The talk will summarize work on several different problems involving particles and bubbles. It will begin with a brief description of the physics underlying a numerical method for the simulation of laminar and turbulent fluid flows with particles. Applications to rotating spheres, sedimenting particles, model porous media and particles in turbulence will be described. The talk will then address some physical aspects of vapor bubbles generated by a pulse of laser or electrical energy, to finish with a gas-bubble-based "acoustic fish".

Dr. Bin Cheng, School of Mathematical and Statistical Sciences, Arizona State University

Time-averages of multiscale PDE systems and applications in geophysical fluid dynamics

Time-averages are common observables in analysis of experimental data and numerical simulations of physical systems. We describe a PDE-theoretical framework for studying time-averages of dynamical systems that evolve in both fast and slow scales. Patterns arise upon time-averaging, which in turn affects long term dynamics via nonlinear coupling. We apply this framework to geophysical fluid dynamics in spherical and bounded domains subject to strong Coriolis force and/or Lorentz force.

Conservative time-discretization for stiff Hamiltonian systems and molecular chain models

A variety of problems in modeling of large biomolecules and nonlinear optics lead to large, stiff, mildly nonlinear systems of ODEs that have Hamiltonian form.

This talk describes a discrete calculus approach to constructing unconditionally stable, time-reversal symmetric discrete gradient conservative schemes for such Hamiltonian systems (akin to the methods developed by Simo, Gonzales, et al), an iterative scheme for the solution of the resulting nonlinear systems which preserves unconditional stability and exact conservation of quadratic first integrals, and methods for increasing the order of accuracy. Some comparisons are made to the more familiar momentum conserving symplectic methods.

As an application, some models of pulse propagation along protein and DNA molecules and related numerical observations will be described, with some consequences for the search for continuum limit PDE approximations.

Prof. Elias Balaras, Department of Mechanical and Aerospace Engineering, George Washington University

Dissecting mechanical hemolysis using direct numerical simulations of whole blood

Mechanical hemolysis is a critical element in the design of cardiovascular devices where abnormal, disturbed flow patterns are often unavoidable. A characteristic example is that of ventricular assist devices (VAD), which are used to treat more than 50000 patients with ailing hearts in the US alone. Today, computational fluid dynamics (CFD) are increasingly used to optimize the hydrodynamic performance of biomedical devices, such as VAD’s, but improvements on blood damage and blood aggregation characteristics is hampered by the lack of predictive mechanical hemolysis and thrombosis models. Although in most of these devices the flow is highly three-dimensional and unsteady, currently available models for hemolysis are usually based steady shearing experiments and utilize global measures of the stress scalar magnitude and duration of exposure. More recent strain-based models are conceptually well suited for unsteady configurations, but still the instantaneous red blood cell (RBC) deformation estimates are as accurate as the steady experiments utilized. In this talk we will first present a brief survey of the existing models, which are based on either “lumped” descriptions of stress or analytical-numerical RBC descriptions relying on simple geometrical assumptions. We will also introduce a new approach, which is based on an existing coarse-grained particle dynamics method. We will then explore the rationale and RBC physics within each method through model “virtual”, numerical experiments. Finally, we apply all models to simulate the expected level of RBC damage using pathlines calculated for a realistic artificial heart valve. As we will show, our results shed light on the strengths and weaknesses of each approach and identify the key gaps that should be addressed in the development of new models.

Factorization of Non-Symmetric Operators and Exponential H-Theorem

We present an abstract method for deriving decay estimates on the resolvents of non-symmetric operators in Banach spaces in terms of estimates in another – typically smaller – reference Hilbert space. We then apply this approach to several equations of statistical physics, such as the Fokker-Planck equation and the linear and non-linear Boltzmann equation. The main outcome of the method is the first constructive proof of exponential decay towards global equilibrium for the fully nonlinear Boltzmann equation for hard spheres, conditionally to some smoothness assumptions.

This is a joint work with Stephane Mischler and Clement Mouhot.

Prof Andre Tits, Electrical and Computer Engineering and the Institute for Systems Research, University of Maryland

Constraint Reduction in Interior-Point Methods for Linear and Convex Optimization

Constraint reduction is a technique by which each search direction is computed based only on a small subset of the inequality constraints (when the problem is expressed in standard dual form), containing those deemed most likely to be active at the solution. A dramatic reduction in computing time may result for severely imbalanced problems.

In this talk, we survey developments made at UMPC over the past few years, in a group led by Dianne O'Leary and the presenter. The power of constraint reduction is demonstrated on classes of randomly generated problems and on real-world applications. Numerical comparison with both simplex and "unreduced" interior point is reported.

Note: This talk has a significant overlap with a talk given by the author in the UMCP NA Seminar Series in April 2012.

One main problem in data processing is the reconstruction of missing data. In the situation of image data, this task is typically termed image inpainting. Recently, inspiring algorithms using sparse approximations and _{1} minimization have been developed and have, for instance, been applied to seismic images. The main idea is to carefully select a representation system which sparsely approximates the governing features of the original image -- curvilinear structures in case of seismic data. The algorithm then computes an image, which coincides with the known part of the corrupted image, by minimizing the _{1} norm of the representation coefficients.

