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ACMS Abstracts: Fall 2013

Rob Sturman (Leeds)

Lecture 1: The ergodic hierarchy & uniform hyperbolicity

Ergodic theory provides a hierarchy of behaviours of increasing complexity, essentially covering dynamics from indecomposable, through mixing, to apparently random. Demonstrating that a (real) system possesses any of these properties is typically difficult, but one important class of system - uniformly hyperbolic systems - make the ergodic hierarchy immediately accessible. Uniform hyperbolicity is a strong condition, and so we also describe its weaker counterpart, non-uniform hyperbolicity. Our chief example in this lecture is the Arnold Cat Map, an example of a hyperbolic toral automorphism.

Lecture 2: Non-uniform hyperbolicity & Pesin theory

The connection between non-uniform hyperbolicity and the ergodic hierarchy is more difficult, but is made possible by the work of Yakov Pesin in 1977, and extensions due to Katok & Strelcyn. Here we illustrate the theory with a linked twist map, a paradigmatic example of non-uniformly hyperbolic system.

Seminar: Rates of mixing in models of fluid flow

The exponential complexity of chaotic advection might be reasonably assumed to produce exponential rates of mixing. However, experiments suggest that in practice, boundaries slow mixing rates down. We give rigorous results from smooth ergodic theory which establishes polynomial mixing rates for linked twist maps, a class of simple models of bounded flows.

Reed Ogrosky (UW)

Long-wave modeling of a viscous liquid film inside a vertical tube

Viscous film flows that coat the interior of a cylinder arise naturally in industrial and biological settings and have a free surface that is unstable to long-wave disturbances. I will discuss a series of recently derived long-wave asymptotic models of these flows, and an accompanying set of experiments, in the (i) absence and (ii) presence of a steady, upwards pressure-driven airflow. In the absence of airflow, the long-wave models show excellent agreement with experiments. In the presence of airflow, the agreement is primarily qualitative in nature, though modifications to the modeling of the free surface stress improve the agreement. In both cases, the long-wave models capture qualitative features of the flow seen in the experiments that are not captured by their popular thin-film model counterparts.

Amit Einav (Cambridge)

One of the most influential equations in the kinetic theory of gases is the so-called Boltzmann equation, describing the time evolution of the probability density of a particle in dilute gas. While widely used, and intuitive, the Boltzmann equation poses an interesting conceptual problem: as an equation, it is irreversible in time. However, one assumes that such equation arises from reversible Newtonian laws, which seems like a contradiction. This raises the following question: can one achieve an irreversible system from a reversible one, in macroscopic time scales? If so, how can it happen? In his 1956 paper, Marc Kac presented an attempt to solve this problem in a particular settings of the spatially homogeneous Boltzmann equation. Kac considered a model of N indistinguishable particles, with one dimensional velocities, that undergo a random binary collision process. Under the property of chaoticity, defined by Kac, he managed to show that when one takes the number of particles to infinity, the limit of the first marginal of the N-particle distribution function satisfies a caricature of the Boltzmann equation. Besides giving an intuitive explanation to how the Boltzmann equation can arise as a mean field limit, Kac hoped to use his model to study the properties of the Boltzmann equation - specifically, the rate of convergence to equilibrium.

We will start our talk by describing the Boltzmann equation and explaining the model Kac proposed. We will then discuss the so-called spectral gap problem, posed by Kac, which attempted, unsuccessfully, to find an exponential rate of conversion to equilibrium of the mean field limit using the natural normed structure. At this point we'll discuss a different, less linear, approach to the problem - the so-called entropy method, and show that this too is unsuccessful, but has more potential to work with more delicate investigation. In our talk we will also mention McKean model, which is an extension of Kac's model to the case where the velocities are not one dimensional. Time permitting we will discuss more relevant concepts such as entropic chaos, and the effects of moments of the mean field limit on chaoticity and entropic chaos.

Shamgar Gurevich (UW)

The incidence and cross methods for efficient radar detection

I will explain a model of radar detection and its digital form. The latter enables us to introduce techniques from Applied Algebra (construction of specific vectors using commutative groups of operators, and generalizations of Fast Fourier Transform techniques) to suggest new efficient algorithms for radar detection. I will explain these methods, and I will demonstrate an application to the Inhomogeneous Radar Scene Problem, formulated in our interaction with engineers from General Motors (GM), who want to develop sensitive radar devices for cars.

This is a joint work with Alexander Fish (Mathematics, Sydney) and is part from a joint project with Igal Bilik (GM), Akbar Sayeed (ECE, Madison), Oded Schwartz (EECS, Berkeley), and Kobi Scheim (GM).

