Professor Yannis G. Kevrekidis, Program in Applied and Computational Mathematics and Department of Chemical
Engineering, Princeton University

Equation-free multi scale computation: Enabling Microscopic Simulators to Perform System Level
Tasks

I will present and discuss a framework for computer-aided multiscale analysis, which enables models at a "fine" (microscopic/stochastic) level of description to perform modeling tasks at a "coarse" (macroscopic, systems) level.
These macroscopic modeling tasks, yielding information over long time and large space scales, are accomplished through appropriately initialized calls to the microscopic simulator for only short times and small spatial domains: "patches" in macroscopic space-time.

Traditional modeling approaches first involve the derivation of macroscopic evolution equations (balances closed through constitutive relations). An arsenal of analytical and numerical techniques for the efficient solution of such evolution equations (usually Partial Differential Equations, PDEs) is then brought to bear on the problem.

Our equation-free (EF) approach, introduced in PNAS (2000) when successful, can bypass the derivation of the macroscopic evolution equations when these equations conceptually exist but are not available in closed form. We discuss how the mathematics-assisted development of a computational superstructure may enable alternative descriptions of the problem physics (e.g. Lattice Boltzmann (LB), kinetic Monte Carlo (KMC) or Molecular Dynamics (MD) microscopic simulators, executed over relatively short time and space scales) to perform systems level tasks (integration over relatively large time and space scales, "coarse" bifurcation analysis, but also optimization and control tasks) directly.
In effect, the procedure constitutes a systems identification based, "closure on demand" computational toolkit, bridging microscopic/stochastic simulation with traditional continuum scientific computation and numerical analysis. We illustrate these "numerical enabling technology" ideas through examples from chemical kinetics (LB, KMC), rheology (Brownian Dynamics), homogenization and the computation of "coarsely self-similar" solutions, and discuss various features, limitations and potential extensions of the approach.
An overview article on the topic of the presentation can be found at: http://www.neci.nj.nec.com/homepages/cwg/eqfree.pdf

October 1

Professor Semyon V. Tsynkov, Department of Mathematics, North Carolina State University and Tel Aviv
University

Optimization of Acoustic Source Strength in the Problems of Active Control of Sound

We consider the problem of eliminating the unwanted time-harmonic noise on a predetermined
region of interest. The desired objective is achieved by active means, i.e., by introducing additional sources
of sound called control sources that generate the appropriate annihilating acoustic signal (anti-sound). The
general solution for controls has been obtained previously in both continuous and discrete formulation of
the problem. In the current talk, we focus on optimizing the overall absolute acoustic source strength of the
control sources. Mathematically, this amounts to the minimization of multi-variable complex-valued
functions in the sense of L_1 with conical constraints, which are only ``marginally'' convex. The
corresponding numerical optimization problem appears very challenging even for the most sophisticated
state-of-the-art methodologies, and even when the dimension of the grid is small, and the waves are long.

Our central result is that the global L_1-optimal solution can, in fact, be obtained without solving the
numerical optimization problem. This solution is given by a special layer of monopole sources on the
perimeter of the protected region. We provide a rigorous proof of the global L_1 minimality for both
continuous and discrete optimization problems in the one-dimensional case. We also provide numerical
evidence that corroborates our result in the two-dimensional case, when the protected domain is a cylinder.
We believe that the same result holds in the general case as well and formulate it as a conjecture in the
end of the talk.

October 8

Dr. Dr. Dmitri Klimov, Institute for Physical Science and Technology, University of Maryland

Aggregation of A$\beta$16-22 amyloid peptides: A molecular dynamics study

Using multiple all-atom molecular dynamics simulations, we investigate aggregation of solvated fragments
(residues 16-22) of amyloid A$\beta$ peptides, which are linked to Alzheimer's disease. Their aggregation resulting in
appearance of A$\beta_{16-22}$ oligomers proceeds in two stages. The first involves the formation of disordered oligomers
and is driven by hydrophobic interactions. Consistent with experiments antiparallel packing of peptides due to favorable
electrostatic interactions emerges during the second stage. Disordered oligomers contain obligatory $\alpha$-helical
intermediate. Dramatic conversion of $\alpha$-helical into $\beta$-strand structures is observed upon further assembly of
A$\beta_{16-22}$ oligomers. Targeted mutations indicate that both, hydrophobic and electrostatic, interactions are critical
for maintaining stable A$\beta_{16-22}$ oligomers. Our results taken in the context of recent experimental observations
imply the existence of universal mechanism of amyloid assembly, the apparent cause of Alzheimer’s disease. Overview of
recent experimental data on amyloid deposition will be presented. The computational aspects of molecular dynamics
simulations will also be discussed.

October 15

Joint CSCAMM/Numerical Analysis Seminar:

Dr. Fabio Nobile, Texas Institute for Computational and Applied Mathematics (TICAM), University
of Texas at Austin

Some issues in the mathematical modeling and numerical simulation of the cardiovascular system

The simulation of the human cardiovascular system presents many challenging aspects in both
modeling and set up of numerical tools. In this talk we will address two of them. The first one concerns the
simulation of blood flow in a large artery when the deformation of the vessel wall is taken into account. We
will present recent results on the so-called "Arbitrary Lagrangian Eulerian" (ALE) formulation, suitable to
simulate fluid flow problems in domains with moving boundaries, and we will discuss the stability of some
coupled fluid/structure algorithms. The second issue we will deal with concerns the possibility to achieve a
global description of the circulatory system by combining models of different complexity and space
dimension: what we called the "multiscale approach". This idea is motivated by the fact that local
phenomena, as the presence of an atherosclerotic plaque or an implanted prosthesis, may have effects on
the whole circulation, which should be predicted by the numerical simulation. We will give an overview on
the different models available in the literature to describe blood circulation and we will present strategies to
couple them in order to obtain a global model that accounts at the same time for local and systemic
features.

