Joint CSCAMM/Mathematics Seminar
January 25th at 2:00PM Math Colloquium Room #3206

Professor Guillaume Bal, Department of Applied Physics and Applied Mathematics, Columbia University

Radiative transfer models: derivation and applications

Radiative transfer equations have long been used to model the energy density of waves in random media, with applications in quantum waves in semiconductors, light propagation through turbulent atmospheres, underwater acoustics, and elastic wave propagation in the Earth's crust.  In this talk, we consider the derivation of such models from first principles, i.e., from equations for high frequency wave fields. Mathematically rigorous derivations are presented in the paraxial and Ito-Schroedinger approximations of wave propagation and in the random Liouville regime. The radiative transfer models are also applied to the understanding of the enhanced refocusing properties of time reversed waves propagating in random media.

The validity of radiative transfer models to predict the energy density of high frequency waves is then investigated numerically. Two-dimensional acoustics systems of equations are solved over large domains (on the order of 500 times 500 wavelengths) using parallel
architectures. We demonstrate the very good accuracy of the macroscopic radiative transfer models and show the relative statistical stability of the wave energy density, i.e., the fact that the latter depends on macroscopic statistics of the random medium and not on its specific realization.
 

*NOTE SPECIAL DAY*

*THURSDAY*, January 26th

Dr. Oleg Musin, Moscow State University

The kissing problem in three and four dimensions

The kissing number k(n) is the maximal number of equal nonoverlapping spheres in n-dimensional space that can touch another sphere of the same size. This problem in dimension three was the subject of a famous discussion between Isaac Newton and David Gregory in 1694. In three dimensions the problem was finally solved only in 1953 by Schutte and van der Waerden. It was proved that the bounds given by Delsarte's method are not good enough to solve the problem in 4-dimensional space.

Delsarte's linear programming method is widely used in coding theory. In this talk we will discuss a solution of the kissing problem in four dimensions which is based on an extension of the Delsarte method.

This extension also yields a new proof of k(3)<13.

*NOTE SPECIAL DAY*

Joint CSCAMM/Mathematics Seminar
*TUESDAY*, January 31st at 2:00PM Math Colloquium Room #3206

Professor Becca Thomases, Courant Institute of Mathematical Sciences, New York University

Nonlinear Elasticity and Viscoelasticity

In this talk I will present a global existence result for small data nonlinear elasticity which also applies to the Oldroyd-B viscoelastic fluid model. These results can be obtained via a general local decay theorem which applies to wide variety of isotropic hyperbolic systems.

While small solutions decay, the problem for large data is much more complicated. I will present some recent numerical work on the Oldroyd-B system which shows that the system develops large stress gradients for certain curvilinear flows. These stress gradients appear to become unbounded as material parameters (in particular the Weissenberg number) are increased.
 

The Terascale Optimal PDE Simulations (TOPS) project is sponsored by the U.S. Department of Energy to research and deploy a collection of open-source, scalable solvers (PETSc, Hypre, SuperLU, etc.) for discrete problems arising in several large-scale applications, including fusion reactor modeling and design. Optimal complexity methods, such as multigrid/multilevel preconditioners, keep the time spent in dominant algebraic kernels close to linear as the applications scale on parallel computers. Krylov accelerators and Jacobian-free variants of Newton's method, as appropriate, are wrapped outside to deliver robustness in multirate, multiscale coupled systems, which are solved either implicitly or in more traditional forms of operator splitting. The TOPS software framework is being extended beyond forward simulation to optimization. We outline the TOPS research agenda and illustrate with a range of applications in magnetically confined fusion energy, as the fusion community gears up for participation in the International Thermonuclear Experimental Reactor (ITER) consortium, the ultimate goal of which is abundant energy production outside of the planetary carbon cycle.

Dr. Hao-Min Zhou, School of Mathematics, Georgia Institute of Technology

Variation Models and PDE Techniques in Wavelet Inpainting

In this talk, I will present a recent work (collaborated with Tony Chan (UCLA) and Jackie Shen (Minnesota)) on image inpainting in wavelet domain. The problem is closely related to the classical image inpainting, with the difference being that the inpainting regions are in the wavelet domain, that brings new challenges to the reconstructions, as there is no geometrically well defined inpainting region in the pixel domain, and the damage is inhomogeneous. We propose new variational models, especially total variation minimization in conjunction with wavelets for the wavelet inpainting problems. The models lead to PDE's, which are Euler-Lagrange equations of the variational formulations, in the wavelet domain and can be solved numerically. The proposed models can have effective and automatic control over geometric features of the inpainted images including sharp edges, even in the presence of substantial loss of wavelet coefficients, including in the low frequencies.

