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ACMS Abstracts: Spring 2013

Arnd Scheel (Minnesota)

Pattern selection in the wake of fronts

Motivated by precipitation patterns such as Liesegang rings, we discuss patterns formed in the wake of moving fronts. It turns out that strikingly simple kinetic mechanisms can lead to a plethora of patterns, from simple bands and stripes to spirals and helices. We'll discuss a number of mathematical problems that arise when one tries to predict which wavenumbers and what type of pattern would arise in the wake of fronts. Applications include elementary chemistry, patterns in bacteria colonies, and self-assembly of nano-scale textures.

Kourosh Shoele (RE Vision Consulting)

Fluid interactions with structures, from fish fins to hydrokinetic devices

Studying the interaction between fluid and structure is a fundamental step in understanding the underpinnings of many engineering and physical phenomena, from energy harvesting to biolocomotion of insects, birds and fishes. The complex nature of these interactions makes the design of computational, experimental, and analytical techniques for modeling such problems challenging. Here I discuss new procedures, both in potential flow and viscous flow, for studying the interactions of a flexible structure with a flow. In particular, I will focus on two particular phenomena, the flow interaction with skeleton-reinforced fish fins and the extraction of ocean energy through oscillating systems.

Andrej Zlatoš (UW)

Reactive processes in inhomogeneous media

We study fine details of spreading of reactive processes (e.g., combustion) in multi-dimensional inhomogeneous media. One often observes a transition from one stable state (e.g., unburned fuel) to another (e.g., burned fuel) to happen on short spatial as well as temporal scales. We demonstrate that this phenomenon also occurs in the simplest model for reactive processes, reaction-diffusion equations with ignition-type reaction functions, under very general assumptions.

Specifically, we show that in up to three spatial dimensions, the width (both in space and time) of the zone where reaction occurs stays uniformly bounded in time for some fairly general classes of initial data, and this bound even becomes independent of the initial datum after an initial time interval. Such results have recently been obtained in one spatial dimension but were unknown in higher dimensions. As one indication of the added difficulties, we also show that three dimensions is indeed the borderline case, and the result is false for general inhomogeneous media in four and more dimensions.

Cary Forest (UW)

Stirring Magnetized Plasma

Recently, a new concept for stirring a hot (T=100000 C), unmagnetized plasma has been demonstrated, making it possible to study the Dynamos and the Magnetorotational Instability (MRI) for the first time in a laboratory plasma. In the Plasma Couette Experiment, plasma is confined by a cylindrical, axisymmetric multicusp magnetic field. The field vanishes rapidly away from the boundaries, leaving a large, unmagnetized plasma in the bulk. Azimuthal flows (6 km/s) are driven with JxB torque using biased, heated filaments located at a single toroidal position at the boundary. Measurements show that momentum couples viscously from the magnetized edge to the unmagnetized core, and that the flow is axisymmetric. In order for the toroidal velocity to couple inward, the collisional ion viscosity must overcome the drag due to ion-neutral collisions. Flow speeds can be adjusted by simply increasing the bias voltage of the electrodes. When flow is driven only from the outer boundary, the plasma rotates as a solid-body and the MRI is stable. However, the addition of electrodes at the inner boundary enables us to drive the sheared flow necessary for destabilizing the MRI. This experiment has already achieved magnetic Reynolds numbers of Rm~50 and magnetic Prandtl numbers of Pm ~0.3–6, which are approaching regimes shown to excite the MRI in local linear analysis and global Hall-MHD numerical simulations. Experiments characterizing the MRI will compare the onset threshold to theoretical and numerical predictions, look for altered velocity profiles due to momentum transport during nonlinear saturation, and identify two fluid effects expected to arise from the Hall term and plasma-neutral interactions (important in protoplanetary accretion disks).

While these experiments have been carried out, have also been constructing a much larger (3 m diameter) dynamo experiment based on a Von Kármán like flow in a sphere that should be capable of generating plasmas with Rm~1500. Several different scenarios have been investigated numerically to show that the experiment has a very good chance of being a dynamo, including steady von Kármán flow that generates an equatorial dipole, and a new Galloway-Proctor like flow that uses time dependent boundary conditions to generate smooth, but chaotic flows that give a fast dynamo. The experiment is now constructed and will be described in this talk. Initial results on plasma formation will be presented.

