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

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.

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.

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.

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.

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.

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.

"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.

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.