Fall 2018 Seminars

The last decade has witnessed tremendous advances in the fabrication of two-dimensional (2D) materials
with novel electronic structures. Celebrated examples of such materials include graphene and black phosphorus.
The surface conductivity in these systems in the infrared frequency regime permits the propagation of fine-scale electromagnetic waves called surface plasmon-polaritons (SPPs).
In this talk, I will discuss macroscopic consequences of the optical conductivity of 2D materials via solutions of classical Maxwell's equations. I will formally discuss the following topics:
(I) Edges of anisotropic 2D materials act as induced sources of SPPs.
(II) Periodic structures made of 2D materials intercalated in conventional dielectrics may allow for the propagation
of homogenized, slowly varying waves with nearly no phase delay (epsilon-near-zero behavior).
(III) The curvature of 2D materials may generate further confinement of SPPs.
(IV) Nonlinearities of the 2D material and the ambient media cause non-intuitive dispersion of SPPs.
Part of this work is jointly with: A. Andreeva (U. Minnesota), E. Kaxiras (Harvard), T. Low (U Minnesota), M. Luskin (U. Minnesota), M. Maier (Texas A&M), A. Mellet (U MD)

In this talk, I will review our recent works on numerical methods and analysis for solving the Dirac equation in the nonrelativistic limit regime, involving a small dimensionless parameter which is inversely proportional to the speed of light. In this regime, the solution is highly oscillating in time and the energy becomes unbounded and indefinite, which bring significant difficulty in analysis and heavy burden in numerical computation. We begin with four frequently used finite difference time domain (FDTD) methods and the time splitting Fourier pseudospectral (TSFP) method and obtain their rigorous error estimates in the nonrelativistic limit regime by paying particularly attention to how error bounds depend explicitly on mesh size and time step as well as the small parameter. Then we consider a numerical method by using spectral method for spatial derivatives combined with an exponential wave integrator (EWI) in the Gautschi-type for temporal derivatives to discretize the Dirac equation. Rigorous error estimates show that the EWI spectral method has much better temporal resolution than the FDTD methods for the Dirac equation in the nonrelativistic limit regime. Based on a multiscale expansion of the solution, we present a multiscale time integrator Fourier pseudospectral (MTI-FP) method for the Dirac equation and establish its error bound which uniformly accurate in term of the small dimensionless parameter. Numerical results demonstrate that our error estimates are sharp and optimal. Finally, these methods and results are then extended to the nonlinear Dirac equation in the nonrelativistic limit regime. This is a joint work with Yongyong Cai, Xiaowei Jia, Qinglin Tang and Jia Yin.

Every organism transmits the information for making a similar organism across bottleneck stages that are considered generational boundaries. The bottleneck stage is minimally a single cell, which has two interdependent but distinct stores of information. One store is the well-understood linear DNA sequence that is replicated during cell divisions. The other is a three-dimensional arrangement of molecules that cycles during development such that it is nearly recreated at the start of each generation. Together they form the cell code for making an organism – a union of ‘genetic’ and ‘epigenetic’ information stores that coevolve. I will discuss the implications of this perspective for our understanding of living systems and the beginnings of a framework for the joint consideration of all the information that is transmitted across generations to perpetuate life.

What to do when the size and complexity of your model essentially prevent you from using it? Well, get a smaller and simpler model... At the heart of this dimension reduction process is the notion of parameter importance which, ultimately, is part of the modeling process itself. Global Sensitivity Analysis (GSA) aims at efficiently identifying important and non-important parameters; non-importance is important! We will present in this talk advances and challenges in GSA; these will include how to deal with correlated variables, how to treat time-dependent problems and stochastic problems and how to analyze the robustness of GSA itself at low cost. The role played by surrogate models will also be discussed. The discussion will be illustrated by an application from neurovascular modeling. Joint work with Alen Alexanderian, Tim David, Joey Hart and Ralph Smith.

I will present the construction of solutions of the 3D Navier-Stokes equations whose initial vorticity is supported on curves (vortex filaments). This is the first instance for 3D Navier-Stokes where the stability of asymptotically (microscopically) self-similar solutions can be proved. This is joint work with J. Bedrossian and B. Harrop-Griffiths.

We discuss a distributed Lagrange multiplier formulation of the Finite Element Immersed Boundary Method for the numerical approximation of the interaction between fluids and solids. The discretization of the problem leads to a mixed problem for which a rigorous stability analysis is provided. Optimal convergence estimates are proved for the finite element space discretization. The model, originally introduced for the coupling of incompressible fluids and solids, can be extended to include the simulation of compressible structures.

We show that a class of spaces of vector fields whose semi-norms involve the magnitude of “directional” difference quotients is in fact equivalent to the class of fractional Sobolev spaces. The equivalence can be considered a Korn-type characterization of fractional Sobolev spaces. We use the result to better understand the energy space associated to a strongly coupled system of nonlocal equations related to a nonlocal continuum model via peridynamics. Moreover, the equivalence permits us to apply classical Sobolev embeddings in the process of proving that weak solutions to the nonlocal system enjoy both improved differentiability and improved integrability.

We present a positive and asymptotic preserving numerical scheme for solving linear kinetic, transport equations that relax to a diffusive equation in the limit of infinite scattering. The proposed scheme is developed using a standard spectral angular discretization and a classical micro-macro decomposition. The three main ingredients are a semi-implicit temporal discretization, a dedicated finite difference spatial discretization, and realizability limiters in the angular discretization. Under mild assumptions, the scheme becomes a consistent numerical discretization for the limiting diffusion equation when the scattering cross-section tends to infinity. The scheme also preserves positivity of the particle concentration on the space-time mesh and therefore fixes a common defect of spectral angular discretizations. The scheme is tested on well-known benchmark problems with a uniform material medium as well as a medium composed from different materials that are arranged in a checkerboard pattern.

We introduce a second-order time discretization method for stiff kinetic equations. The method is asymptotic-preserving (AP) -- can capture the Euler limit without numerically resolving the small Knudsen number; and positivity-preserving -- can preserve the non-negativity of the solution which is a probability density function for arbitrary Knudsen numbers. The method is based on a new formulation of the exponential Runge-Kutta method and can be applied to a large class of stiff kinetic equations including the BGK equation (relaxation type), the Fokker-Planck equation (diffusion type), and even the full Boltzmann equation (nonlinear integral type). Furthermore, we show that when coupled with suitable spatial discretizations the fully discrete scheme satisfies an entropy-decay property. Various numerical results are provided to demonstrate the theoretical properties of the method.

We will describe generalizations of matrix algebras which suggest a considerable broadening of the spectral analysis toolkit and their applications. Including such classical notions such as Parseval identities, Fourier transforms, Spectral decomposition and Singular Value decomposition.