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# Applied and Computational Mathematics

DMS Applied Mathematics Seminar
Feb 22, 2019 02:00 PM
Parker Hall 328

Speaker: Prof. Erkan Nane

Title: Blow-up results for space--time fractional dynamics

DMS Applied Mathematics Seminar
Feb 12, 2019 04:00 PM
Parker Hall 352

Please note the special time and location.

Speaker: Professor Jialin Hong (Institute of Computational Mathematics and Scientific/Engineering Computing, Chinese Academy of Sciences)

Title: Stochastic Symplecticity and Ergodicity of Numerical Methods for Stochastic Nonlinear Schroedinger Equation

Abstract: In this talk we present a review on stochastic symplecticity (multi-symplecticity) and ergodicity of numerical methods for stochastic nonlinear Schroedinger (NLS) equation. The equation considered is charge conservative and has the multi-symplectic conservation law. Based on a stochastic version of variational principle, we show that the phase flow of the equation, considered as an evolution equation, preserves the symplectic structure of the phase space. We give some symplectic integrators and multi-symplectic methods for the equation. By constructing control system and invariant control set, it is proved that the symplectic integrator, based on the central difference scheme, possesses a unique invariant measure on the unit sphere. Furthermore, by using the midpoint scheme, we get a full discretization which possesses the discrete charge conservation law and the discrete multi-symplectic conservation law. Utilizing the Poisson equation corresponding to the finite dimensional approximation, the convergence error between the temporal average of the full discretization and the ergodic limit of the symplectic method is derived.

(In collaboration with Dr. Chuchu Chen, Dr. Xu Wang and Dr. Liying Zhang).

DMS Applied Mathematics Seminar
Feb 08, 2019 02:00 PM
Parker Hall 328

Speaker: Dr. Shelvean Kapita, University of Georgia

Title: Bivariate Spline Solutions to the Helmholtz Equation

Abstract: Although there are many computational methods for solving the Helmholtz equation, e.g., $$hp$$ finite element methods, the numerical solution of the Helmholtz equation still poses challenges, particularly for large wavenumbers. We shall explain how to use bivariate splines to numerically solve the Helmholtz equation in both bounded and unbounded domains. In addition, we shall establish existence, uniqueness and stability of the weak solution of the Helmholtz equation, under the assumption that $$k^2$$, where $$k$$ is the wavenumber, is not a Dirichlet eigenvalue of the associated Poisson equation. With this assumption, the standard assumption that the domain be strictly star-shaped  is no longer needed. Finally, we will explain how to use bivariate splines to solve the exterior domain Helmholtz equation using a PML technique. We demonstrate the effectiveness of bivariate splines for the bounded domain and exterior Helmholtz equation with a variety of numerical examples.

DMS Applied Mathematics Seminar
Feb 01, 2019 02:00 PM
Parker Hall 328

Speaker: Yanzhao Cao (Auburn University)

Title: Robust Robin-Robin Domain Decomposition Algorithms for Stokes-Darcy Interface Problems

Abstract: Many turbulent/porous flow problems can be modeled by Stokes-Darcy interface systems.  In this talk we will discuss two efficient Robin-Robin domain decomposition algorithms to solve these systems. Both convergence analysis and numerical experiments will be presented.

DMS Applied Mathematics Seminar
Jan 25, 2019 02:00 PM
Parker Hall 328

Speaker: Junshan Lin (Auburn University)
Title: Embedded eigenvalues and Fano resonance for metallic structures with small holes

Abstract: Fano resonance, which was initially discovered in quantum mechanics by Ugo Fano, has been extensively explored in photonics since the past decade due to its unique resonant feature of a sharp transition from total transmission to total reflection. Mathematically, Fano resonance is related to certain eigenvalues embedded in the continuum spectrum of the underlying differential operator.  For photonic structures, the quantitative studies of embedded eigenvalues mostly rely on numerical approaches. In this talk, based on layer potential technique and asymptotic analysis, I will present quantitative analysis of embedded eigenvalues and their perturbation as resonances for a periodic array of subwavelength metallic structure. From a quantitative analysis of the wave field for the scattering problem, a rigorous proof of Fano resonance will be given.  In addition, the field enhancement at Fano resonance frequencies will be discussed.

DMS Applied Mathematics Seminar
Jan 11, 2019 02:00 PM
Parker Hall 328

Speaker: Yusuke Asai (Hokkaido University, Japan)

Title: Development of hepatitis C virus dynamics model and numerical methods for random ordinary differential equations with time delay

Abstract: Mathematical modeling by differential equations plays an important role to understand natural sciences. In particular, required data are frequently not obtained in biology and medicine and simulation using mathematical models helps us to analyze system behavior under various scenarios.

