Date

Speaker

Title and abstract

01/18/11

Tuesday

3:30-5:00PM

ACE 6.304

Cory Hauck

(Oak Ridge National Lab)

Host: Irene Gamba

A Collision-Based Hybrid Method for Linear Transport

We present a hybrid method for simulating kinetic equations with multiscale phenomena in the context of linear transport. The method consists of (i) partitioning the kinetic equation into collisional and non-collisional components; (ii) applying a different numerical method to each component; and (iii) re-partitioning the kinetic distribution after each time step in the algorithm. Preliminary results show that, for a wide range of test problems, the combination of a low-order method for the collisional component and a high-order method for the non-collisional component provides a level of accuracy that is comparable to a uniform high-order treatment of the entire system. This work is joint with Ryan McClarren, Texas A&M University.

01/21/11

Friday

3:00-4:00PM

RLM 11.176

Gil Ariel

(Bar Ilan University)

Host: Richard Tsai

Deterministic and stochastic limits of three scale ODE systems

Averaging methods for ordinary differential equations is a well developed and widely used tool for the analysis of the effective behavior of ODEs with two well separated time scales. When three time scales are involved the theory is not as well developed and many questions are still open. In particular, it has been demonstrated that under some conditions the effective behavior of 3-scale ODEs may be modeled by a limiting stochastic process. However, rigorous analysis of such models has only been established in a few particular cases, for example, discrete rapidly mixing maps and the Lorentz attractor. The talk will review some methods and results of multiscale analysis of 3-scale systems and present a new theorem of iterated averaging for oscillatory ODEs.

02/11/11

Friday

1:00-2:00PM

ACE 6.304

Jingwei Hu

(U Wisconsin)

Host: Sergey Fomel

& Lexing Ying

A Numerical Scheme for Quantum Boltzmann Equation Efficient in Fluid Regime

Numerically solving the Boltzmann kinetic equations with the small Knudsen number is challenging due to the stiff nonlinear collision term. A class of asymptotic preserving schemes was introduced in [J. Comput. Phys., 229, 2010] to handle this kind of problems. The idea is to penalize the stiff collision term by a BGK type operator. This method, when applied to the quantum Boltzmann equation, is not practically effective due to the complexity of the quantum
Maxwellians. The new contribution we make is to use a `classical' BGK operator instead of the quantum one, and achieve a scheme that well captures the fluid limit and avoids computing the complicated Bose-Einstein or Fermi-Dirac distribution. A spectral method for the quantum collision operator is also discussed.

02/11/11

Friday

3:00-4:00PM

ACE 6.304

Lexing Ying

(UT Austin)

Host: Kui Ren

Fast algorithms for oscillatory kernels

Computations involving oscillatory kernels arise in many computational problems associated with high frequency wave phenomena. In this talk, we will discuss recent progress on developing fast linear complexity algorithms for several problems of this type. Two common ingredients of these algorithms are discovering new structures with low-rank property and developing new hierarchical decompositions based on these structures. Examples will include N-body problems of the Helmholtz kernel, sparse Fourier transforms, Fourier integral operators, and fast Helmholtz solvers.

02/25/11

Friday

3:00-4:00PM

ACE 6.304

Pierre Degond

(CNRS, France)

Host: Irene Gamba

Current challenges in pedestrian dynamics and crowd modeling

Understanding Pedestrian and crowd dynamics is an important issue in modern societies. Recent crowd disasters (e.g. Duisburg love parade (July 2010) or Cambodia water festival (November 2010)) have put the safety issues in the forward scene. Another important issue the is efficiency of public arenas such as subway corridors, train or airport terminals, or shopping malls. Many different kinds of models of pedestrian and crowd dynamics exist, from the discrete (individual-based) ones to the continuum ones. The talk will review the sallient features of these models. However, most of the existing models suffer from a poor validation against experimental data. There are many difficulties ranging from ethical to technical ones which explain why the data are scarce. We will report on a series of experiments that have been conducted in the 'Pedigree' project in France and the preliminary findings that have resulted from these data. Future modeling directions and challenges will be discussed.

