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ACMS Abstracts: Spring 2014

Adrianna Gillman (Dartmouth)

Fast direct solvers for linear partial differential equations

The cost of solving a large linear system often determines what can and cannot be modeled computationally in many areas of science and engineering. Unlike Gaussian elimination which scales cubically with the respect to the number of unknowns, fast direct solvers construct an inverse of a linear in system with a cost that scales linearly or nearly linearly. The fast direct solvers presented in this talk are designed for the linear systems arising from the discretization of linear partial differential equations. These methods are more robust, versatile and stable than iterative schemes. Since an inverse is computed, additional right-hand sides can be processed rapidly. The talk will give the audience a brief introduction to the core ideas, an overview of recent advancements, and it will conclude with a sampling of challenging application examples including the scattering of waves.

Yaniv Plan (Michigan)

Low-dimensionality in mathematical signal processing

Natural images tend to be compressible, i.e., the amount of information needed to encode an image is small. This conciseness of information -- in other words, low dimensionality of the signal -- is found throughout a plethora of applications ranging from MRI to quantum state tomography. It is natural to ask: can the number of measurements needed to determine a signal be comparable with the information content? We explore this question under modern models of low-dimensionality and measurement acquisition.

Harvey Segur (Colorado)

The nonlinear Schrödinger equation, dissipation and ocean swell

The focus of this talk is less about how to solve a particular mathematical model, and more about how to find the right model of a physical problem.

The nonlinear Schrödinger (NLS) equation was discovered as an approximate model of wave propagation in several branches of physics in the 1960s. It has become one of the most studied models in mathematical physics, because of its interesting mathematical structure and because of its wide applicability – it arises naturally as an approximate model of surface water waves, nonlinear optics, Bose-Einstein condensates and plasma physics.

In every physical application, the derivation of NLS requires that one neglect the (small) dissipation that exists in the physical problem. But our studies of water waves (including freely propagating ocean waves, called “ocean swell”) have shown that even though dissipation is small, neglecting it can give qualitatively incorrect results. This talk describes an ongoing quest to find an appropriate generalization of NLS that correctly predicts experimental data for ocean swell. As will be shown, adding a dissipative term to the usual NLS model gives correct predictions in some situations. In other situations, both NLS and dissipative NLS give incorrect predictions, and the “right model” is still to be found.

This is joint work with Diane Henderson, at Penn State.

Sangtae Kim (Purdue)

The Faxén Laws of Stokes flow and their connection to singularity solutions

Elena D'Onghia (UW)

The origin of spiral arms in galactic disks

The precise nature of spiral structure in galaxies remains uncertain. Using high-resolution N-body simulations, I follow the motions of stars under the influence of gravity, and show that mass concentrations with properties similar to those of giant molecular clouds or clumps of gas in the galactic disk can induce the development of spiral arms through a process termed "swing amplification". However, unlike in earlier work, I will demonstrate that the eventual response of the disk is highly non-linear, significantly modifying the formation and longevity of the resulting patterns. I will discuss how these findings affect phenomena occurring in the stellar disk, like the migration of the Sun from its birth place.

Michael Shelley (Applied Math Lab, Courant Institute, NYU)

Mathematical models of soft active materials

Soft materials that have an "active" microstructure are important examples of so-called active matter. Examples include suspensions of motile microorganisms or particles, "active gels" made up of actin and myosin, and suspensions of microtubules cross-linked by motile motor-proteins. These nonequilibrium materials can have unique mechanical properties and organization, show spontaneous activity-driven flows, and are part of self-assembled structures such as the cellular cortex and mitotic spindle. I will discuss the nature and modeling of these materials, focusing on fluids driven by "active stresses" generated by swimming, motor-protein activity, and surface tension gradients. Amusingly, the latter reveals a new class of fluid flow singularities and an unexpected connection to the Keller-Segel equation.

Paul Hand (MIT)

Evaluating signal recovery algorithms with semirandom models

The planted clique and sparse principal component analysis problems involve identifying a specific ordered structure within a noisy environment. There are many algorithms for such tasks, and it is important to have a theoretical understanding of which algorithms are better for "typical" problem instances. The simplest environment for evaluating these algorithms is a planted-random model, where the signal is buried in some uniform noise. This type of model is often not typical enough, as it affords comparable performance for some robust and non-robust algorithms. Thus, there is a need for more sophisticated models. Semirandom models are like planted-random models, but they allow adversarial alteration. The semirandom model prevents methods from exploiting distribution-specific properties that are likely not true in typical problem instances. Thus, they provide a reasonable context in which to evaluate algorithm performance under a robustness requirement. In this talk, we will present existing results for planted clique with planted random and semirandom models. We will also discuss the potential for semirandom models in two other problems: sparse principal component analysis and the problem of finding the sparsest element in a subspace.

Scott Hottovy (UW)

Modeling with stochastic differential equations: A noisy circuit and thresholds for rainfall

In this talk I will discuss modeling physical systems using stochastic differential equations (SDE) with two specific examples: a noisy electrical circuit and the onset of convection in clouds. SDE are used to model many systems governed by deterministic dynamics with intrinsic noise. For some models, when the noise depends on the state of the system, the SDE depends on the construction of the stochastic integral (Itô, Stratonovich). For example, in a population model, the choice of stochastic integral will lead to growth in the Stratonovich construction but extinction in the Itô choice. I will discuss an interesting experiment of an electrical circuit that shifts the SDE model from obeying Stratonovich calculus to obeying Itô calculus.

In the second part of this talk, I will describe ongoing work with Sam Stechmann on modeling moisture in a one column cloud. Interesting statistics can be solved for exactly by using a simple SDE model, Brownian motion with drift, where the dynamics change when a threshold is reached. I will compare and contrast four different models of the transition to rainfall using random and deterministic thresholds and show that a simplified model can be used as an approximation in some regimes.

