Date

Speaker

Title and abstract

01/25/2010

Friday

Chi-Wang Shu (Brown)

Host: Irene Gamba

Maximum-principle-satisfying high order schemes for scalar conservation laws and passive convection in incompressible flows

We construct uniformly high order accurate schemes satisfying a strict maximum principle for scalar conservation laws. A general framework (for arbitrary order of accuracy) is established to construct a limiter for finite volume schemes (e.g. essentially non-oscillatory (ENO) or weighted ENO (WENO) schemes) or discontinuous Galerkin (DG) method with first order Euler forward time discretization solving one dimensional scalar conservation laws. Strong stability preserving (SSP) high order time discretizations will keep the maximum principle and make the scheme uniformly high order in space and time. One remarkable property of this approach is that it is straightforward to extend the method to two and higher dimensions. The same limiter can be shown to preserve the maximum principle for DG or finite volume schemes solving two-dimensional incompressible Euler equations in the vorticity stream-function formulation, or any passive convection equation with an incompressible velocity field. Numerical tests for both the WENO finite volume scheme and the DG method are reported. This is a joint work with Xiangxiong Zhang.

01/25/2010

Monday

3:00PM

Xuemin Tu (UC Berkeley)

Host: Bjorn Engquist

Implicit sampling for nonlinear filters

Applications of filtering and data assimilation arise in engineering, geosciences, weather forecasting, and many other areas where one has to make predictions based on uncertain models supplemented by a stream of data with noise. For nonlinear problems filtering can be very expensive.  In this talk, a particle-based nonlinear filtering scheme will be presented. This algorithm is based on implicit sampling, a new sampling technique related to chainless Monte Carlo method. Its main features are that the posterior densities are represented by pseudo-Gaussians and a resampling based on normalization constants. This  filter is designed to focus particle paths sharply so as to reduce the number of particles needed for nonlinear problems. Examples will be given.

01/29/2010

Friday

Armando Majorana (University of Catania, Italy)

Host: Irene Gamba

Deterministic and stochastic description of electron flow in semiconductor nano-devices

To model carrier transport in semiconductor devices different techniques, from macroscopic to microscopic scale, have been developed and applied. After an elementary introduction on the main features of molecular dynamics, stochastic and deterministic models for electron transport in semiconductors are described and compared. Numerical schemes for deterministic models, based on the Boltzmann kinetic equation, are depicted and results are shown for both models in the case of Silicon and Gallium Nitride semiconductors.

02/05/2010

Friday

Daniel Onofrei (Utah)

Host: Kui Ren

Cloaking: mathematical results and challenges

Cloaking attracted a lot of interest in the recent years. In the first part of this talk I will briefly introduce the main ideas
behind the transformation based cloaking scheme and discuss about the error estimates associated to its regularization. This will answer to the question of mathematical and physical feasibility of such a scheme. The second part of my talk will be devoted to the new idea of active exterior cloaking. I will highlight the advantages of this scheme and discuss the
mathematical ideas behind it as well as the numerical realization of this cloak in the acoustic regime.
 

02/12/2010

Friday

Jianfeng Lu (Courant Institute, NYU)

Host: Lexing Ying

Electron dynamics in crystals under perturbation

We are going to discuss two results about effective electron dynamics in crystals. For many electrons coupled by Coulomb interaction, we derive an effective dynamical dielectric response when the applied perturbation is macroscopic in space and high frequency in time. For independent electron in a crystal with external macroscopic potential and magnetic perturbation, we give a simple WKB-asymptotics based proof for the effective semiclassical dynamics, including the effects of Berry curvature. The talk is based on joint works with Weinan E and Xu Yang.

02/15/2010

Monday

ACE 4.304

Lin Lin (Princeton)

Host: Lexing Ying

Title: Selected inversion: with application to electronic structure calculation

Abstract: In this talk we are going to discuss the selected inversion technique. This technique allows one to compute selected components of the inverse of a sparse matrix with optimal usage of the sparsity pattern. We will talk about an efficient parallel implementation of the algorithm, as well as application to electronic structure calculation of 2D and 3D metallic systems.

