Fall 2009 Semester:

Modeling and analysis of collective behavior of self-propelled agents Jian-Guo Liu Duke University Collective behaviors of self-propelled agents (representing birds, fishes, cars, etc) such as flocking, swarming, trail formulation, attract much of recent research activities in applied mathematics. In this talk, I will discuss some of the recent developments in modeling and analysis of these emergence behaviors. In particular, I will present some analysis of flocking estimates for Cucker-Smale modes and Vicsek modes for birds and fish. I will also discuss the connection and passage among particle models, kinetic models, and continuum models of these self-propelled agents.

Application of perturbation series Ben Akers University of Illinois at Chicago Applications of perturbation series are presented, with historical perspective. Pertubation methods for roots of polynomials are compared with similar methods for wa- ter wave problems. The role of resonances in perturbation series is explored. Modern boundary perturbation methods for solving the water wave problem are motivated. This talk will be accessible to mathematicians of general background.

Semiclassical limit for weakly coupled nonlinear Schrodinger systems Benedetta Pellacci Università degli Studi di Napoli TBA

Liouville type theorems for nonlinear elliptic equations Patricio Felmer Universidad de Chile TBA

High-order moment equations for simulation of fast magnetohydrodynamics Ammar Hakim Tech-X Corporation Fluid equations are a common tool to study bulk plasma behaviour. Among the most commonly used fluid models are the the single-fluid Magnetohydrodynamics (MHD) model and the Hall-MHD model, often extended to include effects of collisions via transport terms. A more general approach is to treat to the plasma as a mixture of multiple fluid species. In these multi-fluid models each plasma species is described by a set of fluid equations evolving under electromagnetic forces and collisions. The electromagnetic fields are modeled using Maxwell equations of electromagnetism. The multi-fluid models are derived by taking moments of the Boltzmann equation and truncating the resulting hierarchy of equation using a closure approximation. In this talk the physics of high-order moment equations is described. For simplicity, the collision terms are approximated using a BGK collision operator. The hyperbolicity of the resulting system is discussed, taking into account the effect of the heat transport tensor. The inability of the moment equations to describe long mean-free-path dynamics, in particular, phase mixing and collisionless damping is demonstrated using a simple advection model. A possible solution, involving multiple fluid equations per plasma species, is proposed.

High-order and adaptive methods for laser-plasma interaction problems Jeffrey Banks Lawrence Livermore National Laboratory To expand the ranges of laser plasma experiments on the National Ignition Facility, a better understanding of the nonlinear behavior of laser-plasma instabilities, and in particular, thresholds for the onset of nonlinear behavior, is required. Continuum Vlasov simulation is an attractive option but the high cost of full phase space computations presents substantial hurdles. Furthermore, the desire to preserve certain mathematical properties of the contin- uous operators, such as solution positivity and conservation, provides additional complica- tions. To address these challenges we are developing high-order finite volume discretizations to reduce the number of cells required to achieve a given level of error. We employ various non-linear algorithms, such as PPM, WENO, and FCT to enforce known physical properties. Additionally, we have begun to investigate the use of adaptive mesh renement to localize the placement of fine grids and further reduce costs. In this talk I will discuss the algorithms we have developed as well as touch briefly on some of the computer science aspects of the problem as they relate to our ability to use large computational resources in an efficient and scalable manner. I will present results from classical Vlasov-Poisson test problems as well as provide an initial comparison to a traditional semi-Lagrangian approach. Time permitting, Vlasov-Maxwell systems will also be discussed.

Symbolic computation of conservation laws of nonlinear partial differential equations Willy Hereman Colorado School of Mines A method will be presented for the symbolic computation of conservation laws of nonlinear partial differential equations (PDEs) involving multiple space variables and time. Using the scaling symmetries of the PDE, the conserved densities are constructed as linear combinations of scaling homogeneous terms with undetermined coefficients. The variational derivative is used to compute the undetermined coefficients. The homotopy operator is used to invert the divergence operator, leading to the analytic expression of the flux vector. The method is algorithmic and has been implemented in the syntax of the computer algebra system MATHEMATICA. The software is being used to compute conservation laws of nonlinear PDEs occuring in the applied sciences and engineering. The software package will be demonstrated for PDEs that model shallow water waves, ion-acoustic waves in plasmas, sound waves in nonlinear media, and transonic gas flow. The featured equations include the Korteweg-de Vries and Boussinesq equations, the Navier and Kadomtsev-Petviashvili equations, and the Zakharov-Kuznetsov and Khoklov-Zabolotskaya equations.

Using invariant manifolds to classify chaotic transport pathways in mixed phase space Kevin Mitchell University of California Merced We describe how the topological structure of stable and unstable manifolds (so-called homo- or hetero-clinic tangles) embedded within a chaotic phase space can be used to extract a symbolic classification of chaotic transport and escape pathways. We pay particular attention to phase spaces that contain a mixture of both chaos and regularity. For such systems, the dynamics in the vicinity of "stable islands" is known to be particularly troublesome to analyze. We describe a technique that utilizes the structure of invariant manifolds in the vicinity of such stable islands to extract a symbolic model for the islands' influence on the transport process. Though our analysis focuses on Hamiltonian systems of two degrees-of-freedom, we also discuss the extension of our technique to higher dimensional phase spaces. We illustrate this technique with a few specific examples drawn from atomic physics.