03/03/11

Thursday

3:30-5:00PM

ACE 6.304

Markos Katsoulakis

(UMass Amherst)

Host: Irene Gamba

Accelerated Kinetic Monte Carlo methods: hierarchical parallel algorithms and coarse-graining

In this talk we present two intimately related approaches in speeding-up molecular simulations via Monte Carlo simulations. First, we discuss coarse-graining algorithms for systems with complex, and often competing particle interactions, both in the equilibrium and non-equilibrium settings, which rely on multilevel sampling and communication. Second, we address mathematical, numerical and algorithmic issues arising in the parallelization of spatially distributed Kinetic Monte Carlo simulations, by developing a new hierarchical operator splitting of the underlying high-dimensional generator, as means of decomposing efficiently and systematically the computational load and communication between multiple processors. The common theme in both methods is the desire to identify and decompose the particle system in components that communicate minimally and thus local information can be either described by suitable coarse-variables (coarse-graining), or computed locally on a individual processors within a parallel architecture.

03/11/11

Friday

3:00-4:00PM

ACE 6.304

Jianfeng Lu

(Courant, NYU)

Host: Lexing Ying

Multiscale analysis of solids

The multiscale methods for solid mechanics of materials receive intense investigations in recent years. The main spirit of these methods is to couple together models at different physical scales in order to achieve efficient and accurate description of the material. In this talk, we will discuss some recent progress in mathematical understandings of the connection and coupling between microscopic models like density functional theory and atomistic empirical potential models with macroscopic continuum theory.

03/18/11

No seminar scheduled due to Spring Break

03/25/11

Friday

3:00-4:00PM

ACE 6.304

Ioana Dumitriu

(University of Washington)

Host: Lexing Ying

Random Matrices in Numerical Linear Algebra

The relationship between random matrix theory and numerical linear algebra now spans more than three decades, and it is ever growing and expanding. From analyzing the condition number of a "typical" matrix (Demmel '88, Edelman '89), to understanding why certain exponential worst-case algorithms behave very well in practice (Spielman-Teng'01), and from approximating low-rank matrices (Liberty-Woolfe-Martinsson-Rokhlin-Tygert '07, Halko-Martinsson-Tropp '09) to building communication-avoiding algorithms for eigenvalue computations (Ballard-Demmel-Dumitriu '10), random matrix results (along with actual random matrices) have been used to understand, speed up, and even stabilize numerical linear algebra algorithms. We will survey some of the work mentioned above, and present some recent contributions as well as avenues for future work.

03/29/11

Tuesday

3:30-5:00PM

ACE 6.304

Zhiming Chen

(Chinese Academy of Sciences)

Host: Lexing Ying

An adaptive anisotropic perfectly matched layer method for 3-D time harmonic electromagnetic scattering problems

We develop an anisotropic perfectly matched layer (PML) method for solving the time harmonic electromagnetic scattering problems in which the PML coordinate stretching is performed only in one direction outside a cuboid domain. The PML parameters such as the thickness of the layer and the absorbing medium property are determined through sharp a posteriori error estimates. Combined with the adaptive finite element method, the proposed adaptive anisotropic PML method provides a complete numerical strategy to solve the scattering problem in the framework of FEM which produces automatically a coarse mesh size away from the fixed domain and thus makes the total computational costs insensitive to the choice of the thickness of the PML layer. Numerical experiments are included to illustrate the competitive behavior of the proposed adaptive method. This is a joint work with Tao Cui and Linbo Zhang.

