We will describe a simple, direct approach to computing a singular or nearly singular integral, such as a harmonic function given by a layer potential on a curve in 2D or a surface in 3D. The value is found by a standard quadrature, using a regularized form of the singularity, with correction terms added for the errors due to regularization and discretization. These corrections are found by local analysis near the singularity. This technique might be useful in viscous fluid calculations with moving interfaces, since a pressure term due to a boundary force can be written as a layer potential. The accurate evaluation of a layer potential near the curve or surface on which it is defined is not routine, since the integral is nearly singular. In work with M.-C. Lai, we solve boundary value problems in 2D by computing the integral at grid points near the curve as described and using these values to find those at all points. A similar approach works in 3D, with the surface integrals computed in overlapping coordinate grids on the surface. To solve a boundary value problem, we first need to solve an integral equation for the strength of a dipole layer on the surface. We have proved that the solution of the discrete integral equation converges to the exact solution. In related work with G. Baker, we use a special choice of regularization in a boundary integral calculation of an unstable interface in 2D inviscid flow, such as a Rayleigh-Taylor flow with a heavy fluid over a lighter fluid.

In this talk, we report some joint work with colleagues at PSU on a variational phase field approach developed for modeling the transformation of vesicle bio-membranes under the elastic bending energy, with prescribed volume and surface area. Such an approach is substantiated via careful analysis and extensive computation. It can effectively capture various membrane configurations and it is insensitive to topological events. We also discuss the problem of retrieving useful topological information of the membrane from the phase field model for both tracking and control purposes which may be of even broader interests.

In this presentation, a dissipative particle dynamics (DPD) based approach for modeling suspensions is examined. A series of tests are applied comparing simulation results to well established theoretical predictions. The model recovers the dilute limit intrinsic viscosity prediction of Einstein and provides reasonable estimates of the Huggins coefficient for semidilute suspensions. At higher volume fractions, it was necessary to explicitly include lubrication forces into the algorithm as the usual DPD interactions are too weak to prevent overlaps of the rigid bodies and account for other related effects due to lubrication forces. Results were then compared with previous studies of dense hard sphere suspensions using the Stokesian dynamics method and experimental data. Comparison of relative viscosity values determined from strain controlled shearing versus stress controlled shearing simulations are also given. The flow of spheroidal objects is studied. The rotation of a single spheroid under shear is consistent with the predictions of Jeffery. Simulations of sheared spheroids at higher volume fractions produce a nematic phase. An example is given of application of DPD to model flow in another geometry, gravitational driven flow between parallel cylinders, which is of practical interest.

I will describe the behavior of solutions of a steady advection-diffusion problem on a bounded two-dimensional domain with prescribed Dirichlet data when Pe, the Peclet number is very large. The characteristic property of advection by cellular flows is that the fluid motion is separated into flows cells. At high Peclet numbers advection dominates diffusion and the solutions tend to constant in each flow cell. Boundary layers of order Pe^(-1/2) arise near cells' separatrices. In this talk I will discuss this boundary layer structure by means of an asymptotic "diffusion on separatrices" model.

We shall formulate the image compression problem as a mathematical optimization problem and then discuss the role of wavelets and other multiscale families in trying to solve this problem. In particular we shall explain the success of some practical encoders and some of the directions currently pursued in image compression.

The interplay between pseudodifferential operators, Banach algebras, and time-frequency analysis provides a powerful means to analyze and design mobile, wireless communication systems. In particular, we will see that the mobile radio channel can be modeled as pseudodifferential operator whose symbol lies in a specific function space, called Sjoestrand class. It turns out that the ``system matrix'' associated with this pseudodifferential operator belongs to a Wiener-type Banach algebra that has a sparse representation with respect to Gabor bases. I will show how these theoretical results lead very naturally to the design of optimal transmission pulses for multicarrier wireless communication systems. The research presented in this talk has already found practical application in form of patents and modems.

Instabilities of particles and waves lead to chaotic dynamics that can be described by a corresponding kinetic equation and by the moment equations. The phenomenon of chaos is not the same as the Gaussian or Poissonian random process. Numerous "remnants" of regular dynamics in phase space generate a strong intermittent behavior of particles known as Levy flights or similar to them. Under such conditions the kinetics is too far from the regular diffusion and very new type of equations arrive to the scene. In some situations the new kinetics is described by the equations with fractional derivatives in space and time and these equations should replace the regular diffusion equations. This new understanding of the phenomenon of chaos is typical for some problems of the confined plasma. We will discuss some cases of fractional kinetics, some of  new features of this kinetics, and some new requirements to their simulations.