In this talk, we will develop a mathematical framework to analyze why these algorithms succeed and how accurate inpainting can be achieved. We will first present a general theoretical approach. Then we will focus on the situation of images governed by curvilinear structures, in which case we analyze both wavelets as well as shearlets as the chosen representation system. Using the previously developed general theory and methods from microlocal analysis, under certain conditions on the size of the missing parts we will prove that such images can be arbitrarily well reconstructed.

This is joint work with Emily King and Xiaosheng Zhuang.

Many integro-differential equations are used to describe neuronal networks or neural assemblies. Among them, the Wilson-Cowan equations are the most wellknown and describe spiking rates in different locations. Another classical model is the integrate-and-fire equation that describes neurons through their voltage using a particular type of Fokker-Planck equations. It has also been proposed to describe directly the spike time distribution which seems to encode more directly the neuronal information. This leads to a structured population equation that describes at time $t$ the probability to find a neuron with time s elapsed since its last discharge.

We will compare these models and perform some mathematical analysis. A striking observation is that solutions to the I&F can blow-up in finite time, a form of synchronization. We can also show that for small or large connectivity the 'elapsed time model' leads to desynchronization. For intermediate regimes, sustained periodic activity occurs which profile is compatible with observations. A common tool is the use of the relative entropy method.

Thursday
November 1 3:30 PM
MATH 3206 (note time + location)

Two-Phase Flow in Porous Media: Modeling, Travelling Waves and Stability

I will discuss two models of two-phase ﬂuid ﬂow in which undercompressive shock waves have been dis-covered recently. In the ﬁrst part of talk, the focus is on two-phase ﬂow in porous media. Plane waves are modeled by the one-dimensional Buckley-Leverett equation, a scalar conservation law. The Gray-Hassanizadeh model for rate-dependent capillary pressure adds dissipation and a BBM-type dis-persion, giving rise to undercompressive waves. Two-phase ﬂow in porous media is notoriously subject to ﬁngering instabilities, related to the classic Saffman-Taylor instability. However, a two dimensional linear stability analysis of sharp planar interfaces reveals a criterion predicting that weak Lax shocks may be stable or unstable to long-wave two-dimensional perturbations. This surprising result depends on the hyperbolic-elliptic nature of the system of linearized equations. Numerical simulations of the full nonlinear system of equations, including dissipation and dispersion, verify the stability predictions at the hyperbolic level. In the second part of the talk, I describe a phase ﬁeld model of capillary effects in a thin tube, in which a resident ﬂuid is displaced by injected air. PDE simulations reveal the appearance of a combination of rarefaction wave together with an undercompressive shock that terminates at the spherical cap tip of the injected air. The shock can be understood through a singular dynamical system whose trajectories yield travelling wave solutions.

Dr. Amit Einav, Department of Pure Mathematics and Mathematical Statistics, University of Cambridge

Entropy, Entropy Production and trend to equilibrium in Kac's Model of any Dimension

In 1956 Marc Kac introduced a binary stochastic N-particle model from which, under suitable condition on the initial datum (what we now call 'Chaoticity') a caricature of the famous Boltzmann equation, in its spatially homogeneous form, arose as a mean field limit. The ergodicity of the evolution equation resulted in convergence to equilibrium as time goes to infinity, for any N. Kac expressed hopes that investigation of the rate of convergence can be expressed independently in N and result in an exponential trend to equilibrium for his caricature of Boltzmann equation. Later on, in 1967, McKean extended Kac's model to a more realistic d-dimensional one from which the actual Boltzmann equation arose, extending Kac's results and hopes to the real case.

Kac's program reached its conclusion in the 2000s in a series of papers by Janvresse, Maslen, Carlen, Carvalho, Loss and Geronimo, however it was known long before that the linear L^{2} based approach of Kac will not yield the desired result. A new method was devised, one that draws its ideas from a conjecture by Cercignani's for the real Boltzmann equation: investigate the entropy, and entropy production in Kac's model, in hope to get a better rate of convergence.

In our talk we will discuss Kac models of any dimension, recall the spectral gap problem and its conclusions as well as describe Cercignani's many body conjecture. We will show that, while the entropy and entropy production are more suited to deal with Kac's models, in full generality the rate they produce is not much better than that of the linear approach. We will conclude that more restrictions are need, and share a few insights we may have in the subject.

Prof. Henk Dijkstra, Department of Physics and Astronomy, Utrecht University

Interaction of noise and nonlinear dynamics in the wind-driven ocean circulation

Results will be presented of a study on the interaction of noise and nonlinear dynamics in a quasi-geostrophic model of the wind-driven ocean circulation. The recently developed framework of dynamically orthogonal field theory is used to determine the statistics of the flows which arise through successive bifurcations of the system as the ratio of forcing to friction is increased. Focus will be on the understanding of the role of the spatial and temporal coherence of the noise in the wind-stress forcing. For example, when the wind-stress noise is additive and temporally white, the statistics of the stochastic ocean flow does not depend on the spatial structure and amplitude of the noise. This implies that a spatially inhomogeneous noise forcing in the wind stress field only has an effect on the dynamics of the flow when the noise is temporally colored. The latter kind of stochastic forcing may cause more complex or more coherent dynamics depending on its spatial correlation properties.