02/16/2010

Tuesday

3:30PM

Fengyan Li (RPI)

Host: Irene Gamba

Discontinuous Galerkin Methods for Hamilton-Jacobi Equations

Hamilton-Jacobi (H-J) equations provide important mathematical models for many applications. The solutions of such equations may develop discontinuous derivatives even for the smooth initial and boundary data. And the concept of viscosity solution was introduced in the early 1980s. In this talk, I will present our recent work in developing high order
numerical methods for H-J equations.

One part of the talk concerns the design of discontinuous Galerkin (DG) methods for directly solving time-dependent H-J equations. The main difficulty comes from the fact that these equations in general are not in the divergence form. By recognizing and following a weighted-residual or stabilization-based formulation of central DG methods when applied to
hyperbolic conservation laws, a central DG method is designed for H-J equations.   Though the stability and the error estimate are established only for linear cases, the high order accuracy and reliability of the method in approximating the viscosity solutions are demonstrated through general numerical examples. This work is jointly done with S. Yakovlev (RPI).

The other part of the talk focuses on the development of efficient high order methods for static H-J equations. In particular, second order DG-based fast sweeping methods are proposed for one family of H-J equations - Eikonal equations. These iterative methods demonstrate the linear computational complexity, namely, the number of iterations for the convergence is independent of the number of total unknowns.  Besides its accuracy, DG discretization is chosen more for its compactness, which is important for the overall efficiency of the algorithms. This work is collaborated with S. Chen (Indiana U., South Bend), C.-W. Shu (Brown U.), Y.-T. Zhang (Notre Dame) and H.-K. Zhao (UC Irvine).

02/19/2010

Friday

ACE 4.304

Wei Cai (UNC Charlotte)

Host: Lexing Ying

High order and Multi-physics Numerical Methods for surface plasmon polaritons

Surface plasmon polariton (SPP) is an electronic excitation at a metal surface which involves collective motions of electron gas in a metallic material. Understanding and simulation of SPP is important for the calculation of optical properties of metallic materials to external electromagnetic fields for applications such as surface enhanced Raman scattering and near field optics and optical circuits. In this talk, we will first review the basic physical concept of SPPs and some classical models. Then, we will present two numerical methods for simulating plasmons: (1) A dispersive high order discontinuous Galerkin (DG) method, (2) A density functional theory-Maxwell equation coupling method. The dispersive DG methods will be used to simulate the plasmon resonant phenomena of coupled silver nanowires for optical circuit applications. The multi-physics method coupling the density functional theory and Maxwell equations will be used to study the many body quantum effects in the optical responses of SPPs. Under the Thomas-Fermi DFT model, we will derive the Hamilton-Jacobi equations for the electron density and hydrodynamic velocity potential coupled with the Maxwell equations. Numerical issues and boundary conditions are then discussed. In addition to linear analysis of the multi-physics coupling
model, numerical simulation of SPPs  will be discussed.

02/26/2010

Friday

ACE 4.304

Ronny Hadani (Austin)

Host: Lexing Ying

Representation theoretic patterns in three dimensional cryo-electron microscopy

Three dimensional cryo-electron microscopy (3D cryo-EM, for short) is the problem of determining the three dimensional structure of a large molecule from the set of images, taken by an electron microscope, of randomly oriented and positioned identical molecular particles which are frozen in a thin layer of ice. A solution to this problem is of particular interest, since it promises to be an entirely general technique which does not require crystallization or other special preparation stages. Present approaches to the problem fail with particles that are too small, cryo-EM images that are too noisy or at resolutions where the signal-to-noise ratio becomes too small.

In my talk, I will describe a novel algorithm, referred to as the intrinsic reconstitution algorithm, due to Amit Singer and Yoel Shkolnisky, which constitutes a basic step for the solution of the 3D cryo-EM problem. The appealing property of this new algorithm is that it exhibits remarkable numerical stability to noise. My main goal is to give a conceptual explanation, based on representation theory, for the admissibility (correctness) and the numerical stability of the intrinsic reconstitution algorithm. If time permits, I will mention some recent results concerning more elaborate aspects of the cryo-EM problem. This work is joint with Amit Singer (Princeton) and is a part of an ongoing project conducted jointly with Shamgar Gurevich (IAS), Yoel Shkolnisky (Tel-Aviv University) and Fred Sigworth (Yale).