October 22

Dr. John Aston, US Census Bureau, Statistical Research Division

Partial Volume Correction for Neuroimaging using Tensor Based Statistical Algorithms

The partial volume effect in Positron Emission Tomography (PET) is a problem for quantitative
radiotracer studies. These studies can be used to study many well-known diseases such as Epilepsy.
However partial volume effects can cause misinterpretation of the data.

The talk will firstly introduce PET and then discuss the partial volume problem. This results from the limited
spatial resolution of the imaging device (a few mm's) and results in a blurring of the data. Two factors are
involved for pre-defined regions; spillover of radioactivity into neighboring regions and the underlying tissue
inhomogeneity (mixed tissue types) of the particular region.

Linear modeling methods are currently used to correct for this effect on a regional level, using tissue
classification from higher resolution imaging modalities, e.g. Magnetic Resonance Imaging, and
anatomically defined regions which are assumed to contain homogeneous radiotracer (the PET data
source) concentrations. We extend these methods to incorporate the underlying noise structure of the PET
measurements, and develop fast tensor based algorithms to facilitate the computation of true radiotracer
concentration estimates and their associated errors. Computationally efficient algorithms are essential due
to the massive nature of the datasets where there is intrinsic spatial correlation in the data. We also
investigate the possibility of using the developed noise models to infer whether the defined regions were
correctly defined as homogenous, using Krylov subspace approximate estimates for the regional errors
associated with the fits.

November 5

Dr. Yuri Godin, CeLight Inc.

Mathematical Modeling of Photonic Crystals

Photonic crystal is an artificial material designed for guiding the electromagnetic
waves. It has a periodic structure such that photons of light behave the same way electrons do in
semiconductors, whose crystalline structure forbids the passage of electrons in a well-defined
energy range, known as the band gap.

To fabricate a material with a band gap for light requires creating a photonic crystal, which has a
unique periodic structure that will reflect and refract light of specific wavelengths. In the talk, we
present two- and three-dimensional models of photonic crystals and discuss a generic mechanism
that leads to formation of spectral bands and gaps. Numerical results will be shown.

November 19

Dr. Yen-Hsi Richard Tsai, Department of Mathematics, Princeton University

Visibility in an implicit framework

We will discuss a new mathematical formulation for obtaining visibility
information in the presence of obstacles. We study the dynamics of the shadow
boundaries in the case of moving vantage point or deforming obstacles.
Furthermore, we consider a set of variational problems that has possible
applications in fields ranging from optimal control to computer graphics and
computer vision.

November 26

Dr. William Dorland, Department of Physics, University of Maryland

Astrophysical Gyrokinetics: Turbulent heating, fluctuation signatures and more

"Gyrokinetics" is a well-established research area in the magnetic confinement fusion
research community. The term refers to the rigorous kinetic study of low-frequency, highly
anisotropic instabilities and turbulence in hot, magnetized plasmas. Gyrokinetic turbulence limits
the achievable densities and temperatures in laboratory fusion experiments. Consequently, much
time (two decades) and energy (many scientists) has been invested in the study of gyrokinetic
dynamics. Here, I will describe astrophysical applications of this research. Generically, shear
Alfven waves in incompressible magnetohydrodynamic turbulence cascade down to scales
comparable to the ion Larmor radius. Simulations and theory support the idea that the turbulence
in this regime is highly anisotropic, so that the wavelengths along the ambient magnetic field are
much greater than those in the perpendicular directions. For a broad range of conditions, the
resulting turbulence is gyrokinetic. Several interesting questions can therefore be addressed with
gyrokinetic simulations, such as how the properties of the MHD cascade change as kinetic effects
like Barnes damping, Landau damping, and FLR orbit averaging become important, and the
relative efficiencies of turbulent ion and electron heating. We present such simulations and relate
the results to observations of fluctuations in the interstellar medium and to Chandra observations
of radiatively inefficient accretion flows.

December 10

Professor C. William Gear, NEC Laboratories

Macro scale numerical analysis from microscopic simulators

Accurate modeling of some phenomena may only be possible at the microscopic level
because we do not (yet) know the macroscopic-level equations or because all we have is a
microscopic-level simulator based on empirical data. If the model closes on a lower-dimensional
description (of moments of the more detailed description) we would like to compute the
macroscopic-level behavior of that lower-dimensional description. Computer resources limit the
simulation of microscopic models to small spatial domains and short time periods, so it is
necessary to bridge the gaps between small space and large space, and between short time and
long time. We show how the numerical solutions of a microscopic model over a collection of
small, physically separated micro-patches can be combined to generate an approximate solution
of the macroscopic model. The result will be a “time stepper” that deals with large spatial domains
over long time steps. Once that is available, it can be used to perform long time-scale
integrations, steady state analysis, or bifurcation studies.

A frequent feature of complex systems is the emergence of macroscopic, coherent behavior from
the interactions of microscopic “agents” - molecules, cells, individuals in a population. The
implication is that macroscopic rules (description of behavior at a high level) can somehow be
deduced from microscopic ones (description of behavior at a much finer level). For some
problems - like Newtonian fluid mechanics - the successful macroscopic description (the Navier-
Stokes equations) predated its microscopic derivation from kinetic theory. In most current
problems, however, the physics are known at the microscopic/individual level, but the closures
required to translate them to a high-level macroscopic description are simply not available. Our
procedure can only be successful if a macroscopic description is conceptually possible.

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