04/07/11

Thursday

3:30-5:00PM

ACE 6.304

Chi-Wang Shu

(Brown)

Host: Irene Gamba

Inverse Lax-Wendroff Procedure for Numerical Boundary Conditions of Hyperbolic Equations

We develop a high order finite difference numerical boundary condition for solving hyperbolic Hamilton-Jacobi equations and conservation laws on a Cartesian mesh. The challenge results from the wide stencil of the interior high order scheme and the fact that the boundary intersects the grids in an arbitrary fashion. Our method is based on an inverse Lax-Wendroff procedure for the inflow boundary conditions. We repeatedly use the partial differential equation to write the normal derivatives to the inflow boundary in terms of the tangential derivatives and the time derivatives (for time dependent equations). With these normal derivatives, we can then impose accurate values of ghost points near the boundary by a Taylor expansion. At the outflow boundaries, we use Lagrange extrapolation or least squares extrapolation if the solution is smooth, or a weighted essentially non-oscillatory (WENO) type extrapolation if a shock is close to the boundary. Extensive numerical examples are provided to illustrate that our method is high order accurate and has good performance when applied to one and two dimensional scalar or system cases with the physical boundary not aligned with the grids and with various boundary conditions including the solid wall boundary condition. This is a joint work with Ling Huang and Mengping Zhang (for the Hamilton-Jacobi equations) and with Sirui Tan (for the time dependent conservation laws).

04/14/11

Thursday

3:30-5:00PM

ACE 6.304

Armando Majorana

(University of Catania)

Host: Irene Gamba

An application of the discontinuous Galerkin method to numerical models of kinetic equations

We propose new deterministic numerical models, based on the discontinuous Galerkin (DG) method, for solving kinetic equations as the linear or nonlinear Boltzmann equation for rarefied gases, the Boltzmamn equation for charge transport in semiconductor, the radiative transport equation. The unknown of these equations is the distribution function, which, in general, depends on time, space coordinates and particle velo\-city. We are interested to show a partial application of the DG method, which, in general, will give a set of partial differential equations, where the unknowns are integrals, with respect to the velocity, of the distribution function multiplies assigned test functions and the independent variables are the time and the space coordinates. For instance, a set of partial differential equations is derived and analyzed in the case of the classical nonlinear Boltzmann equation for mono-atomic gases. In this case the model guarantees the conservation of the mass, momentum and energy for homogeneous solutions

References

A. Majorana: A numerical model of the Boltzmann equation related to the discontinuous Galerkin method. Kinetic and Related models, 4, pp 139–151, 2011.

04/15/11

Friday

3:00-4:00PM

ACE 6.304

Anna Gilbert

(University of Michigan)

Host: Allison Moore

Host: Kui Ren

Designing optimal sparse signal recovery systems

A sparse signal recovery system consists of a measurement matrix and a decoding algorithm. Given a signal, the system first acquires (linear) observations of the signal via the measurement matrix and then the decoding algorithm takes those measurements and produces an approximation to the original signal. If the signal is compressible or sparse, then the number of measurements we need is considerably fewer than the length of the signal and we say that we have compressive sensing of a signal. The optimal number of measurements is a function of the sparsity and the log of the signal length. A decoding algorithm that outputs a good approximation to the signal (shorter than the original signal) and does so in time that scales sub-linearly with the signal length is an extremely efficient one. In this talk, I will address the question whether we can design compressive sensing systems that achieve the best of all worlds---extremely efficient (sub-linear time) algorithms and using as few signal measurements as possible. This is joint work with Yi Li, Ely Porat, and Martin Strauss.

04/22/11

Friday

3:00-4:00PM

ACE 6.304

Sujay Sanghavi

(UT Austin)

Host: Kui Ren

A New Approach to Robust and Flexible High-dimensional Statistics

Dimensionality reduction is core to modern statistics and learning; several methods therein are based on structural assumptions on the data. Popular examples of such structural models include sparsity (e.g. in LASSO, Compressed Sensing etc.), rank deficiency (in PCA and derivative methods), sparse Markov structure, etc. This talk introduces the notion of dimensionality reduction via the simultaneous use of more than structural model. Our approach yields methods that are much more widely applicable, and significantly more robust, than existing ones - often with only slightly larger computational complexity. We present new methods, and corresponding analytical results, for (a) PCA in the presence of arbitrary outliers (b) PCA in the presence of grossly corrupted data (c) Robust Collaborative filtering (d) Multiple sparse regression with partially shared sparsity (e) Graph/correlation clustering Our methods are based on convex optimization. The talk aims to be self-contained.

