Applied Mathematics Colloquium - Fall 2017

Maxence Cassier, Columbia University
On the Limiting Amplitude Principle for Maxwell’s Equations at the Interface of a Metamaterial

In this talk, we are interested in a transmission problem between a dielectric and a metamaterial. The question we consider is the following: does the limiting amplitude principle hold in such a medium? This principle defines the stationary regime as the large time asymptotic behavior of a system subject to a periodic excitation. An answer is proposed here in the case of a two-layered medium composed of a dielectric and a particular metamaterial (Drude model). In this context, we reformulate the time-dependent Maxwell’s equations as a Schrödinger equation and perform its complete spectral analysis. This permits a quasi-explicit representation of the solution via the ”generalized diagonalization” of the associated unbounded self-adjoint operator. As an application of this study, we show finally that the limiting amplitude principle holds except for a particular frequency, called the critical frequency, characterized by a ratio of permittivities and permeabilities equal to -1 across the interface. This frequency is a resonance of the system and the response to this excitation blows up linearly in time.

Robert Pego, Carnegie Mellon
Microdroplet Instablity in a Least-Action Principle for Incompressible Fluids

The least-action problem for geodesic distance on the `manifold' of fluid-blob shapes exhibits an 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.

Alex Townsend, Cornell University
Why are There so Many Matrices of Low Rank in Computational Math?

Matrices that appear in computational mathematics are so often of low rank. Since random ("average") matrices are almost surely of full rank, mathematics needs to explain the abundance of low rank structures. We will give a characterization of certain low rank matrices using Sylvester matrix equations and show that the decay of singular values can be understood via an extremal rational problem. We will give another characterization involving the Johnson-Lindenstrauss Lemma that partially explains the abundance of low rank structures in big data.

Gennady Gor, NJIT
How to Interpret Ultrasonic Experiments on Fluid-Saturated Nanoporous Media?

Ultrasound is a versatile tool which has been used for investigation of matter for decades. Since the wavelength of ultrasound in solids exceeds the sizes of nanopores by several orders of magnitude, ultrasound cannot resolve any individual features on the pore scale, but it provides information on the sample as a whole. In particular, the velocity of ultrasound propagation in a fluid-saturated sample gives the values of the sample's elastic moduli. In my presentation I will discuss recently published experimental data on ultrasound propagation in nanoporous glass saturated with liquid argon. I will show the results of a Monte Carlo molecular simulation for this system, providing a relation between the measured bulk modulus of a fluid-saturated sample and the pore size. These findings suggest that ultrasound can be effectively used as a characterization tool for nanoporous materials.

Andrew Bernoff, Harvey Mudd College
Energy Driven Pattern Formation in Thin Fluid Layers: The Good, the Bad and the Beautiful

A wide variety of physical and biological systems can be described as continuum limits of interacting particles. Many of these problems are gradient flows and their dynamics are governed by a monotonically decreasing interaction energy that is often non-local in nature. We show how to exploit these energies numerically, analytically, and asymptotically to characterize the observed behavior. We describe three such systems. In the first, a Langmuir layer, line tension (the two-dimensional analog of surface tension) drives the fluid domains to become circular and the rate of relaxation to these circular domains can be used to deduce the magnitude of the line tension forces. In the second, a Hele-Shaw problem, vexing changes in topology are observed. The third system models the formation of the convoluted fingered domains observed experimentally in ferrofluids for which pattern formation is driven by line tension and dipole-dipole repulsion. We show that noise in this system plays an unexpected but essential role and deduce an algorithm for extracting the dipole strength using only a shape's perimeter and morphology.

Daniel Szyld, Temple University
Multiple Preconditioned GMRES for Shifted Systems, with Applications to Hydrology and Matrix Functions

GMRES with multiple preconditioners (MPGMRES) is a method that allows for several preconditioners to be used simultaneously. In this manner, the effect of all preconditioners is applied in each iteration. An implementation of MPGMRES is proposed for solving shifted linear systems with shift-and-invert preconditioners. With this type of preconditioner, the Krylov subspace can be built without requiring the matrix-vector product with the shifted matrix. Furthermore, the multipreconditioned search space is shown to grow only linearly with the number of preconditioners. This allows for a more efficient implementation of the algorithm. The proposed implementation is tested on shifted systems that arise in computational hydrology and the evaluation of different matrix functions. The numerical results indicate the effectiveness of the proposed approach.

Peter Monk, University of Delaware
Finite Element Methods for Maxwell’s Equations

In the last few decades, finite element methods for approximating the time harmonic Maxwell system governing electromagnetic wave propagation have undergone profound changes. Whereas in the 1980's there was confusion about how to choose the finite elements to obtain a robust solver, it is now clear not only how to discretize the Maxwell system using edge elements, but also how to analyze the resulting method. Spin-offs from this analysis include the Finite Element Exterior Calculus and the realization that the discrete de Rham diagram is a useful tool to guarantee conservation of charge. More recently several open source implementations of common finite element families for the Maxwell system have appeared making edge elements much more accessible. I shall give a historical survey of the finite element method for Maxwell's equations and comment on the main open problems still facing users. I shall also present current uses of finite elements in computational photonics, particularly light transport in nanoscale diffraction gratings where unusual surface phenomena have been observed. A particular application of this analysis is in the design of thin film solar cells.

