Center for Scientific Computation and Mathematical Modeling Center for Scientific Computation and Mathematical Modeling

Spring 2017 Seminars

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Jan. 25, 2:00 p.m.

Joint Numerical Analysis-CSCAMM seminar

Dr. Lise-Marie Imbert-Gerard, Courant Institute, NYU
Variable coefficients and numerical methods for electromagnetic waves

In the first part of the talk, we will discuss a numerical method for wave propagation in inhomogeneous media. The Trefftz method relies on basis functions that are solution of the homogeneous equation. In the case of variable coefficients, basis functions are designed to solve an approximation of the homogeneous equation. The design process yields high order interpolation properties for solutions of the homogeneous equation. This introduces a consistency error, requiring a specific analysis.

In the second part of the talk, we will discuss a numerical method for elliptic partial differential equations on manifolds. In this framework the geometry of the manifold introduces variable coefficients. Fast, high order, pseudo-spectral algorithms were developed for inverting the Laplace-Beltrami operator and computing the Hodge decomposition of a tangential vector field on closed surfaces of genus one in a three dimensional space. Robust, well-conditioned solvers for the Maxwell equations will rely on these algorithms.

Feb. 1, 2:00 p.m.
Prof. Sandra Cerrai, Department of Mathematics, University of Maryland
Fast advection asymptotics for two-dimensional stochastic incompressible viscous fluids and SPDEs on graphs

Fast advection asymptotics for a stochastic reaction-diffusion-advection equation in a two-dimensional bounded domain will be discussed. To describe the asymptotics, one should consider a suitable class of SPDEs defined on a graph, corresponding to the stream function of the underlying incompressible flow.

Feb. 8, 2:00 p.m.
Prof. Leonid Berlyand, Dept. of Mathematics & Materials Research Institute, Pennsylvania State University
Phase Field and Free Boundary Models of Cell Motility

We study two types of models describing the motility of eukaryotic cells on substrates. The first, a phase-field model, consists of the Allen-Cahn equation for the scalar phase field function coupled with a vectorial parabolic equation for the orientation of the actin filament network. The two key properties of this system are (i) presence of gradients in the coupling terms and (ii) mass (volume) preservation constraints. We pass to the sharp interface limit to derive the equation of the motion of the cell boundary, which is mean curvature motion modified by a novel nonlinear term. We establish the existence of two distinct regimes of the physical parameters and prove existence of traveling waves in the supercritical regime.

The second model type is a non-linear free boundary problem for a Keller-Segel type system of PDEs in 2D with area preservation and curvature entering the boundary conditions. We find an analytic one-parameter family of radially symmetric standing wave solutions (corresponding to a resting cell) as solutions to a Liouville type equation. Using topological tools, traveling wave solutions (describing steady motion) with non-circular shape are shown to bifurcate from the standing waves at a critical value of the parameter. Our bifurcation analysis explains, how varying a single (physical) parameter allows the cell to switch from rest to motion.

Feb. 15, 2:00 p.m.
Dr. Flavien Leger, Courant Institute of Mathematical Sciences, New York University
A new approach to bounds on mixing

We consider mixing by incompressible flows. In 2003, Bressan stated a conjecture concerning a bound on the mixing achieved by the flow in terms of an L1 norm of the velocity field. Existing results in the literature use an Lp norm with p>1. In this paper we introduce a new approach to prove such results. It recovers most of the existing results and offers new perspective on the problem. Our approach makes use of a recent harmonic analysis estimate from Seeger, Smart and Street.

Feb. 22, 2:00 p.m.
Prof. Toan Nguyen, Department of Mathematics, Penn State University
Boundary layers: the Good, the Bad, and the Ugly

I will review recent advances concerning the Prandtl's conjecture: slightly viscous flows can be decomposed into inviscid flows, plus a Prandtl's layer near solid boundary, in the inviscid limit.

Mar. 1
No seminar.
Mar. 8, 2:00 p.m.
Dr. Carson Chow, Mathematical Biology Section, National Institutes of Health
Kinetic theory of coupled oscillators

Coupled oscillators arise in contexts as diverse as the brain, synchronized flashing of fireflies, coupled Josephson junctions, or unstable modes of the Millennium bridge in London. Generally, such systems are either studied for a small number of oscillators or in the infinite oscillator, mean field limit. The dynamics of large but finite networks of oscillators is largely unknown. Kinetic theory was developed by Boltzmann and Maxwell to show how microscopic Hamiltonian dynamics of particles could account for the thermodynamic properties of gases. Here, I will show how concepts of kinetic theory and statistical field theory can be applied to deterministic coupled oscillator and neural systems to compute dynamical finite system size effects.

Mar. 15, 2:00 p.m.
Prof. Robert Pego, Department of Mathematical Sciences, Carnegie Mellon University
Microdroplet instablity for a least-action principle for incompressible droplets

The least-action problem for geodesic distance on the `manifold' of fluid-blob shapes exhibits instability due to microdroplet formation. This reflects a striking connection between Arnold's least-action principle for incompressible Euler flows and geodesic paths for Wasserstein distance. A connection with fluid mixture models via a variant of Brenier's relaxed least-action principle for generalized Euler flows will be outlined also. This is joint work with Jian-Guo Liu and Dejan Slepcev.

Mar. 22
No seminar. UMD Spring Break.
Mar. 29, 2:00 p.m.
Prof. Kayo Ide, Department of Atmospheric and Oceanic Science, University of Maryland
Title TBA

Apr. 5, 2:00 p.m.
Prof. Andrej Zlatos, Department of Mathematics, University of California San Diego
Title TBA

Apr. 12, 2:00 p.m.
Dr. Paul Patrone, Mathematical Analysis and Modeling Group, National Institute of Standards and Technology
Title TBA

Apr. 19,

Ki-Net Workshop

Apr. 26,

Cancer Workshop

May 3, 2:00 p.m.
Prof. Perinkulam Krishnaprasad, Institute For Systems Research & Electrical and Computer Engineering, University of Maryland
Title TBA

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