03/12/2010

Friday

Bill Symes (Rice)

Host: Lexing Ying

Source synthesis for inverse problems in wave propagation

Inverse problems in wave propagation can provide apparently redundant data. For example, active source seismology is a source of such problems, and seismologists have long experimented with synthesis of the earth's response to extended or nonphysical energy sources, to enhance the effectiveness of data acquisition and processing in various ways. Recently, extended synthetic sources have been suggested as a means to to substantially reduce the computational complexity of least squares data-fitting (``waveform inversion''). Most proposed synthetic sources for waveform inversion are random in some way. In contrast, I suggest a deterministic choice of extended source, one that maximizes the difference between the data predicted by the current inversion iterate and the response synthesized from redundant target data. Optimal source synthesis in this sense has proven effective in several non-seismic inverse problems. I will describe the concept, and present some preliminary evidence concerning its performance in inversion of reflection seismograms.

03/19/2010

Friday

Sping Break

03/26/2010

Friday

Guillaume Bal (Columbia)

Host: Kui Ren

Inverse Transport Problems and Photoacoustics.

Inverse transport consists of reconstructing the optical parameters in a transport equation from knowledge of a measurement operator. We review several uniqueness and stability results obtained in the context of various boundary measurements as they arise e.g. in optical tomography, a medical imaging modality. Accurate numerical reconstructions
obtained  by carefully capturing the singularities of the measurement operator are also briefly presented. In many practical settings, the typical boundary measurements available in inverse transport do not allow us to reconstruct the optical parameters with sufficient resolution. In the second part of the talk, I present recent results obtained for the inverse transport problem with internal controls as they arise in the application of photoacoustic tomography, a recent hybrid medical imaging modality that combines the large contrast observed in optical parameters with the high resolution of ultrasounds.

These are joint works with Alexandre Jollivet, Francois Monard, and Gunther Uhlmann

03/30/2010

Tuesday

Jeff Haack (Wisconsin)

Host: Irene Gamba

An all-speed asymptotic-preserving scheme for the low Mach number limit of the isentropic Euler and Navier-Stokes equations

In this talk, I will present an asymptotic preserving (AP) numerical scheme for the low Mach number limit of the compressible isentropic Euler and Navier-Stokes equations that is uniformly stable and accurate for all Mach numbers. Numerical computation in this limit is challenging due to the expensive time and space resolution constraints needed for the stiff acoustic waves in the system, which are not important in the limit. We propose splitting the system into two parts, one of which is a linear system that contains the stiff acoustic dynamics. This implicit system automatically turns into an incompressible projection step when the Mach number is small, capturing the correct incompressible dynamics.  I will present numerical results in one and two dimensions supporting these results.

04/02/2010

Friday

Maria Cameron (NYU)

Host: Sergey Fomel

Computing transition paths for rare events

The overdamped Langevin equation is often used as a model in molecular dynamics.  At low temperatures, a system evolving according to such an SDE spends most of the time near the potential minima and performs rare transitions between them.  A number of methods have been developed to study the most likely transition paths. I will focus on one of them: the MaxFlux functional. The MaxFlux functional has been around for almost thirty years but not widely used because it is challenging to minimize. Its minimizer provides a path along  which the reactive flux is maximal at a given
finite temperature.  I will show two ways to derive it in the framework of transition path theory: the lower bound approach and the geometrical approach. I will  present an efficient way to  minimize the MaxFlux functional numerically.  I will demonstrate its application to the problem  of finding the most likely transition paths in the Lennard-Jones-38 cluster between the  face-centered-cubic and icosahedral structures.
 

04/07/2010

Wednesday

Jack Xin (UC Irvine)

Host: Irene Gamba

Blind source separation methods and applications

Blind source separation refers to recovering source signals from their mixtures with no knowledge of mixing environment. The source separation criteria can be based on either statistics or features of source signals. The statistical methods rely on independence assumption of source signals from which a non-convex optimization problem is formulated. A soft-constrained nonlocally weighted dynamic iterative method will be introduced for improved consistency and stability,
and robust performance in adverse conditions. The feature based methods rely on sparseness of source signals in natural
or transformed coordinates. The methods do not perform well when sparseness condition is violated. We show that proper post-processing utilizing multiple data samples and coherence of source signals allows one to remove the errors resulting from lack of sparseness. We illustrate our methods for recorded sounds in reverberant rooms and spectroscopy data of organic molecules.