Joint CSCAMM/Mathematics Seminar
February 15th at 2:00PM Math Colloquium Room #3206

Professor Wei Cai, Department of Mathematics, University of North Carolina at Charlotte

Numerical Methods for Modeling Photonics and Nano-Electronics

Computational modeling has established itself as an indispensable tool for studying new physical phenomena and behaviors in micro-to-mesoscopic systems. Numerical modeling based on classical and quantum physics provides unparalleled possibilities in investigating physical processes not accessible to experimental explorations. Due to the intrinsic multi-scale and multi-physics properties in devices operating in far from equilibrium involving ultrafast kinetics and high field transport, there is a great demand for efficient and accurate numerical algorithms for fast simulations. In this talk, we will present our research in computational methods for the following two problems: (1) Energy transfer in resonant systems such as coupled microcavity dielectric resonators and coupled resonant plasmon silver nanowires, (2) Carrier transport in quantum dots and nanotransistors.  

Several numerical approaches will be studied. In order to compute the optical field propagation in heterogeneous media, we have developed 4^th order upwinding embedded boundary methods and discontinuous spectral element methods for dispersive lossy Maxwell’s systems. For the transport in quantum dots and nano-transistors, we have developed fast integral equation methods for layered media based on acceleration algorithms for Green’s functions and 2-D FMM. Issues of boundary conditions for open quantum systems using Green’s functions will also be addressed. Numerical methods for calculating self-energy and quantum conductance with the Landauer formula in nano-MOSFET, including the geometric effects of the quantum devices, will be presented. Finally, future research issues will be discussed.

Joint CSCAMM/Mathematics Seminar
February 22nd at 2:00PM Math Colloquium Room #3206

Professor Hongkai Zhao, Department of Mathematics, University of California-Irvine

The Fast Sweeping Method for Hyperbolic Problems

An efficient iterative algorithm, the fast sweeping method, for steady hyperbolic equations will be presented. I will show that the iterative algorithm has an optimal complexity, i.e., the number of iterations is finite and is independent of mesh size, for Eikonal equation which is a nonlinear hyperbolic boundary value problem. I will also explain different convergence mechanisms for hyperbolic and elliptic problems. Extensions to more general Hamilton-Jacobi equations and hyperbolic conservation laws will also be discussed. If time permits, applications to computer vision and image processing will be shown.
 

By Nail A. Gumerov and Ramani Duraiswami

It is hard to overestimate the practical importance of development of computationally efficient methods for solution of Maxwell’s equations for large scale problems. Computation of propagation and scattering of electromagnetic waves is a key issue for antenna design, radar applications, wireless networks, optical devices and materials, diffractional tomography, and many more emerging areas. In the present study we develop a computational method based on a scalar potential representation, which efficiently reduces the solution of Maxwell’s equations to the solution of two scalar Helmholtz equations.

One of the major challenges in developing our formulation was the lack of an existing theory for the translation of such a representation, since the form of the decomposition is not invariant with respect to translations. We developed such a theory by introducing the concept of “conversion” operators, which enables representation of the electric and magnetic vector fields via scalar potentials in an arbitrary reference frame. This speeds up methods such as the Fast Multipole Method, since only two Helmholtz equations need be solved, and moreover, the divergence free constraints are satisfied automatically by construction.

For illustration of the method we implemented an algorithm for the solution of the Mie-type scattering problem from a system of spherical objects of different sizes and dielectric properties using a variant of T-matrix method and GMRES. Results of the computations agree well with the previous theoretical and experimental results. We also discuss opportunities for application of the method of scalar potentials to solution of other boundary value problems for Maxwell’s equations.

Professor Ping Lin, Department of Mathematics, The National University of Singapore

A quasi-continuum approximation and its analysis

In many applications materials are modeled by a large number of particles (or atoms) where any one of particles interacting with all others through a pair potential energy. The equillibrium configuration of the material is the minimizer of the total energy of the system. The computational cost is high since the number of atoms is huge. Recently much attention has been paid to a so-called quasicontinuum (QC) approximation which is a mixed atomistic/continuum model.

The QC method solves a fully atomistic problem in regions where the material contains defects (or larger deformation gradients), but uses continuum finite elements to integrate out the majority of the atomistic degrees of freedom in regions where deformation gradients are small. However, numerical analysis is still at its infancy. In this talk we will conduct a convengence analysis of the QC method in the case that there is no serious defect or that the defect region is small. The difference of our analysis from conventional finite element analysis is that our exact solution is not a solution of a continuous partial differential equation but a discrete atomic scale solution which is not simply related to any conventional partial differential equation. We will consider both one dimensional and two dimensional cases. Some thoughts about the dynamical case may be mentioned as well. The QC method may be related to some other fields such as model reduction and pre-conditioning.

NO SEMINAR

University of Maryland Spring Break

We present deterministic FEM for the solution of elliptic problems with coefficients which are spatially inohomogeneous random fields. The FEM is based on M-term on a principal component, Karhunen-Loeve type expansion of input data, computed by a generalized FMM and on sparse wavelet resp. spectral approximations of meshwidth h resp. order p for the random solution's joint probability densities, parametrized in the principal components of the input data. Numerical analysis of the random solution's regularity and of the complexity of the method as h→0 resp. p→∞ simultaneously with M→∞ are given.

Joint work with R.A. Todor and P. Frauenfelder of ETH.
 