For review see: G.M. Zaslavsky, Physics Report,371, 461 (2002).

The evolution of the subject of cloud radiative transfer from 1D to 3D will be reviewed, starting from the 1970s when the first papers using simple cubic clouds were published and continuing to the present where simple multifractal models of cloud structure have become the norm. The role of data about cloud structure will be emphasized, including the types of instruments used and their advantages and disadvantages, and the possibility of tomography in the future. The other theme will be the extraction of simple results from all the complexity of 3D radiation modeling, including so-called "radiative smoothing" and the new multiple-scattering lidar concepts it has led to.

The dual weighted residual method (DWR) has been attracting increasing attention over the past years as a paradigm for so called goal oriented numerical schemes primarily in connection with Finite Element discretizations. In many complex numerical simulation tasks one is not interested in the whole solution of a PDE but merely in some local functional of the solution like point values, local averages or integrals over a lower dimensional manifold. "Goal oriented" then means to compute this value within some target accuracy tolerance at possibly low numerical cost. In particular, this could mean that the solution is only very poorly approximated in parts of the domain which results in significant savings in comparison with numerical schemes that first determine a global approximation of the solution and then apply the functional to that approximation. The approach is based on error representations derived from duality arguments. Since this representation can typically be bounded by sums of local quantities of residual type the equilibration of these quantities gives rise to adaptive resolution schemes. While this has been observed to work often very well in numerical applications no rigorous error estimates or convergence proofs seem to be known, not even for linear model problems. A principal obstruction is that the error representation involves the unknown dual solution. The relevant information on the dual solution has to be inferred from approximations to the dual problem. The need to understand the effect of the unresolved scales in those approximations indicates the multiscale nature of the problem. This talk reviews some recent joint work with A. Kunoth which aims at deriving rigorous convergence results by recasting the DWR method in a wavelet framework and exploiting recent advances in the development of adaptive wavelet solvers.

In this talk I will discuss a stabilized finite element (FE) formulation for reacting flow applications. This formulation is used to simulate reacting flow applications for incompressible, variable density and low Mach number compressible fluid flow, heat transfer and mass transfer with non-equilibrium chemical reactions. These systems are characterized by highly nonlinear, multiple time and length scale physics governed by coupled systems of PDEs. The numerical solution of these systems can be very challenging.

In this presentation I will overview a number of the important solution methods that we have applied in the computational simulation of transport/reaction systems. These include, fully-implicit time integration, direct-to-steady-state solvers, continuation, bifurcation and linear stability methods using parallel Newton-Krylov techniques. The resulting large sparse linear systems are solved by the application of Krylov methods using parallel additive Schwarz domain decomposition (DD) preconditioners with subdomain ILU solvers. Representative results for the 1-level methods and preliminary results for 2-level geometric DD and algebraic multi-level preconditioners will also be presented. In this discussion I will also present some recent collaborative work on approximate block factorization methods with Howard Elman and Robert Shuttleworth of U. MD.

To demonstrate the capability of these solution methods I will present simulation results for some representative low heat release and high heat release transport / reaction simulations. In this context we briefly discuss robustness, efficiency and scaling of the solution methods.

*This work has been partially funded by the DOE Office of Science, Mathematical, Information, and Computational Sciences Division, and the DOE/NNSA ASC program. This work was carried out at Sandia National Laboratories operated for the US Department of Energy under contract no. DE-ACO40-94AL85000.

Joint Wavelet-Harmonic Analysis/CSCAMM Seminar

THURSDAY 11th Nov at 4:00PM Math #1311

Modern computational harmonic analysis and signal/image processing algorithms use localized reconstructions for the high order approximation of piecewise smooth objects. In this talk I construct and analyze algorithms for two unrelated problems, the classical resolution of Gibbs' phenomena, and the application oriented problem of recovering a bandlimited signal from its bunched samples. In both cases I will discuss how high resolution accuracy is obtained from localized reconstructions by designing appropriate smooth Fourier space filters.

Although global projections such as truncated Fourier series yield exponentially close approximations for smooth functions, a single discontinuity introduces O(1) spurious oscillations, Gibbs' Phenomena, and reduces the high order convergence rate to first order. A family of filters have been developed over the last century to localize the global approximation to a smooth region, and by doing so accelerate the convergence rate. However, the filter selection for a particular problem has remained largely heuristic, adversely affecting the accuracy of those methods utilizing filters, such as the numerical methods for time dependent problems. Using ideas from time-frequency analysis I construct a spatially adaptive filter which is shown to achieve optimal (exponential) accuracy for this class of methods, and as a result definitively resolved the question of filter selection. Portions of this work were joint with Eitan Tadmor.