Michael Cardiff (UW)

The hydrogeologic inverse problem: characterizing aquifer heterogeneity with multi-frequency stimulations

The parameters that control the flow of water and transport of contaminants underground in aquifers are extraordinarily heterogeneous, often varying by 2-3 orders of magnitude even in aquifer deposits that are considered relatively “homogeneous”. Estimation of the spatial distribution of these parameters is often the key prerequisite for performing accurate predictive modeling of hydrogeologic systems. In this talk, I will introduce the hydrogeologic inverse problem and discuss various approaches that have been suggested in order to gain information about the heterogeneity in the subsurface. I will then discuss a modern strategy, Hydraulic Tomography (HT), which helps to constrain aquifer heterogeneity through cross-well pressure interference tests. In particular, I will first discuss the mathematical approach, which relies on large-scale forward modeling and analysis of sensitivities using an adjoint state approach, coupled to a geostatistical inversion framework. I will then discuss practical issues associated with actual application of tomographic methodologies in the field, and present a modified strategy, Oscillatory Hydraulic Tomography, which has practical, analytical, and computational benefits beyond traditional HT.

Ronen Avni (UW)

Mathematical model for cell motility driven by active gel

Cell crawling is a highly complex integrated process involving three distinct activities: protrusion adhesion and contraction, and also three players: the plasma membrane (car body), the actin network engine) and the adhesion points (clutch). The actin network consists of actin polymers and many other types of molecules, e.g. molecular motors, which dynamically attach to and detach from the network, making it a biological gel. Furthermore, energy is consumed in the form of ATP due to both the activity of molecular motors and the polymerization at the filament tips; thus the system is far from thermodynamic equilibrium. These characteristics make the above system unique and responsible for a wide range of phenomena (different force-velocity relationships) and behaviors (contraction, elongation, rotation, formation of dynamic structures)

Like the story on the blind men and the elephant, previous works considered only parts of the complex process, neglecting other sub-processes, or using unrealistic assumptions. Our goal was to derive a mathematical model for the whole system that can predict the rich variety of behaviors. For this purpose we had to identify the major players and integrate previous works into a one coherent mathematical model with (almost) no arbitrary constraints, adding our own mathematical description where needed. The model we derived consists of several temporal and spatial scales, molecular processes e.g. capping / branching to macro processes; furthermore, we used a hydrodynamic approach, hence could relate local dynamic events on the boundary to the bulk inside the domain. We focused on the processes near the leading edge that drive the system, i.e. the complexity comes in the b.c., and termed this filaments-membrane dynamics “the polymerization machinery”.

In my talk I will describe the mathematical model we derived and its relation to previous works. I will also describe a numerical simulation we derived for a free-surface flow of complex fluid in arbitrary geometries. Finally I will discuss open questions and opportunities in this line of research.

Shilpa Khatri (UNC)

Settling and rising in stratified fluids: fluid-structure interactions and multiphase flow

The fluid dynamics of particles settling and droplets rising is vital to understanding the effect of stratification in marine settings. Whether studying marine snow aggregates settling or oil jets rising, similar small scale dynamics are observed. I will present three different cases of rising and settling in stratified fluids: (1) porous particles settling, (2) oil droplets rising, and (3) dense-core miscible vortex rings settling. Each case has been studied using experiments and by developing models and finding solutions using numerical and analytical techniques. Challenges exist in developing methodology to handle the boundary conditions at fluid-structure and fluid-fluid interfaces. We will present model results which accurately capture experimental data and give insight into the importance of entrainment.

Junping Wang (National Science Foundation)

Weak Galerkin Finite Element Methods for PDEs

In this talk, the speaker will discuss the basic principles for a newly developed finite element technique, called weak Galerkin finite element methods, for partial differential equations. The discussion will be made for a model second order elliptic problem. The speaker will make a comparison between the weak Galerkin method with existing finite element methods. The talk will further be extended to other model PDEs such as the Stokes equation, the biharmonic and Maxwells equations. The talk should be accessible for graduate students with basic training in computational mathematics or PDEs.

Douglas Zhou (Shanghai Jiao Tong)

Causal and structural connectivity of neuronal networks

Current experimental techniques usually cannot probe the global interconnection pattern of a network. Thus, reconstructing or reverse-engineering the network topology of coupled nodes based upon observed data has become a very active research area. Most existing reconstruction methods are based on networks of oscillators with generally smooth dynamics. However, for nonlinear and non-smooth stochastic dynamical systems, e.g., neuronal networks, the reconstruction of the full topology remains a challenge. Here, we present a noninterventional reconstruction method, which is based on Granger causality theory, for the widely used conductance-based, integrate-and-fire type neuronal networks. For this system, we have established a direct connection between Granger causal connectivity and structural connectivity.

Igor Aronson (Argonne)

Living liquid crystals

Placement of swimming bacteria in lyotropic liquid crystal, a water dispersion of elongated aggregates of organic molecules, produces a new class of biomechanical systems – living liquid crystal (LLC). This new hybrid material synergistically combines the properties of both constituents: biological response to external stimuli and long-range order due to anisotropy. LLC displays a wide range of fundamentally new phenomena, from the emergence of self-organized textures caused by bacterial swimming to direct optical visualization of flagella rotation and the liquid crystal-controlled trajectories of bacterial motion. LLC sheds new light on self-organization in active biomechanical systems and can possibly lead to valuable biomedical applications.