Many models of geophysical flow arising in oceanography, meteorology or climatology belong to the class of hyperbolic balance laws. We present a new well-balanced finite volume method within the framework of the finite volume evolution Galerkin (FVEG) schemes. The methodology will be illustrated for the shallow water equations with source terms modeling the bottom topography, friction effects and Coriolis forces. Results can be generalized to more complex systems of balance laws.

The FVEG methods belong to the class of genuinely multidimensional methods. They couple a finite volume formulation with approximate evolution operators. The latter are constructed using the bicharacteristics of the multidimensional hyperbolic systems, such that all of the infinitely many directions of wave propagation are taken into account explicitly. We derive a well-balanced approximation of the integral equations and prove that the FVEG scheme is well-balanced for the stationary steady states as well as for the steady jets in the rotational frame. Several illustrating examples will confirm this property experimentally, too.

Further, we compare the results obtained by the FVEG scheme with the solution of the finite element methods, which are typically used in the river engineering. We also present results of extensive numerical study of accuracy and computational efficiency of both methods.

Polymer-induced turbulent drag reduction has been the subject of intense investigations ever since its accidental discovery by Tomms and Mysels during the Second World War, with important applications, most well known of which is the facilitation of the oil transfer though pipelines, such as in the Alaskan pipeline.  Recent developments in numerical methods, have allowed us for the first time to be able to obtain accurate and stable simulations of highly drag reduced (more than 60%) turbulent channel flows of dilute polymer solutions.  The data confirm earlier results in our group at lower drag reduction values whereby the primary mechanism for drag reduction is the decreased intensity of the wall eddies which is the result of a significant increase to extensional deformations contributed by the polymer additives, exactly as proposed earlier by Metzner and Lumley.  Recent work has been able to more systematically investigate the changes to the flow structure effected due to the polymers, and in particular the coherent structures, using Karhunen-Loeve (K-L) Proper Orthogonal Decomposition (POD) analysis of the data. This demonstrated a dramatic enhancement of the importance of large scale motions with increased viscoelasticity and an equally dramatic decrease in the K-L dimension of the flow (an order of magnitude) as viscoelasticity increases versus similar Newtonian results.  More recent work aims at building up better numerical techniques able to probe even higher drag reductions, close to the maximum drag reduction (Virk) limit.  Towards that goal we recently developed an exponential mapping for the viscoelastic conformation tensor that allows us to preserve its positive definiteness under all flow conditions exactly.  This preservation avoids the formation of numerically-induced Hadamard instabilities (thus offering exceptional stability to the numerical calculations) and allows always the obtainment of physically meaningful results under all flow conditions.  These numerical capabilities can prove essential for the understanding of turbulence modification through polymer additives that can potentially lead to the development of better drag reducers.

Joint CSCAMM/Norbert Wiener Center Seminar

Anna Gilbert, Department of Mathematics, University of Michigan

The interplay of analysis and algorithms

It has recently been observed that sparse and compressible signals can be sketched using very few nonadaptive linear measurements in comparison with the length of the signal. This sketch can be viewed as an embedding of an entire class of compressible signals into a low-dimensional space. In particular, d-dimensional signals with m nonzero entries (m-sparse signals) can be embedded in O(m log d) dimensions. To date, most algorithms for approximating or reconstructing the signal from the sketch, such as the linear programming approach proposed by Cand`es–Tao and Donoho, require time polynomial in the signal length. I will talk about new methods for sketching both m-sparse and compressible signals and novel signal recovery algorithms. I will also talk about the types of statistical information which can be recovered from these sketches and how sparse representations are obtained more generally.
 

I will discuss a robust, coherent interferometric approach for array imaging in cluttered media, in regimes with significant multipathing of the waves by the inhomogeneities in clutter. In such scattering regimes, the recorded traces at the array have long and noisy codas and classic imaging methods give unstable results. Coherent interferometry is essentially a very efficient statistical smoothing technique that exploits systematically the spatial and temporal coherence in the data to obtain stable images. I will show that in coherent interferometry, there is a delicate balance between having stable and sharp images and achieving the optimal resolution. This balance depends on two clutter dependent decoherence parameters. I will explain briefly how we can estimate these parameters efficiently during the image formation process. The robustness of the proposed imaging method will be illustrated with several results.

The rate of Fourier approximation of a given function is determined by its regularity. For functions with singularities, even very simple, like the Heaviside step function, the convergence of the Fourier series is slow, and their reconstruction from the truncated Fourier data involves systematic errors (“Gibbs effect”).

It was recently discovered in the work of D. Donoho, E. Candes and others that an accurate reconstruction from the sparse measurements data (in particular, from the truncated Fourier data) is possible not only for regular functions, but rather for “compressible” ones - those possessing a sparse representation in a certain basis.

In many important applications (like Image Processing) the linear representation of the data in a certain fixed basis may be not the most natural starting point. Instead, we can approximate the data with geometric models, explicitly incorporating nonlinear geometric elements like edges, ridges, etc.

Accordingly, instead of “linear sparseness” we use another notion of “simplicity”, based on the rate of the best approximation of a given function by semialgebraic functions of a prescribed degree.

The main subject of the present talk is that such ”simple” functions can be accurately reconstructed (by a non-linear inversion) from their truncated Fourier or Moment data.