DEPARTMENT OF MATHEMATICS AND STATISTICS

02/03/14 David Shirokoff, McGill, High order penalty methods: a Fourier approach to solving PDE's on domains with curved boundaries .

Penalty methods offer an attractive approach for solving partial differential equations (PDEs) on domains with curved or moving boundaries. In this approach, one does not enforce the PDE boundary conditions directly, but rather solves the PDE in a larger domain with a suitable source or penalty term. The new penalized PDE is then attractive to solve since one no longer needs to actively enforce the boundary conditions. Despite the simplicity, these methods have suffered from poor convergence rates which limit the accuracy of any numerical scheme (usually to first order at best). In this talk I will show how to systematically construct a new class of penalization terms which improve the convergence rates of the penalized PDE, thereby allowing for higher order numerical schemes. I will also show that the new penalized PDE has the added advantage of being solved in a straightforward manner using Fourier spectral methods. Finally, I demonstrate that the method is very general and works for elliptic (Poisson), parabolic (heat), and hyperbolic (wave) equations and can be applied to practical problems involving the incompressible Navier-Stokes equations and Maxwell equations.

03/10/14 Young-Ju Lee, Texas State University, A Solver-Friendly Hybrid Mixed Finite Element Method.

We present a new hybrid mixed finite element method to compute the flux variable accurately and efficiently. The method is a kind of two--step method, based on a system of first-order equations for second-order elliptic partial differential equations. On a coarse mesh, the primary variable is approximated by a standard Galerkin method. Then, on a fine mesh, an H(div}) projection of the dual variable is sought as an accurate approximation for the flux variable. We show that the mesh size "h" for the finer mesh can be taken as a square of the coarse mesh size "H" for lowest order approximation spaces. This means that the computational cost for the coarse-grid solution is negligible compared to that for the fine-grid solution. Our approach requires to solve two coercive systems, and thus it does not rely on the framework of traditional mixed formulations. Utilizing that the pair of finite element spaces is free from the requirement of inf-sup stability condition, one can take full advantages of well-developed fast solvers, such as multigrid methods for two completely decoupled problems. In addition, our approach is shown to provide an asymptotically exact a posteriori error estimator for the potential variable in $H^1$ norm. This is the joint-work with JaEun Ku at Oklahoma State University and Doongwoo Sheen at Seoul National University.

03/24/14 Maurizio Porfiri, NYU Poly, Fish'n robots: not a take-out food.

Engineering design of robots is often inspired by nature; recently developed bioinspired robots accurately imitate various aspects of their live counterparts. Yet, the relationship between engineering and nature has often been one-directional: engineers borrow ideas from nature to build more efficient, more appealing, and better performing robotic systems for use in traditional human-centered applications. In some cases, these systems are used as proxies for studying the natural system, but whether these devices can be integrated within the ecological niche inspiring their design seldom is experimentally tested. An even more elemental research question pertains to the feasibility of modulating spontaneous behavior of animal systems through bioinspired robotics. In this talk, we discuss recent research findings at the Dynamical Systems Laboratory of New York University Polytechnic School of Engineering on the feasibility of integrating robotic fish in experimental paradigms to investigate the behavior of social fish. In this context, robots can be used to offer consistent, customizable, and controllable stimuli for decomposing the complex nature of information sharing and decision making in fish. This talk will address fundamental scientific questions such as: is fish behavioral response influenced by a robotic fish and, if so, what is the consistency of such response? What are the determinants of attraction or repulsion of a robotic fish and can they be modulated through the administration of psychoactive compounds? What is the role of hydrodynamic effects and visual cues? Does the behavior and motion of a robotic fish influence fish social response? Do fish interact differently with a robotic fish depending on their personality?

03/31/14 Junshan Lin, Auburn, Scattering Resonances for Photonic Structures and Schrodinger Operators .

