Dr. Ian H. Sloan, University of New South Wales, Australia
Good approximations on the sphere, with applications to geodesy and the scattering of sound
The theme of this talk is that polynomial approximations on the sphere are important for
applications, but that successful applications of high-degree polynomials need a good understanding of
underlying approximation properties. We illustrate with two case studies.
First, for applications in geodesy, there is good reason to use cubature rules that have a high degree of
polynomial accuracy .The stability, and even the computability, of such rules depends critically on the
properties of the underlying polynomial interpolants. Second, a recent spectral approach to the scattering of
sound by three dimensional objects needs for its analysis good approximation properties of the
‘hyperinterpolation’ polynomial approximation scheme. In the course of this talk the existing state of knowledge
for both interpolation and hyperinterpolation will be reviewed.
Dr. Pierre A. Gremaud, Department of Mathematics, North Carolina State University
Numerical and mathematical issues in bulk solid handling
Granular materials are routinely handled in many industries. However, due to the
incomplete understanding of such materials, solids-processing plants typically operate much less
efficiently than their liquids-processing counterparts. When dealing with granular materials,
engineers use models based on phenomenological relations, such as Coulomb's law of sliding
friction. Generalizing such a law to multidimensional and/or dynamic settings has proved to be
problematic in several respects (it may lead to ill-posedness, for instance).
In this talk, we will study and review several corresponding phenomena and models involving both
static granular assemblies and steady-state flows. The mathematical structure of those problems
often exhibit nonstandard features, such as the presence of both differential and algebraic
constraints. Concrete examples involving an Eikonal formulation (pile formation), system of
hyperbolic conservation laws and elliptic systems (granular flows) will be introduced and analyzed.
Numerical methods and results will be discussed.
Dr. Ramani Duraiswami, UMIACS, University of Maryland
Creating Virtual Audio via Scientific Computing and Mathematical Modeling
Humans are remarkably adept at processing the sound they receive at their two ears to
perceive the spatial location (and other attributes) of sources in an acoustic scene. We are interested in
creating auditory virtual reality over headphones. To create convincing sound at specified spatial locations,
the cues in the sound used to localize the source must be reintroduced into the sound that is played. While
binaural cues (that arise from the differences in the sound received at the two ears) are important, they are
incomplete. Additional cues that arise from the scattering of sound off a persons body (external ears, head,
and torso), and off the environment are also essential.
The cues that arise due to scattering off the listener are encoded in the so-called "Head Related Transfer
Function" (HRTF), while the cues that arise from room scattering are encoded in a Room Transfer Function
(RTF). The HRTF exhibits considerable variation from person-to-person Hitherto, the HRTF had to be
measured individually, in a tedious measurement procedure that made it impractical to use them widely.
We take an approach to modeling the scattering process, which relies on physical modeling and
decomposition of the problem, and the use of fast algorithms for computation.
In this talk I will attempt to describe the mathematical formulations of some problems that arise, and some
solutions we have proposed. (Joint work with Dmitry Zotkin & Nail Gumerov. The support of NSF and ONR
is gratefully acknowledged)
Dr. Xiangdong Ji, Department of Physics, University of Maryland
Lattice QCD for Pedestrians
Quantum Chromodynamics (QCD) is a fundamental theory for physics of strong interactions
which hold the mysteries of the origin of mass, the formation of proton, neutron, and atomic nuclei, and the
quark-gluon plasma in the early stage of the Universe. At the moment, the only direct approach to solve
this strongly-coupled quantum field theory is simulate it on a discrete spacetime lattice. In this talk, I
discuss the progress, challenges, and prospects of lattice QCD.
Dr. Shi Jin, University of Wisconsin, Madison
Computations of Multivalued Solutions of Nonlinear PDEs
Many physical problems arising from oscillatory waves, dispersive waves or Hamiltonian
systems require the computations of multivalued solutions which cannot be described by the viscosity
methods. In this talk I will review several recent numerical methods for such problems, including the
moment methods, kinetic equations and the new level set method with S. Osher. Applications to the
semiclassical Schroedinger equation and Euler-Piosson equations with applications to modulated electron
beams in Klystrons will be discussed.
Numerical Methods for Maxwell's Equations in Inhomogeneous Media
We will discuss three numerical methods for Maxwell equations in inhomogeneous media with
applications in photonic devices. First, we will discuss recently developed upwinding embedded boundary
methods on a Cartesian grid to handle arbitrary material interface. Secondly, we will discuss several issues
in discontinuous Galerkin methods such as construction of numerical fluxes, uniaxial PML boundary
conditions. Thirdly, we will discuss a new fast integral solver for scattering in multilayered media. Numerical
results will also be presented.
Dr. Wojciech Czaja, Department of Mathematics, University of Maryland
Singularity Detection in Images Using Dual Local Autocovariance
Differences in the eigenvalues of an autocovariance matrix indicate
directions at which the local Fourier power spectrum of a function is slowly
decreasing. This provides a technique to discriminate edge-like singularities
from other features in images.
Dr. Robert Pego, Department of Mathematics, University of Maryland
Domain coarsening - Modeling, Analysis, and a bit of Computation.
Models of domain coarsening during dynamic processes such as phase transitions
in materials science involve many scales and levels of description. A great
challenge is to explain the dynamic scaling laws seen in experiment. The
simplest classical theory that yields predictions involves a conservation law
for the size distribution of a family of particles.
Analyzing this law, we have recently obtained an improved well-posedness
theory for measure-valued size distributions with finite mass, using a
physically natural topology given by a Wasserstein distance between size
distributions. The analysis establishes convergence of a physically meaningful
numerical method for the problem.
We also analyze the long-time behavior of solutions. Rigorous analysis shows
that this model does not yield the universal self-similar behavior that was
classically predicted. Instead, long-time behavior depends sensitively on the
initial distribution of the largest particles. E.g., for a dense set of initial
data, convergence to any self-similar solution is impossible.
This is joint work with Barbara Niethammer.
Dr. Tai-Ping Liu, Stanford University
Gas flows with shocks
We will survey the recent progresses on shock wave theory. There is a deep
theory for system of hyperbolic conservation laws in one spacedimension, and
serious efforts for multi-dimensional gas flows. We will report on these and
raise some future possibilities.
Dr. Hailiang Liu, Department of Mathematics, Iowa State University
KdV Dynamics in the Plasma-Sheath Transition
The plasma-sheath transition is a fundamental problem in plasma physics, where
the typical length scale can be predicated by dimensional analysis, but the
sheath transition and inner layer are determined by a complex interplay of the
internal dynamics. Mathematically it provides a challenge to the applied analyst
in that there are multiple scales which must be resolved to obtain an adequate
description of the physical process.
In a recent work with M. Slemrod we formulate a new mathematical model to
catch the dynamics hidden in the plasma-sheath transition layer and the inner
sheath layer for planar motion of a plasma. It is shown that the rescaled
potential in the plasma-sheath transition layer and inner layer is governed by a
perturbed KdV equation, through which some of the complex interactions and
couplings among physical mechanisms acting in the plasmasheath formation process
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