Deterministic models have been long investigated and applied to natural phenomena; however, some factors might be ignored in model building process or we encounter random effect from environment in practice. To handle such uncertain effect, random ordinary differential equations (RODEs) can be an ideal tool because of its simplicity in model building as well as the regularity of the corresponding noise processes. Recently, several classes of numerical methods for RODEs have been developed and applied to real problems, yet time delay has been ignored in those applications. To capture and understand system behavior more accurately, we need to handle both of randomness and time delay simultaneously.

In this talk, development of hepatitis C virus (HCV) dynamics models is briefly introduced and then numerical methods for RODEs with time delay are systematically constructed. The developed methods are applied to the introduced HCV model with target cells, infected cells and viruses compartments, and eclipse phase, the time elapsed between cell infection and virus production, and their behavior will be investigated.

DMS Applied Mathematics Seminar
Dec 07, 2018 02:00 PM
Parker hall 328

Speaker: Habib Najm (Sandia National Lab)

Title: Uncertainty Quantification in Computational Models of Large Scale Physical Systems

Abstract: Uncertainty quantification (UQ) in large scale computational models of complex physical systems faces the two key challenges of high dimensionality and high sample computational cost. Such models often involve a large number of uncertain parameters, associated with various modeling constructions, as well as uncertain initial/boundary conditions. Exploring such high dimensional spaces typically necessitates the use of a large number of computational samples, which, given the cost of large scale computational models, is prohibitively expensive and thus infeasible.  I will discuss a set of UQ methods and a UQ workflow to address this challenge. The suite of methods includes global sensitivity analysis (GSA) with polynomial chaos (PC) regression and compressive sensing, coupled with multilevel Monte Carlo (MLMC) and/or multilevel multifidelity (MLMF) methods. The combination of these tools is often useful to reliably cut down dimensionality with feasible computational costs, identifying a lower dimensional subspace on the uncertain parameters where subsequent adaptive sparse quadrature PC methods can be employed with accurate estimation of predictive uncertainty. I will illustrate this UQ workflow on model problems and on an application involving high-speed turbulent reacting flow.

DMS Applied Mathematics Seminar
Nov 30, 2018 02:00 PM
Parker Hall 328

Speaker: Lianzhang Bao (Jilin University, China)

Title: Dynamics in the logistic type chemotaxis models with a free boundary

Abstract: This talk is concerned with the dynamics in the logistic type chemotaxis with a free boundary. In the first section, a free boundary problem via Fick's law will be derived to describe the spreading of certain species and some current results of the minimal chemotaxis model, chemotaxis model with logistic terms on fixed bounded and unbounded domain will be reviewed. In the second section, the global bounded solution of the free boundary problem and its asymptotic dynamics will be investigated. Some open problems and future works will also be discussed.

This is a joint work with Professor Wenxian Shen.

DMS Applied Mathematics Seminar
Nov 16, 2018 02:00 PM
Parker Hall 328

Speaker: Ismail Abdulrashid (Auburn University)

Title: Effects of Delays in Mathematical Models of Cancer Chemotherapy

Abstract: Two mathematical models of chemotherapy cancer treatment are studied and compared, one modeling the chemotherapy agent as the predator and the other modeling the chemotherapy agent as the prey. In both models constant delay parameters are introduced to incorporate the time lapsed from the instant the chemotherapy agent is injected to the moment it starts to be effective. For each model, the existence and uniqueness of non-negative bounded solutions are first established. Then both local and Lyapunov stability for all steady states are investigated. In particular, sufficient conditions dependent on the delay parameters under which each steady state is asymptotically stable are constructed. Numerical simulations will be presented in order to illustrate the theoretical results.

DMS Applied Mathematics Seminar
Nov 09, 2018 02:00 PM
Parker Hall 328

Speaker: Mozhgan Entekhabi (Florida A&M University)

Title: Inverse Source Problems for Wave Propagation

Abstract: Inverse source scattering problem arises in many areas of science. It has numerous applications to surface vibrations, acoustical and bio-medical industries, and material science. In particular, inverse source problem seeks the radiating source which produces the measured wave field. This research aims to provide a technique for recovering the source function of the classical elasticity system and the Helmholtz equation from boundary data at multiple wave numbers when the source is compactly supported in an arbitrary bounded C 2 − boundary domain, establish uniqueness for the source from the Cauchy data on any open non empty part of the boundary for arbitrary positive K, and increasing stability when  wave number K is getting large. Various studies showed that the uniqueness can be regained by taking multifrequency boundary measurement in a non-empty frequency interval (0, K) noticing the analyticity of wave-field on the frequency. One of important examples is recovery of acoustic sources from boundary measurement of the pressure. This type of inverse source problem is also motivated by the wide applications in antenna synthesis, medical imaging and geophysics.

Last Updated: 09/25/2015