04/09/2010

Friday

Shi Jin (Wisconsin)

Host: Lexing Ying

Eulerian computational methods in quantum dynamics

In this talk I will present our recent Eulerian quantum-classical coupling computational methods for quantum tunneling, diffraction, and surface hopping. These methods are based on classical Liouville equations, with interface conditions that account for quantum scattering or transition.

04/14/2010

Wednesday

Francis Filbet (Université Claude Bernard, Lyon)

Host: Irene Gamba

Analysis and numerical simulations to the Boltzmann equation - Multi-scale Problems

We propose a general time discrete framework to design asymptotic preserving schemes for initial value problem of the Boltzmann kinetic and related equations. Numerically solving these equations are challenging due to the nonlinear stiff collision (source) terms induced by small mean free or relaxation time. We propose to penalize the nonlinear collision term by a BGK-type relaxation term, which can be solved explicitly even if discretized implicitly in time. Moreover, the BGK-type relaxation operator helps to drive the density distribution toward the local Maxwellian, thus naturally imposes an asymptotic-preserving scheme in the Euler limit. The scheme so designed does not need any nonlinear iterative solver or the use of Wild Sum. It is uniformly stable in terms of the (possibly small) Knudsen number, and can capture the macroscopic fluid dynamic (Euler) limit even if the small scale determined by the Knudsen number is not numerically resolved. It is also consistent to the compressible Navier-Stokes equations if the viscosity and heat conductivity are numerically resolved. The method is applicable to many other related problems, such as hyperbolic systems with stiff relaxation, and high order parabolic equations.  Work in collaboration with C. Mouhot and S. Jin.

04/23/2010

Friday

 Alex Vladimirsky (Cornell)

Host: Lexing Ying

Homogenization and randomly-terminated optimal control -- computational challenges


I will present two recent projects related to front propagation & optimal control.

The first of these (joint with A. Oberman and R. Takei) deals with 2-scale and 3-scale computations in geometric optics. We propose a new & efficient method to homogenize first-order Hamilton.Jacobi PDEs. Unlike the prior cell-problem methods, our algorithm is based on homogenizing the related geodesic distance function. We illustrate by computing the effective velocity profiles for a number of periodic and "random" composite materials.

The second project (joint work with J. Andrews) deals with deterministic optimal control of processes with probabilistically specified fixed-horizon.  Subject to additional technical assumptions on cost & dynamics, this problem can be converted to an infinite-horizon obstacle problem.  Despite the occurrence of non-trivial free boundary, we show that causal numerical algorithms (e.g., Fast Marching, Ordered Upwind) are still applicable.  We illustrate our method using examples from optimal idle-time processing.

04/30/2010

Friday

Liliana Borcea (Rice)

Host: Kui Ren 

Detection and imaging with waves in heterogeneous, strongly backscattering media

Objects that are buried deep in heterogeneous media produce faint echoes which are difficult to distinguish from the backscattered waves. Sensor array imaging in such media cannot work unless we filter out the backscattered echoes and enhance the coherent arrivals that carry information about the objects that we wish to image. We study such filters for imaging in strongly backscattering, random media.  I will begin with a brief review of such filters for finely layered media, based on travel time transformation of the array data, the normal move-out used frequently in seismic imaging. Then, I will present a new approach that is based on the spectral decomposition of the scattering matrix in time windows that are to be selected as part of the problem.  This new approach applies to a large class of random media, but I will present the theory only in layered media.

05/06/2010

Thursday

ACE 4.304

Albert Fannjiang (UC Davis)

Host: Kui Ren 

Compressive Imaging by Sparse Measurements

We examine inverse scattering problem from the perspective of
compressed sensing theory. A key to achieving compressive imaging is the design of sampling schemes. We demonstrate several sampling schemes that have a guaranteed performance with sparse measurement.