Matthew Johnston (UW)

Characterization of Steady States of Mass Action Systems by Correspondence to Weakly Reversible Networks

An elementary reaction is given by a set of reactants which reacts together at some kinetic rate to form some set of (stoichiometrically distinct) products (for example, [math]\displaystyle{ 2H + O_2 \to 2H_2O }[/math]). Under a few simplifying assumption, such as continuous-stirring and mass action kinetics, we may model the dynamics of a network of such reactions as a system of autonomous, polynomial (nonlinear), ordinary differential equations.

In general, determining properties of the steady states of such systems is difficult due to the nonlinearity of the equations. A surprising result of chemical reaction network theory is that, in many cases, properties on the reaction graph alone suffice to characterize the steady state set. That is to say, we can write the network of interactions down and, without any consideration of governing dynamics at all, we may determine such properties as the number and stability of steady states, their location in state space, and their dependence on the rate constants.

In this talk, I will introduce a new method for characterizing the steady states of mass action systems for which this graph theoretical method is not directly applicable by corresponding the underlying networks to ones with "better" graphical structure. The talk should be accessible to anybody with a background in basic ordinary differential equations. No prior knowledge of chemical reaction networks or graph theory is assumed.

Vu Hoang (Karlsruhe)

Nonexistence of bound states for a periodic waveguide problem

I consider a two-dimensional periodic medium, which is perturbed by a line defect (a waveguide). In this situation, one expects that the associated partially periodic self-adjoint operator has no bound states on the whole space. This fact is surprisingly hard to prove and the the result was achieved only recently.

In this talk, I give an introduction to the analysis of periodic elliptic operators and discuss the nonexistence of bound states based on a classical argument by L. Thomas. Finally, I give a few details of our proof. (Joint work with Maria Radosz)

Nigel Boston (UW)

Invariant-Based Face Recognition

After a brief review of recent striking applications of algebra to engineering and computer science, the currently significant problem of face recognition is addressed. We introduce a new approach to obtaining invariants of Lie groups adapted to this problem and describe its success in implementations.

Shreyas Mandre (Brown)

Linear stability of time-dependent flows: dissolution-driven convection

We present a general theory for the linear stability of non-autonomous systems and present an example of its use in fluid dynamics. The technique essentially identifies the spectral radius of the propagator of the linear operator as the appropriate measure for the amplification of initial perturbations by the linearized dynamics. The technique unifies the classical modal stability theory using eigenvalues, the non-modal approaches using optimal growth of energy and the frozen coefficient analysis. In particular, we show that by choosing to use the energy which gives the least possible optimum growth, the results of the classical modal theory are recovered.

The dissolution-driven convection in porous media as an example of our general technique. Such a flow is motivated by processes involved in the geological carbon dioxide sequestration. The solute is introduced at some horizontal level in a porous medium filled with a liquid, and it gradually continues to diffuse into the underlying liquid. The dissolved solute makes the liquid heavier, thus making the system susceptible to gravitational instability and convection. In this case as well, the background state is unsteady corresponding to the diffusion of the solute.

We settle the long-standing debate about the interpretation and validity of frozen coefficient analysis for convection in porous media, and demonstrate the existence of a threshold time for convection to commence. We also show that asymptotically the amplification rate increases as exp(C t^{1/2}), for a dimensionless constant C and the most amplified wavenumber decays as t^{-1/2}.

Ben Recht (UW)

A convex perspective on spectral methods in signal processing

Spectral methods that leverage the singular value decomposition to reject noise and identify models are ubiquitous in signal processing. Despite their widespread adoption, little is understood about their finite-sample convergence properties in the presence of noise. In this talk, I will provide a general framework, called atomic norm denoising, for understanding these spectral methods in the context of convex optimization. The resulting optimization problems have generic, mean-squared-error guarantees and reduce to familiar soft-thresholding algorithms in the context of sparse approximation. I will specialize these techniques to provide a convex approach to spectrum estimation, estimating the frequencies and phases of a mixture of complex exponentials from noisy or missing data. These results serve as an extension of compressed sensing to the case when frequencies are not confined to lie on a discrete grid but can assume any value in a continuous interval.

Silas Alben (Michigan)

Optimizing snake locomotion in the plane

We develop a numerical scheme to determine which planar snake motions are optimal for locomotory efficiency, across a wide range of frictional parameter space. For a large coefficient of transverse friction, we show that retrograde traveling waves are optimal. We give an asymptotic analysis showing that the optimal wave amplitude decays as the -1/4 power of the coefficient of transverse friction. This result agrees well with the numerical optima. At the other extreme, zero coefficient of transverse friction, we propose a triangular direct wave which is optimal. Between these two extremes, a variety of complex, locally optimal, motions are found. Some of these can be classified as standing waves (or ratcheting motions).