Bo Li (UCSD)

Variational implicit solvation of biomolecules

The structure and dynamics of biomolecules such as DNA and proteins determine the functions of underlying biological systems. Modeling biomolecules is, however, extremely challenging due to their enormous complexity. Recent years have seen the initial success of variational implicit-solvent models (VISM) for biomolecules. Central in VISM is an effective free-energy functional of all possible solute-solvent interfaces, coupling together the solute surface energy, solute-solvent van der Waals interactions, and electrostatic contributions. Numerical relaxation by the level-set method of such a functional determines biomolecular equilibrium conformations and minimum free energies. Comparisons with experiments and molecular dynamics simulations demonstrate that the level-set VISM can capture the hydrophobic hydration, multiple dry and wet states, and many other important solvation properties. This talk begins with a description of the level-set VISM and continues to present new developments around the VISM. These include: (1) the coupling of solute molecular mechanical interactions in the VISM; (2) the effective dielectric boundary forces; and (3) the solvent fluid fluctuations. Mathematical theory and numerical methods are discussed, and applications are presented. This is joint work mainly with J. Andrew McCammon, Li-Tien Cheng, Joachim Dzubiella, Jianwei Che, Zhongming Wang, Shenggao Zhou, and Zuojun Guo.

Jingwei Hu (Texas)

An asymptotic-preserving scheme for the semiconductor Boltzmann equation toward the energy-transport limit

We design an asymptotic-preserving scheme for the semiconductor Boltzmann equation which leads to an energy-transport system for electron mass and internal energy as mean free path goes to zero. To overcome the stiffness induced by the convection terms, we adopt an even-odd decomposition to formulate the equation into a diffusive relaxation system. New difficulties arise in the two-scale stiff collision terms, whereas simple BGK penalization does not work well to drive the solution to the correct limit. We propose a variant by introducing a threshold on the stiffer collision term such that the evolution of the solution resembles a Hilbert expansion at the continuous level. Formal asymptotic analysis and numerical results confirm the efficiency and accuracy of the new scheme. This is joint work with Li Wang from UCLA.

Marko Budisic (UW)

Visualizing invariant structures of dynamical systems using the ergodic quotient

For practical purposes, ergodic measures can be understood as distributions of points in orbits of a dynamical system. Studying families of ergodic measures is an alternative approach to studying the phase portrait of a dynamical system, emphasizing an "average behavior" of a trajectory over "instantaneous behavior".

The ergodic measures can be computationally represented in the space of their Fourier-Stieltjes coefficients as a direct consequence of the Birkhoff Ergodic Theorem. This representation, termed "ergodic quotient", is an analogue of the phase portait. Its geometry is analyzed using Diffusion Maps, a numerical machine-learning algorithm. It identifies continuous segments in the ergodic quotient, which correspond to families of orbits belonging to macro-scale dynamical features, e.g., elliptic islands and chaotic seas. Several examples of ergodic quotients will be presented to demonstrate the approach. This project was the topic of speaker's dissertation at UC Santa Barbara.

Yoichiro Mori (Minnesota)

Modeling electrodiffusion and osmosis in physiological systems

Electrolyte and cell volume regulation is essential in physiological systems. After a brief introduction to cell volume control and electrophysiology, I will discuss the classical pump-leak model of electrolyte and cell volume control. I will then generalize this to a PDE model that allows for the modeling of tissue-level electrodiffusive and osmotic phenomena. This model will then be applied to the study of cortical spreading depression, a pathophysiological phenomenon in the brain that is thought to underlie migraine aura and other pathologies.

Cheng Koay (UW)

Symmetry and uniformity considerations in Magnetic Resonance Imaging acquisitions

The problem of finding the minimum energy configurations of n electrons on the surface of the unit sphere, which is known as the Thomson problem, is extremely challenging for large n. The minimum energy configurations from the Thomson problem have been found useful as a modeling tool in a variety of problems from modeling spherical viral structures to designing imaging acquisitions in Magnetic Resonance Imaging (MRI). The Thomson problem has also inspired the development of deterministic and efficient approaches capable of generating configurations that are in good approximation to the minimum energy configurations. In this talk, the focus will be on a variant of the Thomson problem, which is to find the minimum energy configurations that are endowed with antipodal symmetry. It will be argued that these new minimum energy and antipodally symmetric configurations are more appropriate for MR acquisition design due to the fact that the raw MR data (in spatial frequency domain) and its image data (in real spatial domain) are related through the Fourier relationship and the image data, to a good approximation, is real-valued. It has taken the speaker several years with several failed attempts to establish the above claim. This talk chronicles in part the speaker's journey, a few tools that were crafted along the way, and his final exit at the end of that proverbial tunnel.