New PDE reduced models for geophysical flows Mark Remmel UW-Mathematics New intermediate PDE models for the rotating shallow water equations (RSW) are derived by considering the nonlinear interactions between subsets of the eigenmodes for the linearized equations. It is well-known that the two-dimensional (2D) quasi-geostrophic (QG) equation results when the nonlinear interactions are restricted to include only the vortical eigenmodes. Continuing past QG in a non-perturbative manner, the new models result by including subsets of interactions which include inertial-gravity wave (IG) modes. The simplest such model adds nonlinear interactions between one IG mode and two vortical modes. In sharp contrast to QG, the latter model behaves similar to the full RSW equations for decay from balanced initial conditions as well as unbalanced, random initial conditions with divergence-free velocity. Quantitative agreement is observed for statistics that measure structure size, intermittency, and cyclone/anticyclone asymmetry. In particular, dominance of anticyclones is observed for Rossby numbers Ro in the range 0.1 < Ro < 1 (away from the QG parameter regime Ro -> 0). A hierarchy of models is explored to determine the e ects of wave-vortical and wave-wave interactions on statistics such as the skewness of vorticity in decaying turbulence. Possible advantages over previously derived intermediate models include (i) the non-perturbative nature of the new models (not restricting them a priori to any particular parameter regime), and (ii) insight into the physical and mathematical consequences of vortical-wave interactions.

Some results on fast algorithms of electron structure calculation Lexing Ying University of Texas at Austin Ab-initio computation in the framework of density functional theory has been widely used to study large quantum systems. The development of fast algorithms for insulating systems has been very successful, however these algorithms perform poorly on metallic systems. In this talk, we discuss some recent progress on developing fast algorithms for electron structure calculation of large metallic systems at low temperature. More specifically, we focus on an optimal rational expansion of the Fermi operator and a fast algorithm for extracting the diagonal of the shifted one-particle effective Hamiltonian. This is a joint work with Lin Lin, Jianfeng Lu, Weinan E, and Roberto Car.

Attenuation compensation and boundary segmentation in ultrasound images Jue Wang Union College Spatial variations of attenuation across tissue layers can result in shadowing and enhancement in ultrasound B-scan images. These artifacts affect the underlying signal backscatter which is the main component of ultrasound images and has clinical significance in detecting diseases and tumors. We present a method based on the variational principle to compensate for attenuation artifacts via functional minimization. Pair of pathological useful backscatter and attenuation fields are reconstructed along with segmented anatomic structures. A three-step alternating minimization procedure is adopted to compute the numerical solutions. Furthermore, the existence, uniqueness, stability and convergence of the minimization problems are proved.

Traveling fronts in combustible media Andrej Zlatos University of Chicago Traveling fronts are special solutions of reaction-diffusion equations which model phenomena such as propagation of species in an environment or spreading of flames in combustible media. In this talk we will address questions of existence, uniqueness, and stability of traveling fronts in general inhomogeneous media. We will show that in certain circumstances they are global attractors of the corresponding parabolic evolution, thus describing long time dynamics for very general solutions of the PDE.

Optimization-based closures for radiation transport Cory Hauck Oak Ridge National Lab In this talk, I will present two new moment closures for solving radiative transport equations. These closures modify the well-known the spherical harmonic closure, which is known to produce non-physical oscillations in the kinetic distribution and even negative values for the particle concentration. The first closure is derived from the solution to a constrained, quadratic optimization problem. The second closure uses a standard filter which is derived from an unconstrained, quadratic optimization problem but includes derivative information in the cost functional. Preliminary comparisons with current methods will be shown in one and two dimensions.

Fragmentation and aggregation of bacterial emboli David Bortz University of Colorado at Boulder Klebsiella pneumoniae is one of the most common causes of intravascular catheter infections, potentially leading to life-threatening bacteremia. These bloodstream infections dramatically increase the mortality of illnesses and often serve as an engine for sepsis. Our current model for the dynamics of the size-structured population of aggregates in a hydrodynamic system is based on the Smoluchowski coagulation equations. In this talk, I will discuss the progress of several investigations into properties of our model equations. In particular, I will focus on: a) accurate characterization of the fractal properties for the aggregates, b) a novel approach to fragmentation modeling, and (time permitting) c) self-similar solutions to the equations.

Semi-Eulerian and higher order Gaussian beam methods for the Schrödinger equaiton in the semiclassical regime Xu Yang Princeton University This is a follow-up study of the Eulerian Gaussian beam method developed for efficiently computing the Schrödinger equation within the presence of caustics. It dealt with the issue of easy numerical implementation and generalized the method to have higher order accuracy. The difference of this approach is that we solve the ray tracing equations to determine the centers of the beams and then compute quantities of interests only around these centers. This effectively yields a local level set implementation, and the beam summation can be carried out in physical space instead of the higher-dimensional phase space. As a consequence, it reduces the computational cost and also avoids the delicate issue of beam summation around caustics in the Eulerian Gaussian beam method.

Numerical computation for low-frequency dynamics in magnetized plasma Carl Sovinec UW-Engineering Physics Extended magnetohydrodynamic simulations for magnetically confined plasma must address a 'gang of four,' in addition to the usual motivations for numerical computation, nonlinearity and geometry. The 'gang' includes separations of temporal and spatial scales, extreme anisotropy, and the magnetic divergence constraint. The influence each issue has on numerical simulation will be described after background on applications for magnetic confinement fusion. Regarding time-scale separation in the two-fluid plasma model, basic analysis and heuristic information from differential approximation are presented for an implicit leapfrog algorithm. Results show that the algorithm is unconditionally stable with respect to time-step, provided that advective and physically dispersive terms are centered in the advances for each field.

TBA Henri Berestycki Centre d'Analyse et de Mathématique Sociales, Paris, France TBA