Johannes Tausch, Southern Methodist University
Fast Galerkin BEM for Parabolic Moving Boundary Problems

Time dependence in boundary integral reformulations of parabolic PDEs is reflected in the fact that the layer potentials involve integrals over time in addition to integrals over the boundary surface.

This implies that in the numerical solution a time step involves the summation over space and the complete time history. Thus the naive approach has order N^2 M^2 complexity, where N is the number of unknowns in the spatial discretization and MM is the number of time steps. However, with a space-time version of the fast multipole method the the complexity can be reduced to nearly NM.

The talk will focus on the application of the methodology to problems with time dependent geometries. Two different situations will be considered: In the first the boundary at time tt is a differomorpic image of a fixed reference geometry. Here the discretization consists of simple space-time tensor product finite element spaces. Since this setting does not allow for topology changes in time, we will also consider a full space time discretization to handle more general situations.

The eventual goal of this work is to solve free surface problems, such as the Stefan problem, which describes the evolution of a phase change interface. The talk will conclude with a new approach to solve one dimensional problems with shape optimization techniques.

Javier Diez, UNICEN
Wettability Dynamics of Thin Liquid Films on Solid Substrates

We report experimental, theoretical and numerical results on the hydrodynamic instabilities involved in the breakup of patterned thin liquid films on solid substrates, such as single filaments and bidimensional grids formed by the intersection of them. We focus on the effects of contact angle hysteresis on the flow dynamics and the type of drops patterns that are formed by these unstable flows. In particular, we describe in detail the motion of the contact line in the region nearby the end of a retracting liquid filament. Since the flow develops under conditions of partial wettability, the filament end recedes and forms a bulge (head) which lately develops a thinning neck behind it. This neck finally breaks up giving place to the generation of a separated drop, while the rest of the filament repeats the cycle. The contact angle hysteresis plays a fundamental role in this type of processes since, not only defines the dynamics, but also the geometrical features and size of the resulting drops as well as the distance between them. We use a combined model for the relation between the contact line velocity and contact angle, which takes into account the hydrodynamic viscous dissipation and the molecular kinetics at the liquid-solid interface. This relationship allows to reproduce the experimental results by means of numerical simulations of the full Navier-Stokes equation. We also develop a simple hydrodynamic model to account for the observed dynamics as well as to determine the number of drops resulting from the breakup of a filament of given length. These studies are applied to the nanometric scale in experiments with metalic filaments on a silicon oxide substrate, which are melted by short laser pulses. This is work joint with Alejandro G. González, Pablo D. Ravazzoli, Ingrith Cuellar.

Mike O'Neil, New York University
Fast Direct High-Order Methods for Electromagnetic Scattering from Bodies of Revolution

Full three-dimensional high-order codes for electromagnetic scattering from complex geometries are currently out of reach in most cases. However, in particular (non-trivial) geometries very efficient algorithms can be constructed. In this talk, we will detail a separation of variables method along bodies of revolution that results in a sequence of boundary integral equations along only a one-dimensional curve. The overall scheme, accelerated by the FFT, results in an overall O(N^3/2) direct method for solving scattering problems from perfect electric conducting and dielectric bodies of revolution. Several numerical examples will be included, as well as an overview of the evaluation of “modal Green’s functions” and the various numerical tools needed in the construction of this solver.

Michael Shelley, New York University / Flatiron Institute
Active Mechanics in the Cell

Many fundamental phenomena in eukaryotic cells -- nuclear migration, spindle positioning, chromosome segregation -- involve the interaction of (often transitory) mechanical structures with boundaries and fluids. I will discuss the recent interactions of mathematical modeling and simulation with experimental measurements of active biomechanical processes within the cell. This includes studying how the spindle finds its proper place prior to cell division, and trying to explain coherent fluctuations within the nucleus.

Mette Olufsen, North Carolina State University
Alteration in Cardiovascular Regulation in Disease - A Model Based Analysis

The state of the cardiovascular system can be assessed from time-series signals including heart rate and blood pressure. Characteristics of these signals are used to determine pathophysiology. Experienced clinicians can visually inspect signals and with high level of certainty diagnose patient outcome, yet they may do so without knowing the cascade of events triggering the outcome. This talk will address how mathematical modeling can be adapted to predict patient specific behavior, and how the optimized system equations can be used to predict emergent behavior. Focus will be on studying dynamics observed in patients diagnosed with postural orthostatic tachycardia (increased heart rate brought on by change in posture, head-up tilt) and/or vasovagal syncope (fainting in response to overaction to some trigger) observed in patients diagnosed with functional somatic syndrome. Data analyzed here are from girls experiencing side effects (dizziness and fatigue) after vaccination against the human papilloma virus (HPV). It has been hypothesized that this condition can be ex­­plained by the presence of agonistic autoantibodies against beta2 adrenoceptors and M2 mus­carinic re­ceptors inducing slow heart rate and blood pressure oscillations emerging during head-up tilt. In this talk we will use modeling to stimulate this condition as well as describe how to induce syncope (fainting) by switching the stable negative feedback to a positive feedback triggered by reduced filling of the heart.