04/29/11

Friday

3:00-4:00PM

ACE 6.304

Charles Jackson

(UT Austin)

Host: Kui Ren

What makes uncertainty quantification in climate prediction so hard?

The problem of estimating uncertainties in climate prediction is not well defined. While one can express its solution within a Bayesian statistical framework, the solution is not necessarily correct. One must confront the scientific issues for how observational data is used to test various hypotheses for the physics of climate. Moreover, one also must confront the computational challenges how estimates the posterior distribution without the help of a statistical emulator of the forward model. I will present results of a recently completed estimate of the uncertainty in specifying 15 parameters important to clouds, convection, and radiation of the Community Atmosphere Model. We learned that the maximum posterior probably is not in the same region of parameter space as the minimum log-likelihood. We have interpreted these differences to the existence of model biases and the potential that the minimum log-likelihood, which are often the desired solutions to data inversion problems, are over-fitting the data. Such a result highlights the need for a combination of scientific and computational thinking to begin to address uncertainties for complex multi-physics phenomena. 

05/02/11

Monday

1:00-2:00PM

RLM 6.118

Fengyan Li

(RPI)

Host: Irene M. Gamba

Exactly divergence-free central DG methods for ideal MHD equations

Ideal MHD system consists of a set of nonlinear hyperbolic conservation laws, with a divergence-free constraint on the magnetic field. Though such constraint holds for the exact solution as long as it does initially, neglecting this condition numerically may lead to instability. In this talk, I will report our recent work in developing high order central DG methods with the exactly divergence-free magnetic field for ideal MHD equations.

05/10/11

Tuesday

3:30-4:30PM

ACE 6.304

Per-Gunnar Martinsson

(Univ. of Colorado Boulder)

Host: Lexing Ying

Fast Direct Solvers for Elliptic PDEs

That the linear systems arising upon the discretization of elliptic PDEs can be solved very rapidly is well-known, and many successful iterative solvers with linear complexity have been constructed (multigrid, Krylov methods, etc). Interestingly, it has recently been demonstrated that it is possible to directly compute an approximate inverse to the coefficient matrix in linear (or close to linear) time. The talk will survey some recent work in the field and demonstrate that direct solvers have several advantages, including improved stability and robustness, and dramatic improvements in speed (in certain environments). Moreover, the direct schemes we propose have very low communication costs, and are better suited to parallel implementations than many previously proposed "fast" solvers.

05/13/11

Friday

3:00-4:00PM

ACE 6.304

Xudong Chen

(National Univ. of Singapore)

Host: Kui Ren

Subspace-based Optimization Methods for Solving Electromagnetic Inverse Scattering Problem

This talk presents a numerical method to solve inverse scattering problems (ISP). The recently proposed subspace-based optimization method (SOM) is found to be effective in solving ISP. The essence of the SOM is that a part of the secondary source is determined from the spectrum analysis without using any optimization, whereas the rest is determined by an optimization method. Since the optimization is carried out in a smaller dimensional space, the algorithm significantly speeds up the convergence. There is a great flexibility in partitioning the space of secondary source into two orthogonally complementary subspaces: the signal subspace and the noise subspace. This flexibility enables the algorithm to perform robustly against noise. On the basis of the SOM, a twofold SOM (TSOM) and its variation, the FFT-TSOM, are proposed to solve in a more stable and more efficient manner the two-dimensional (2D) and three-dimensional (3D) electromagnetic ISP. Numerical simulations validate the efficacy of the proposed method: robustness against noise, fast convergence, high resolution, and the ability to deal with scatterers of special shapes. The SOM can be also applied to solve electric impedance tomography (EIT) problem and transport-based imaging problems.