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• Jan. 24, 2014
Peijun Li
Department of Mathematics, Purdue University
Location and Time: Parker Hall 356, 2pm-3pm
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.

• Jan. 31, 2014 (Cancelled)
Xiaoying Han
Department of Mathematics and Statistics, Auburn University
Location and Time: Parker Hall 356, 2pm-3pm
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.

• Feb. 14, 2014
Yanzhao Cao
Department of Mathematics and Statistics, Auburn University
Location and Time: Parker Hall 356, 2pm-3pm
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.

• Feb. 20, 2014
Yuesheng Xu
Sun Yat-sen University, China and Syracuse University
Location and Time: Parker Hall 356, 4pm-5pm
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.

• Feb. 28, 2014
Paul Schmidt
Department of Mathematics and Statistics, Auburn University
Location and Time: Parker Hall 356, 2pm-3pm
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.

• March 28, 2014
Dmitry Glotov
Department of Mathematics and Statistics, Auburn University
Location and Time: Parker Hall 356, 2pm-3pm
Title: Slow coarsening in the Allen-Cahn model

Abstract: Coarsening refers to the evolution of patterns of clusters in which the area of the interfaces tends to decrease over time. This phenomenon is manifested in the models for foams, grain structure in allows, molecular beam epitaxy, etc. The rates of coarsening are physically relevant since they are readily observable both empirically and in the models. We study the rates of coarsening in the Allen-Cahn model and will present estimates that indicate that these rates follow a power law. The method is based on the framework developed by Kohn and Otto which links the length scale of the system with its energy. The method relies on an interpolation inequality, dissipation inequality, and an ODE argument and produces time-averaged one-sided estimates of the energy.

• April 11, 2014
Leo Rebholz
Department of Mathematics, Clemson University
Location and Time: Parker Hall 356, 2pm-3pm
Title: Efficient, stable, and accurate finite element discretizations for approximate deconvolution models of turbulent flow

Abstract: The talk discusses discretization strategies for the Stolz-Adams approximate deconvolution model (ADM) of turbulent flow. After an introduction to the Navier-Stokes equations and Large Eddy Simulation, we derive the ADM and discuss difficulties in constructing efficient, stable, and accurate numerical schemes for it which use finite elements for the spatial discretization. We then show how a small change to the model can resolve this critical numerical issue, and provide several numerical experiments that demonstrate the effectiveness of the modified model/scheme.

• April. 18, 2014
Xiaoying Han
Department of Mathematics and Statistics, Auburn University
Location and Time: Parker Hall 356, 2pm-3pm
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.

• April 25, 2014
Tin-Yau Tam
Department of Mathematics and Statistics, Auburn University
Location and Time: Parker Hall 356, 2pm-3pm
Title: Inverse spread limit of nonnegative matrix and its application

Abstract: Given a nonnegative $n\times n$ matrix $A$, we introduce the notion of inverse spread $s(A)$. We study the asymptotic behavior of $s(A^m)$, that is, the behavior of $s(A^m)$ as $m\to \infty$. The study arises from evolutionary biology. The study involves Perron-Frobenius Theory, graph theory, DNA, etc.