sun Yat-sen University, China and Syracuse University

Title: Fixed-point Algorithms for Emission Computed Tomography Reconstruction

Abstract: Emission computed tomography (ECT) is a noninvasive molecular imaging method that finds wide clinical applications. It provides estimates of the radiotracer distribution inside a patient.s body through tomographic reconstruction from the detected emission events. In this talk, we propose a fixed-point algorithm - preconditioned alternating projection algorithm (PAPA) for solving the maximum a posteriori (MAP) ECT reconstruction problem. Specifically, we formulate the reconstruction problem as a constrained convex optimization problem with the total variation (TV) regularization via the Bayes law. We then characterize the solution of the optimization problem and show that it satisfies a system of fixed-point equations defined in terms of two proximity operators of the convex functions that define the TV-norm and the constraint involved in the problem. This characterization naturally leads to an alternating projection algorithm (APA) for finding the solution. For efficient numerical computation, we introduce to the APA a preconditioning matrix (the EM-preconditioner) for the large-scale and dense system matrix. We prove theoretically convergence of the PAPA. In numerical experiments, performance of our algorithms, with an appropriately selected preconditioning matrix, is compared with performance of the conventional expectation-maximization (EM) algorithm with TV regularization (EM-TV) and that of the recently developed nested EM-TV algorithm for ECT reconstruction. Based on the numerical experiments performed in our work, we observe that the APA with the EM-preconditioner outperforms significantly the conventional EM-TV in all aspects including the convergence speed and the reconstruction quality. It also outperforms the nested EM-TV in the convergence speed while providing comparable reconstruction quality.

Friday, February 14, 2014

Speaker: Yanzhao Cao

Title:  Steady and Quasi-static Flow in a Deformable Poroelasticitic Medium

Abstract: We are surrounded by poroelastic solid materials: natural (e.g., living tissue: plant or animal, rocks, soils) and manmade (e.g., cement, concrete, filters, foams, ceramics). Because of their ubiquity and unique properties poroelasticity materials are of interest to natural scientists,  and engineers. Applications of poroelasticity include reservoir engineering, biomechanics and environmental engineering.

In this talk we present models for a steady and quasi-static flow in a saturated deformable porous medium. In particular, the application of poroelasticity modeling to hydraulic fracking will be discussed. We will present results on well-posedness, regularity and numerical solutions of the governing PDEs.

Friday, January 31, 2014 ﻿CANCELED

Speaker: Prof. Xiaoying Han

Title:  Dynamics of Stochastic Fast-Slow Chemical Reaction Systems

Abstract: Motivated by the need for dynamical analysis and model reduction in stiff stochastic chemical systems, we focus on the development of methodologies for analysis of the dynamical structure of singularly-perturbed stochastic dynamical systems.  We outline a formulation based on random dynamical systems theory. We demonstrate the analysis for a model two-dimensional stochastic dynamical system built on an underlying deterministic system with a tailored fast-slow structure, and an analytically known slow manifold, employing multiplicative Brownian motion noise forcing.

Friday, January 24, 2014

Speaker: Peijun Li,  Department of Mathematics, Purdue University

Title: Near-Field Imaging of Rough Surfaces

Abstract: In this talk, we consider a class of inverse surface scattering problems in near-field optical imaging, which are to reconstruct the scattering surfaces with resolution beyond the diffraction limit. The scattering surfaces are assumed to be small and smooth deformations of a plane surface. Analytic solutions are derived for the direct scattering problems by using the transformed field expansion, and explicit reconstruction formulas are deduced for the inverse scattering problems. The methods require only a single incident field with a fixed frequency and are realized efficiently by the fast Fourier transform. An error estimate is derived with fully revealed dependence on such quantities as the surface deformation parameter, noise level of the scattering data, and the regularity of the exact scattering surfaces. Numerical results show that the methods are simple, stable, and effective to reconstruct scattering surfaces with subwavelength resolution. Some ongoing and future work will be highlighted along the research line of near-field imaging.

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

No Seminar--Colloquium with Dr. Thomas Caraballo; click here

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: 02/24/2014