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# Applied and Computational Mathematics

**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.

**DMS Applied Mathematics Seminar**

Nov 02, 2018 02:00 PM

Parker Hall 328

Speaker: **Vu Thai Luan** (Southern Methodist University)

Abstract: In recent years, exponential integrators have emerged as an efficient alternative to standard time integrators for stiff PDEs. They are fully explicit and do not suffer from the stability restriction imposed by the CFL condition for the linear part. It has also been shown that exponential integrators can take much larger time steps than implicit/IMEX methods while maintaining the same level of accuracy. Thus they can offer significant computational savings, particularly for large-scale stiff systems where no efficient preconditioner is available. In this talk, I will present the basic idea of constructing exponential integrators, derive new methods, and show their applications in meteorology and visual computing.

**DMS Applied Mathematics Seminar**

Oct 26, 2018 02:00 PM

Parker Hall 328

Speaker: **Jiguang Sun** (Michigan Technological University)

Title: Finite Element Methods for Eigenvalue Problems

Abstract: The numerical solution of eigenvalue problems is of fundamental importance in many scientific and engineering applications, such as structural dynamics, quantum chemistry, electrical networks, magnetohydrodynamics, and control theory. Due to the flexibility in treating complex structures and rigorous theoretical justification, finite element methods, including conforming finite elements, non-conforming finite elements, mixed finite elements, discontinuous Galerkin methods, etc., have been popular for eigenvalue problems of partial differential equations. In this talk, we shall introduce finite element approximations for several typical problems, including the Dirichlet eigenvalue problem, the biharmonic eigenvalue problems, Maxwell's eigenvalue problem, and the new quad-curl eigenvalue problem. Furthermore, we shall discuss two non-selfadjoint eigenvalue problems: the transmission eigenvalue problem and the Steklov eigenvalue problem. To solve the resulting matrix eigenvalue problems, a new algebraic eigensolver is developed and some recent progresses are presented.

**DMS Applied Mathematics Seminar**

Oct 19, 2018 02:00 PM

Parker Hall 328

Speaker: **Basiru Usman**

Title: Hopfield lattice model

Abstract: In this talk I will introduce the Hopfield neural network (HNN) proposed by J. J. Hopfield in 1984 which consists of n system of differential equations. The dynamics of HNN resembles that of neurobiology so it is natural to think of n been very large, so then I will introduce the Hopfield lattice model which is basically the extension of n system of differential equation to infinite. To incorporate the environmental noise we introduce randomness to the input term. Existence and uniqueness of solution, long term behavior are investigated.

**DMS Applied Mathematics Seminar**

Oct 05, 2018 02:00 PM

Parker Hall 328

Speaker: **Dan Nguyen** (University of Alabama)

Title: A Multi-scale Approach to Limit Cycles with Random Perturbations Involving Fast Switching and Small Diffusion

Abstract: This talk is devoted to multi-scale stochastic systems. The motivation is to treat limit cycles under random perturbations involving fast random switching and small diffusion, which are represented by the use of two small parameters. Associated with the underlying systems, there are averaged or limit systems. Suppose that for each pair of the parameters, the solution of the corresponding equation has an invariant probability measure $\mu^{\eps,\delta}$, and that the averaged equation has a limit cycle in which there is an averaged occupation measure $\mu^0$ for the averaged equation. Our main effort is to prove that $\mu^{\eps,\delta}$ converges weakly to $\mu^0$ as $\eps \to 0$ and $\delta \to 0$ under suitable conditions. We also examine an application to a stochastic predator-prey model. Moreover, some numerical examples will also be reported.

**DMS Applied Mathematics Seminar**

Sep 28, 2018 02:00 PM

Parker Hall 328

Speaker: **Shitao Liu** (Clemson University)

Title: Inverse Hyperbolic Problems

Abstract: Inverse hyperbolic problems arise naturally in many practical applications such as medical imaging and geophysical exploration. Mathematically, the problems consist of determining the coefficients of second-order hyperbolic equations from the boundary measured data. In this talk we will discuss the single measurement and many measurements—two of the most important formulations—of the inverse hyperbolic problems. In particular, we will make a thorough study on the uniqueness and stability of reconstructing the wave speed from acoustic boundary measurement based on different approaches that are linked to the boundary control theory for hyperbolic equations.

**DMS Applied Mathematics Seminar**

Sep 21, 2018 02:00 PM

Parker Hall 328

Speaker: **Thi Thao Phuong Hoang** Auburn University

Title:Conservative explicit local time-stepping schemes for the shallow water equations

Abstract: We present explicit local time-stepping (LTS) schemes of second and third order accuracy for the shallow water equations. The system is discretized in space by a C-grid staggering method, namely the TRiSK scheme adopted in MPAS-Ocean, a global ocean model with the capability of resolving multiple resolutions within a single simulation. The time integration is designed based on the strong stability preserving Runge-Kutta (SSP-RK) methods, but different time step sizes can be used in different regions of the domain and are only restricted by respective local CFL conditions. The proposed schemes preserve some important physical quantities in the discrete sense, such as exact conservation of the mass and potential vorticity and conservation of the total energy within time truncation errors. Moreover, they inherit the natural parallelism of the original explicit global time-stepping schemes. Extensive numerical tests are presented to illustrate the performance of the proposed algorithms.

Last Updated: 09/25/2015