Mike Steel (University of Canterbury)

Tractable models for some discrete random processes arising in evolutionary biology

Underlying the study of evolutionary biology is a foundation that draws upon multiple branches of mathematics. In this talk, I will provide a brief overview of certain problems that can be modeled by random processes involving finite graphs. These include evolutionary tree reconstruction, speciation-extinction modeling, ancestral state reconstruction, and the emergence of autocatalytic biochemistry in early life. Some recent new results and open problems will also be described.

Laura McLay (UW)

Covering optimization models for emergency response, security, and resilience

Covering models are used in a broad spectrum of applications. This talk will first overview various covering models and then focus on covering models and algorithms for emergency medical services. These emergency medical service models reflect multiple partial coverage. Lastly, I will also talk about covering formulations for security and disaster applications.

Shengqian "Chessy" Chen (UW)

Dynamics of internal waves and stratified flows in shallow water

Internal waves are gravity waves in density-stratified fluids. They are ubiquitous in the ocean and atmosphere, and often have large amplitudes. The stratified flows can be governed by Euler equations, whose solutions, however, are amenable to numerical method only for the most part. Hence, in order to gain insight into wave dynamics, it is important to study asymptotic models built on the long wave (shallow water) assumption.

In this talk, the asymptotic models, both strongly and weakly nonlinear ones are discussed to predict solitary wave occurrence from an internal "dam-break" setup. Although the initial setup does not necessarily satisfy the long-wave assumption, the strongly nonlinear models are capable to capture solitons from the Euler system through numerical computation. Weakly nonlinear models have an additional small-amplitude assumption. In this class, some systems are solvable by the Inverse Scattering Transform (IST), thus to provide analytical prediction for the "dam-break" problem. However, these models may have the drawback of being ill-posed, or developing highly oscillatory wave trains in the solution such as for the Korteweg de Vries equation. To treat the ill-posedness, a new model is proposed to regularize the two-layer Kaup model. Numerical evidence for the new model shows that the regularization has little influence on the prediction from IST.

Marie-Pascale Lelong (NorthWest Research Associates)

Two scenarios involving inertia-gravity waves and eddies on scales of 100 m to 10 km in the ocean

The unifying theme of this two-part talk is the role of inertia-gravity waves (IGWs) and eddy motions in the dynamics of rotating, stratified flows, with an emphasis on oceanic applications on scales of 100 m to 10 km.

In the first part, I will present a mathematical solution based on multiple time-scale expansions demonstrating the generation of inertia-gravity waves following a resonant interaction between a tidal current and a field of geostrophic eddies. The generated IGWs exhibit the frequency of the tide and the vertical scale of the eddies while their horizontal wavelength is determined by the sum/difference of tidal and eddy horizontal scales. Resonance is strongest when the horizontal scale of the tidal current is within a factor of 2 of the eddy scale. Numerical simulations of broadband eddy fields forced by tidal currents confirm the occurrence of this resonance.

In the second part, I will address the lateral dispersion of a passive tracer in oceanic ambient conditions dominated by (i) small-scale (submesoscale) eddies, (ii) spectra of weakly nonlinear waves and (iii) a combination of both. Surprisingly, we find that significant dispersion, of O(1 m^2/2), is possible even in cases such as (ii) through a mechanism akin to Stokes drift.

Jon Wilkening (UC Berkeley)

Nearly time-periodic water waves

We compute new families of time-periodic and quasi-periodic solutions of the free-surface Euler equations involving extreme standing waves and collisions of traveling waves of various types. A Floquet analysis shows that many of the new solutions are linearly stable to harmonic perturbations. Evolving such perturbations (nonlinearly) over tens of thousands of cycles suggests that the solutions remain nearly time-periodic forever. We also discuss resonance and re-visit a long-standing conjecture of Penney and Price that the standing water wave of greatest height should form wave crests with sharp, 90 degree interior corner angles. We conclude with a geophysical application in which nearly-coherent standing waves at the ocean surface can lead to rapidly-moving pressure zones at the sea floor. These pressure zones can generate resonant elastic waves believed to be partially responsible for microseisms, the background noise observed in earthquake seismographs.

Christel Hohenegger (Utah)

Immersed particle dynamics in fluctuating fluids with memory

Multibead passive microrheology aims at characterizing fluid properties via statistically measurable quantities (e.g. mean-square displacement, auto-correlation). Understanding how these material properties relate to biological quantities (e.g. exit time, first passage time through a layer) is of crucial importance. To correctly model the correlations due to the fluid's memory, it is necessary to include a thermally fluctuating force in the Stokes equations. We present such a model for an immersed particle passively advected by a fluctuating Maxwellian fluid. We then apply our model to find the signal due to the fluid's memory in the statistics of the particles velocity autocorrelation. We compare our results on simulated data both with standard multibead microrheology techniques and a Langevin description of the particle's motion. Our approach can be applied to a Stokes fluid with memory created by a large suspension of active swimmers or to the diffusion of a particle in a crowded environment.

James Meiss (Colorado)

Finite-time transport between distinct flow regions is of great relevance to many scientific applications, yet quantitative studies remain scarce to date. The primary obstacle is computing the evolution of material volumes, which is often infeasible due to extreme interfacial stretching. We present a framework for describing and computing finite-time transport in n-dimensional (chaotic) volume-preserving flows that relies on the reduced dynamics of an (n-2)-dimensional "minimal set" of fundamental trajectories. This approach has essential advantages over existing methods: the regions between which transport is investigated can be arbitrarily specified; no knowledge of the flow outside the finite transport interval is needed; and computational effort is substantially reduced. We demonstrate our framework in 2D for an industrial mixing device.