Abstracts

Plenary Speakers

Invited Speakers

Abstracts

Title:
Multiscale computations for flow and transport in heterogeneous porous media

Speaker: Tom Hou, CalTech

Abstract:
Many problems of fundamental and practical importance contain multiple scale solutions. Composite materials, flow and transport in heterogeneous porous media, and turbulent flow are examples of this type. Direct numerical simulations of these multiscale problems are extremely difficult due to the wide range of length scales in the underlying physical problems. In this talk, I describe some of our recent efforts in developing multiscale computational methods to upscale two-phase flows in strongly heterogeneous porous media. For some challenging problems with long range scale interaction arising from engineering applications, we show how to use limited global information to improve the accuracy of the multiscale method. Another important application is how to quantify uncertainty in modeling the heterogeneous random media. We show that by using a coarse multiscale model to precondition the Markov Chain Monte Carlo method, we can significantly improve the efficiency of the MCMC method in generating the probability distribution of the random media subject to some production data. Finally, we introduce a multiscale method for convection dominated incompressible flow with multiscale solutions.

------------------------------------------------------
Title: Coherent semiclassical transport models for thin quantum barriers

Speaker: Shi Jin, University of Wisconsin-Madison

Abstract:
We present time-dependent semiclassical transport models for mixed state scattering with thin quantum barriers. The idea is to use a multiscale approach to connect regions for which a classical description of the system dynamics is valid across regions for which the classical description fails, such as when the gradient of the potential is undefined. We do this by first solving a stationary Schrodinger equation in the quantum region to obtain the scattering coefficients. These coefficients allow us to build the interface condition to the particle flux, that bridges the quantum region, connecting two classical regions. Away from the barrier, the problem may be solved by traditional numerical methods. The overall numerical cost is roughly the same as solving a classical barrier.

By using quantum scattering data and complex Liouville equations we are even able to handle wave interferences across the barrier.

This is a joint work with Kyle Novak.

------------------------------------------------------
Title: Shape Optimization for Elliptic Eigenvalue Problems

Speaker: Chiu-Yen Kao, Department of Mathematics, Ohio State University

Abstract:
Identification or optimization of shapes arises in many science and engineering applications. In this talk, we focus on the optimal shape design related to elliptic eigenvalue problems. Specific applications for identifying structures of photonic crystal, optimization of quality factor of an acoustic resonator, and determining the optimal spatial arrangement of favorable and unfavorable regions for a species to survive will be discussed.

------------------------------------------------------
Title: Shock Wave Propagation in Tissue and Bone

Speaker: Prof. Randall J. LeVeque,
Department of Applied Mathematics, University of Washington

Abstract:
Studying the physical and biological mechanisms of extracorporeal
shock wave therapy (ESWT) requires modeling the propagation of strong shock waves through tissue and bone. Interfaces between different biological materials lead to reflections and focusing of shock waves and the creation of strong rarefaction zones and cavitation fields. I will discuss recent numerical work using high-resolution finite volume methods in which each grid cell is allowed to have distinct material properties. Sharp interfaces either occur at cell edges (if an appropriate geometry-conforming grid can be obtained) or are represented by averaging the material properties over grid cells on a Cartesian grid. In either case, logically rectangular grids with adaptive mesh refinement are used to efficiently deal with multiscale problems where the medium has heterogeneities at various length scales.

------------------------------------------------------

Title: The Ebb & Flow of Multi-scale Modeling at The Dow Chemical
Company

Speaker: Dr. P. Kip Mercure,
Engineering & Process Sciences,
The Dow Chemical Company

Abstract:
Multi-scale modeling is an ongoing effort at The Dow Chemical Company, and it has had periods of increasing and decreasing utility. Three major areas of application will be described: polymer property prediction, meta-material design, and process modeling. The systems described must bridge spatial scales from atomic dimensions to the inside diameters of pipes, and time scales from control system actions in seconds to plant time constants of days. For each area, the complexity tradeoffs for effective use will be described. The utility of multi-scale modeling in industry has changed over time as various techniques demonstrate commercial utility or not. Some of the drivers for commercial interest will be described in the context of present and possible future products.

------------------------------------------------------
Title: New Algorithms in Information Science

Speaker: Stan Osher, Department of Mathematics, UCLA.

Abstract:
The past few years have seen an incredible explosion of new (or revival of old) fast and effective algorithms for various imaging and information science applications. These include: nonlocal means, compressive sensing, graph cuts, Bregman iteration, as well as relatively old favorites such as the level set method and PDE based image restoration. I'll give my view of where we are, hopefully giving credit to all the creators of these new and exciting multiscale techniques.

