Math/ICES Center of Numerical Analysis Seminar (Fall 2014)

Time and Location: Friday, 3:00-4:00PM, POB 6.304 (Previously known as ACE 6.304). Special time and locations are indicated in color.

If you are interested in meeting a speaker, please contact Kui Ren (

Here are the links to the past seminars: Spring 2014 Spring 2013 Fall 2012, Spring 2012, Fall 2011, Spring 2011, Fall 2010, Spring 2010, Fall 2009



Title and Abstract


POB 6.304

Alexander Bobylev,
(Karlstad University, Sweden and
Keldysh Institute of Applied Mathematics, Moscow)

Boltzmann equation and hydrodynamics at the Burnett level

We present a review of some results on Burnett-type hydrodynamic equations derived from the Boltzmann equation. The well-known problem here is connected with regularization of classical (ill-posed) Burnett equations[1-5]. There are several ways to deal with this problem. We discuss in detail one of the approaches, proposed in [1] and further developed in [2-4]. Our approach is based on infinitesimal changes of variables, it shows that the way of truncation of the Chapman-Enskog series is not unique. It is the only approach which does not use any information beyond the classical Burnett equations.  We show how to derive a two-parameter family of stable Generalized Burnett Equations (GBEs) [2] and discuss the optimal choice of the parameters. Surprisingly the resulting well-posed equations are simpler than the original Burnett equations. The equations are derived for arbitrary intermolecular forces. Some special properties of (a) stationary problems and (b) linear non-stationary problems are discussed in more detail.  Finally we present some recent results on the shock-wave structure [4], which show that GBEs yield certain improvement of the Navier-Stokes results for moderate Mach numbers. Some open questions are also discussed.


[1] A.V.Bobylev, Instabilities in the Chapman-Enskog expansion and hyperbolic Burnett equations, J.Stat.Phys. 124, 371 (2006).
[2] A.V. Bobylev, Generalized Burnett hydrodynamics, J.Stat.Phys. 132, 569 (2008).
[3] M.Bisi, M.P.Cassinari and M.Groppi, Qualitative analysis of the Generalized Burnett Equations and applications to half-space problems, Kinet. Relat. Models 1, 295 (2008).
[4] A.V.Bobylev, M.Bisi, M.P.Cassinari and G.Spiga, Shock wave structure for generalized Burnett equations, Phys. of Fluids 23, 1 (2011).
[5] A.V.Bobylev and A.Windfall, Boltzmann equation and hydrodynamics at the Burnett level, Kinet. Relat. Models 5, No.2 (2012).


2:00-3:00 PM
RLM 10.176

Seong Jun Kim
(Mathematics, Georgia Tech)

Optimal path for a scan of the entire environment under limited sensing range

Abstract: We propose a computational strategy to find the optimal path for a mobile sensor with limited range to traverse a cluttered region and achieve complete coverage of the environment. We first pose the problem in the level set framework, and consider a related question of placing multiple stationary sensors to obtain the full surveillance of the environment. The locations of the stationary sensors are then used for the initialization of the path for the moving sensor. The path is optimized by following the gradient flow of the connecting points, which is a system of ODEs, to shrink its length while maintaining the complete coverage of the environment. Furthermore, we use intermittent diffusion, which converts the ODEs into SDEs, to find the global optimal solution. In addition, we introduce two techniques, disentanglement and removing redundant connecting points in the system of SDEs, to reduce the dimension of the system and improve the efficiency of the computation. This is a joint work with Haomin Zhou and Sung Ha Kang.


3:30-5:00 PM
POB 6.304

Alexander Mamonov
(Schlumberger, Houston)

Krein-Gelfand-Levitan algorithm for inverse hyperbolic problems via spectrally matched finite-difference grids.

We present a method for the numerical solution of inverse problems for coefficients of hyperbolic PDEs based on the spectrally matched finite-difference grids (a.k.a. Gaussian quadrature rules or optimal grids). The method is built around an algorithm for interpolation of the measured time domain data. Once an interpolant is obtained, it can be expressed in terms of Stieltjes continued fraction or its matrix generalization. The use of S-fraction coefficients in inversion is twofold. First, they can be used to reformulate the traditional optimization-based approaches to drastically improve the objective functional, which addresses issues such as local minima and slow convergence. Second, the coefficients provide a way to obtain direct, non-iterative reconstructions on the spectrally matched grids. We
supplement the theoretical considerations with numerical results. This is a joint with V. Druskin and M. Zaslavsky.


POB 6.304

Armando Majorana

University of Catania, Italy

Deterministic and stochastic simulations of electron transport in graphene

Graphene is a gapless semiconductor made of a sheet composed of a single layer of carbon atoms 
arranged into a honeycomb hexagonal lattice. In view of application in graphene-based electron devices, it is crucial to understand the basic transport properties of this material. A physically accurate model is given by a semiclassical transport equation whose scattering terms have been deeply analyzed recently. Due to the computational difficulties, the most part of the available solutions have been obtained by direct Monte Carlo simulations. The aim of this work is to use a numerical scheme based on the discontinuous Galerkin method for finding deterministic (non stochastic) solutions of the electron Boltzmann transport equation in graphene. The same method has been already successfully applied to a more conventional semiconductor material like Silicon or Gallium Arsenide.  A n-type doping or equivalently a high value of the Fermi potential is considered. Therefore we neglect the inter band scatterings but retain all the main  electron-phonon scatterings.  Simulations in graphene nano-ribbons are presented and a comparison with stochastic simulations are presented. We will show that standard techniques for direct Monte Carlo simulations require important adjustments, and we propose a new stochastic scheme, which can be also validated by means of deterministic numerical solutions. Work in collaboration with Vittorio Romano.


3:30-5:00 PM
POB 6.304

Qin Li

(Department of Computing and Mathematical Sciences, Caltech)

Spectral method for linear half-space kinetic equations

Understanding the coupling of physical models at different scales is important and challenging. In this talk, we focus on the issue of kinetic-fluid coupling, in particular, the half-space problems for kinetic equations coming from the boundary layer. We will present some recent progress in algorithm development and analysis for the linear half-space kinetic equations, and its application in coupling of neutron transport equations with diffusion equations. This is a joint work with Jianfeng Lu and Weiran Sun.