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. I will begin with a study on the finite symmetric photoinc structure to illustrate the convergence behavior of resonances. Then a general perturbation approach will be introduced for the analysis of near bound-state resonances for both cases. In particular, it is shown that, for a finite one dimensional photonic crystal with a defect, the near bound-state resonances converge to the point spectrum of the infinite structure with an exponential rate when the number of periods increases. An analogous exponential decay rate also holds for the Schrodinger operator with a potential function that is a low-energy well surrounded by a thick barrier. The analysis also leads to a simple and accurate numerical approach to approximate the near bound-state resonances. This is a joint work with Prof. Fadil Santosa in University of Minnesota.

04/7/14 Sangwoo Shin, Princeton, Growth of nanowires in porous structures: dynamics, morphological instabilities, and a control strategy .

The synthesis of nanowires for future electronics and energy applications requires scalability, density, reproducibility and cost-effectiveness. In this regard, template-assisted electrodeposition offers distinct advantages over other synthesis methods. However, the growth of nanowires using this technology is unstable, which presents a major challenge for such widespread applications. Here we show theoretically and experimentally that the dynamics of this process is diffusion-limited, which results in morphological growth instabilities. Moreover, we use our findings to devise a method to control the growth instability. By introducing a temperature gradient across the porous template, the spatial dependence in ion diffusion triggers a self-controlled growth of the nanowires that can control the growth instability. This strategy significantly increases the fraction of long nanowires by reducing the length variation between them. In addition to shedding light on a key nanotechnology, our results provide fundamental insights into a variety of interfacial processes in materials science such as crystal growth and tissue growth in scaffolds.

04/22/14 Philippe Guyenne, Delaware, Waves in ice sheets and in the bone.

This talk will be on two different problems. The first part is about the modeling and numerical simulation of waves in ice sheets, e.g. as occurring in polar regions. A Hamiltonian formulation for ice sheets deforming on top of an ideal fluid is presented and nonlinear wave solutions are examined. In certain asymptotic regimes, analytical solutions are derived and compared with fully nonlinear solutions obtained numerically by a pseudospectral method. In the second part, a viscoelastic model for ultrasound propagation through cancellous bone is proposed. The trabecular matrix of cancellous bone is described as an isotropic viscoelastic material, while the interstitial fluid is modeled by Stokes flow. To simulate bone samples with complicated microstructure, a representative volume element is constructed by using a 2D random distribution of fluid and solid particles. The equations are solved numerically by a finite-difference scheme. The phenomenon of ultrasound attenuation through cancellous bone is examined with the model.

04/28/14 Mike Haslam, York, A High Order Method for the Scattering Problem from Layered Conducting Dielectric Media.

The problem of evaluating the electromagnetic response of a periodic surface to an incident plane wave is of great importance in science and engineering. Applications of the theory exist in several fields of study including solar energy research, optical instrument design, and remote sensing. We discuss the extension of our previous methods to treat the problem of scattering from layered dielectric surfaces, where the conductivity may contain a lossy component. The generalization of our methods is not straight-forward, and involves the careful treatment of certain hyper-singular operators which arise in the formulation of the problem in terms of surface integral equations. We demonstrate the rapid convergence of our methods for classically difficult cases in the resonance regime.

05/5/14 Vlad Vicol, Princeton, Long time behavior of forced critical Surface Quasi-Geostrophic equation (SQG) .

We address the long-time dynamics of smooth solutions to the forced critical SQG equation \begin{align*} \partial_t \theta + {\mathcal R}^\perp \theta \cdot \nabla \theta + \Lambda \theta = f \end{align*} where $f$ is a time-independent force, ${\mathcal R}^\perp = (-{\mathcal R}_2,{\mathcal R}_1)$ are Riesz transforms, and $\Lambda = (-\Delta)^{1/2}$ is the Zygmund operator. Using a nonlinear lower bound for the fractional Laplacian, we give a new proof that the solution instantly becomes Holder continuous, without the use of DeGiorgi iteration. The nonlinear lower bound for $\Lambda$ also shows that the solution is in fact classical. We prove that for $t \gg 1$ the norm of the solution measured in a sufficiently strong topology depends solely on $f$, and use this fact to show the existence of a global attractor, with finite fractal dimension. Additionally, we prove there is no anomalous dissipation in the long time averages for the viscous perturbations of the system. This is joint work with P. Constantin and A. Tarfulea.

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