------------------------------------------------------
Title: Applications of Gaussian Beams

Speaker: James Ralston, Department of Mathematics, UCLA

Abstract:
Gaussian beams are asymptotic solutions to high frequency wave propagation problems that are localized near a single ray path in space-time. Superpositions of these beams are useful in approximating the behavior of solutions near places where rays focus such as caustics. I will discuss this method and some recent applications.

------------------------------------------------------
Title: Discontinuous Galerkin Methods for MHD, Two-Fluid Plasma, and Beyond

Speaker: James Rossmanith,
Department of Mathematics,
University of Wisconsin - Madison

Abstract:
The solar wind can be largely viewed as a collisionless plasma, and in most circumstances, it can be modeled using the magnetohydrodynamic (MHD) equations. After briefly describing this model, we will present a discontinuous Galerkin (DG) method for numerically solving it. An important aspect of this work is the treatment of the divergence-free condition on the magnetic field. We will then show how the MHD equations can fail to produce physically correct results in the case of magnetic reconnection. A more accurate physical model, the two-fluid system, is then introduced. We will argue that the computational expense of solving the two-fluid equations is significantly higher than that of MHD. Finally, we will describe efforts to develop a multiscale method for coupling the MHD and two-fluid models.

------------------------------------------------------
Title: A New Spectral-Galerkin Method for High-Dimensional PDEs:
Algorithms, Analysis and Applications

Speaker: Jie Shen, Department of Mathematics, Purdue University

Abstract:
Many scientific, engineering and financial applications require solving high-dimensional PDEs. However, traditional tensor product based algorithms suffer from the so called "curse of dimensionality". We shall present a new Chebyshev-Galerkin method for non-periodic problems and/or in the whole space. The method is based on two basic ingredients: (i) Choosing the frequencies of the trial functions from the "hyperbolic cross"; (ii) Using a lattice rule or sparse grid to perform the numerical integration. It is shown that with this combination, the "curse of dimensionality" can be broken to some extent. We shall present rigorous error estimates and numerical results supporting this statement.

------------------------------------------------------
Title: Multiscale Methods Based on the Discontinuous Galerkin Scheme

Speaker: Chi-Wang Shu, Division of Applied Mathematics, Brown University

Abstract:
We present two multiscale methods based on the discontinuous Galerkin finite element scheme. The first method uses the heterogeneous multiscale method (HMM) framework, for the numerical simulation of dynamics of crystalline solids. The method couples nonlinear elastodynamics as the continuum description and molecular dynamics as another component at the atomic scale. The governing equations on the macroscale are solved by the discontinuous Galerkin method, which is built up with an appropriate local curl-free space to produce coherent displacement field. The constitutive data are based on the underlying atomistic model: it is either calibrated prior to the computation or obtained from molecular dynamics as the computation proceeds. The decision to use either the former or the latter is made locally for each cell based on suitable criteria. This part is a joint work with Wei Wang and Xiantao Li. The second method uses non-polynomial approximation spaces based on WKB asymptotics, to simulate the one-dimensional stationary Schrodinger-Poisson problem. The stationary Schrodinger equation is discretized by the WKB local discontinuous Galerkin (WKB-LDG) method, and the Poisson potential equation is discretized by the minimal dissipation LDG (MD-LDG) method. The WKB-LDG method provides a significant reduction of both the computational cost and memory in solving the Schrodinger equation. Comparing with traditional continuous finite element Galerkin methodology, the WKB-LDG method has the advantages of the DG methods including their flexibility in h-p adaptivity and allowance of complete discontinuity at element interfaces. A major advantage of the WKB-LDG method is its potential for two-dimensional devices.

This part is a joint work with Wei Wang.

------------------------------------------------------

Title: Rapid Computation of Elastic Energy Differences with Application to Heteropitaxial Growth

Speaker: Prof. Peter Smereka, University of Michigan

Abstract:
When simulating heteropitaxial growth using kinetic Monte Carlo one needs to compute the elastic energy difference between two similar configurations a vast number of times. This talk will describe several tools to allow one to accomplish this efficiently. A Fourier-multigrid method which allows one rapidly compute the displacement field for an arbitrary film profile coupled to semi-infinite substrate will be discussed. Next, the principle of energy localization will be stated which combined with the expanding box method allows one to accurately compute changes elastic energy using local calculations. Finally some results showing the formation of self-assembled stacked quantum dots will be shown.