Second I consider a practical, application driven problem in communication, the recovery of a bandlimited signal from its bunched sampling. The classical Shannon sampling theorem gives an exact relationship between a bandlimited signal and an infinite number of its equidistant samples. Yet, for practical applications only a finite number of samples are available, requiring a localized reconstruction to obtain high order accuracy. Moreover, in many emerging applications, due to the increase in information being transmitted or complications from miniaturization, the fast uniform sampling required by the ever-increasing bandwidth cannot be obtained practically by a single device. To obtain a similar overall sampling rate, and gain flexibility in the sampling structure, a group of less expensive, slower, sampling chips are combined. Here I present an efficient and robust high order algorithm for the "bunched" sampling structure which results from the combination of multiple sampling devices. Extensions to images allows for combining multiple images to achieve sub-pixel super-resolution. This portion of research is joint with Thomas Strohmer.

MONDAY 15th Nov at 2:00PM

We introduce a new iterative regularization procedure for inverse problems based on the use of Bregman distances, with particular focus on problems arising in image processing. We are motivated by the problem of restoring noisy and blurry images via variational methods, by using total variation regularization. We obtain rigorous convergence results and effective criteria for the general procedure. The numerical results for denoising and deblurring appear to give significant improvement over standard models. We compare our technique with earlier work of Scherzer and Groetsch, and Tadmor Nezzar, Vese. Next, by taking the regularization parameter very small and the number of iteration steps large we are led to a new paradigm for restoration based on inverse scale space flows, instead of variational methods.

Joint Numerical Analysis/CSCAMM Seminar

TUESDAY 16th Nov at 3:30PM Math Colloquium Room #3206

In this talk, we present efficient and stable numerical methods for the generalized Zakharov system (GZS) describing the propagation of Langmuir waves in plasma. The key point in designing the methods is based on a time-splitting discretization of a Schroedinger-type equation in GZS, and to discretize a nonlinear wave-type equation by pseudospectral method for spatial derivatives, and then solving the ordinary differential equations in phase space analytically under appropriate chosen transmission conditions between different time intervals or applying Crank-Nicolson/leap-frog for linear/nonlinear terms for time derivatives. The methods are explicit, unconditionally stable, of spectral-order accuracy in space and second-order accuracy in time. Moreover, they are time reversible and time transverse invariant if GZS is, conserve the wave energy as that in GZS, give exact results for the plane-wave solution and possesses `optimal' meshing strategy in `subsonic limit' regime. Extensive numerical tests are presented for plane waves, solitary-wave collisions in 1D of GZS and 3D dynamics of GZS to demonstrate efficiency and high resolution of the numerical methods.

Finally the methods are extended to vector Zakharov system for multi-component plasma and Maxwell-Dirac system (MD) for time-evolution of fast (relativistic) electrons and positrons within self-consistent generated electromagnetic fields.

We enumerate and classify jammed configurations that occur at zero temperature in small 2D and 3D periodic systems composed of monodisperse and polydisperse particles that interact via hard-sphere and soft finite-range potentials. Jammed configurations are created using two algorithms: 1) random displacements of individual hard particles followed by particle growth and 2) collective moves of soft particles based on potential energy minimization followed by compression. In algorithm 1, configurations are jammed when a given particle cannot be displaced when all other particles are held fixed. In algorithm 2, configurations are jammed when no group of particles can be displaced simultaneously. We find that jammed states occur in continuous topological families when only single-particle moves (algorithm 1) are allowed. However, when collective moves (algorithm 2) are allowed, jammed states are discrete, i.e. each possesses a distinct contact network. We decompose the frequency distribution of jammed states into distributions for each topology and then calculate the density of jammed states and their basins of attraction. We believe that determining the types of jammed configurations and the frequency at which they occur at zero temperature may give insight into slow relaxations in disordered systems at finite temperature.

Thanksgiving

We review and compare classical WKB techniques and modern phase-space approaches, based on Wigner transforms, for highly oscillatory PDEs. In particular we discuss semiclassical limit problems, linear homogenisation problems and transport of energy density in periodic media (quadratic homogenisation). Typical examples are linear and nonlinear Schroedinger equations in the vacuum and in crystals, the Maxwell system, and acoustic wave equations. Moreover we review numerical techniques for Schroedinger equations and discuss their application to Bose-Einstein Condensation.

We discuss regularity conditions  for  solutions of the 3D Navier-Stokes equations and the 2D quasigeostrophic equations with powers of the Laplacian, which incorporates the vorticity direction and its  magnitude simultaneously. For the proof of the result we use geometric properties of the nonlinear term.