This is joint work with Giovanni Russo and Tim Schulze.

------------------------------------------------------
Title : Diffractive optical properties of metallic micro-structures

Speaker: Faouzi Triki, Universite Joseph Fourier, France

Abstract :
In the talk I will present a study of the electromagnetic diffraction by a perfectly conducting planar interface which contains micro-cavities. The goal is to understand how cavities smaller than the wavelength contribute to the enhancement and confinement of electromagnetic fields. The approach is based on an asymptotic analysis of integral equations.

------------------------------------------------------
Title: Electromagnetic Particle-In-Cell simulations of plasmas with (Adaptive) Mesh Refinement. Reduction of the range of scale in a Lorentz boosted frame.

Speaker: Dr. Jean-Luc Vay, Lawrence Berkeley National Lab.

Abstract:
Modeling of systems that involve a wide range of scales in space and/or time is challenging. The Adaptive Mesh Refinement technique has been applied successfully to various fields but its applicability to the modeling of wave propagation has proven difficult. We will discuss the issues in the context of electromagnetic Particle-In_Cell simulation of plasmas, such as spurious reflection of waves or self-forces at or near refinement interfaces. Solutions and examples of applications will be presented. For the interaction of relativistic species, we have discovered that the range of space and time scales spanned by the entire system can be reduced by orders of magnitude if choosing the right Lorentz boosted frame as the frame of calculation. We will present the finding and its consequences on the modeling of relativistic systems, and illustrate by a few examples of application.

------------------------------------------------------
Title: Multiscale Theory of Fluctuating Interfaces

Speaker: Christoph A. Haselwandter and Dimitri D. Vvedensky,
The Blackett Laboratory, Imperial College, London SW7 2BZ

Abstract:
We have developed a general methodology for the multiscale analysis of atomistic models of fluctuating interfaces. Beginning with an exact analytic formulation of kinetic Monte Carlo simulations in the form of a lattice Langevin equation, we derive stochastic partial differential equations for the smoothed lattice models by regularizing the transition rules. Subsequent coarse-graining is accomplished by calculating renormalization-group (RG) trajectories from initial conditions determined by the regularized atomistic models.

For homoepitaxial growth we derive a general microscopic continuum equation that is applicable to a wide class of lattice models. The RG analysis of this equation shows that the morphological manifestation of a given atomistic relaxation mechanism can depend qualitatively on the length and time scales considered as well as on the dimensionality of the fluctuating interface. Moreover, our analytic theory allows the systematic study of the interplay between different atomistic processes for general experimental input parameters.

We have also used the above ideas to describe the self-organization of heteroepitaxial nanostructures. Our method produces an equation similar to that obtained by Golovin et al. from classical elasticity. But there are crucial differences between our equation and this earlier work. Most important is that the coefficients in our Langevin equation have a direct relation to the underlying atomistic processes. Since the transition rates of these processes can be calculated with density functional methods, we have the basis for a genuine multiscale description of heteroepitaxial morphological evolution. Another important difference is the presence of noise terms that reflect the randomness of the deposition and diffusion processes. These are central for the quantitative description of the morphological evolution observed in experiments. On the other hand, Golovin et al. explicitly include the effect of a wetting layer in their formulation. This issue remains for further investigation.

------------------------------------------------------
Title: Resonance Problems in Linear and Nonlinear Photonics

Speaker: Michael I. Weinstein, Columbia University

------------------------------------------------------
Title: Grid Based Particle Method For Interface Problems

Speaker: Hong-Kai Zhao, Department of Mathematics, UC Irvine

Abstract:
We develop a particle method based on Eulerian grid for moving interface problem. In this framework the interface is represented by unconnected particles and each particle is associated with an underlying grid point for Eulerian reference. The underlying grid provides a quasi-uniform sampling and neighborhood information among particles on the interface. The unconnected particles provide an easy and accurate way to track the motion of the moving interface. This formulation avoids the requirement for a nice global parametrization of the interface and can deal with topological change with control. Adaptivity can also be implemented easily. Extensive moving interface problems will be tested.

------------------------------------------------------
Title: Some recent advances on identifiability in inverse obstacle scatterings

Speaker: Jun Zou, Dept of Mathematics, The Chinese University of Hong Kong

Abstract:
In this talk we shall review some recent advances on identifiability in inverse acoustic and electromagnetic obstacle scattering problems. The work was fully supported by Hong Kong RGC grants (Projects 404105 and 404606).

 













Contact Us

© 2008 Michigan State University
DECS - Division of Engineering Computing Services