Math/ICES Center of Numerical Analysis Seminar (Spring 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 2013 Fall 2012, Spring 2012, Fall 2011, Spring 2011, Fall 2010, Spring 2010, Fall 2009



Title and Abstract


Chris White
(Mathematics, UT Austin)
An eigenvalue optimization problem for graph partitioning

We begin by reviewing the graph partitioning problem and its applications to data clustering. Motivated by a geometric problem, we then introduce a new non-convex graph partitioning objective where the optimality criterion is given by the sum of the Dirichlet eigenvalues of the partition components. A relaxed formulation is identified and a novel rearrangement algorithm is proposed, which we show converges in a finite number of iterations to a local minimum of the relaxed objective function. We will discuss applications, as well as intriguing connections to other problems such as Nonnegative Matrix Factorization and Reaction Diffusion Equations. This is joint work with Braxton Osting and Edouard Oudet.


Canceled due to campus close

Can Evren Yarman
A new inversion method for NMR signal processing

In this talk we present a new, semi-analytic inversion method for nuclear magnetic resonance (NMR) log measurements. Our method represents multiwait-time measurements via short sums of exponentials. The resulting sparse T2 distribution requires fewer T2 relaxation times than present in linearized inversion methods. The T1 relaxation times, and corresponding amplitudes are estimated via convex optimization and a semi-analytic algorithm. We obtain an efficient way to represent the NMR data that can be utilized to estimate petrophysical properties and for compression in logging-while-drilling applications.



Di Liu
(Michigan State University)

A multiscale method for optical responses of nano structures

We introduce a new framework for the multiphysical modeling and multiscale compu- tation of nano-optical responses. The semi-classical theory treats the evolution of the electromagnetic field and the motion of the charged particles self-consistently by coupling Maxwell equations with Quantum Mechanics. To overcome the numerical challenge of solving high dimensional many body Schr̈odinger equations involved, we adopt the Time Dependent Current Density Functional Theory (TD-CDFT). In the regime of linear responses, this leads to a linear system of equations determining the electromagnetic field as well as current and electron densities simultaneously. A self-consistent multiscale method is proposed to deal with the well separated space scales. Numerical examples are presented to illustrate the resonant condition.


Finite Element Rodeo Conference in ICES

Adam Oberman
(McGill University)
Numerical Methods for a class of nonlinear elliptic PDEs with emphasis on the Monge-Ampère equation

Nonlinear elliptic and parabolic Partial Differential Equations (PDEs) have applications to image processing, first arrival times in wave propagation, homogenization, mathematical finance, stochastic control and games theory. In order to capture geometric features such as folds and corners, and avoid artificial singularities which arise from bad representations of the operators, it is important to use convergent numerical schemes.

In the first part of the talk, I will introduce the class of equations, and present some of the examples mentioned above. I'll also explain how to build simple nonlinear finite difference methods for the obstacle problem. In the second part of the talk, I'll focus on a specific equation and explain the method. The (elliptic) Monge-Ampère Partial Differential Equation is a classical nonlinear PDE arising in geometry. It has been studied recently due to its connections with Optimal Transportation theory. Starting with the Dirichlet problem, I will present a finite difference scheme which is the only scheme proven to converge to weak (viscosity) solutions. Building on the original discretization, I'll describe modifications which improve the accuracy and solution speed. Finally, I will show how to solve the problem with Optimal Transportation boundary conditions.

This is joint work with Jean-David Benamou and Brittany Froese.



No Seminar Due to Spring Break

Hongyu Liu
(UNC Charlotte)
Recovery by a single far-field measurement

In this talk, I will describe the recent theoretical and computational progress on recovering  electromagnetic scatterers by using a single far-field measurement. We establish the uniqueness in determining PEC obstacles of general polyhedral type. We also develop several direct imaging schemes, which can work in an extremely general setting, assuming the uniqueness holds true.



Kazuo Aoki
(Kyoto University)
Some topics in classical kinetic theory of gases

Kinetic theory of gases is a classical subject in non-equilibrium statistical physics. Nowadays, however, gases in non-equilibrium state play important roles in many fields, such as micro fluid dynamics and vacuum technology. Kinetic theory of gases is a principal tool to deal with the behavior of such non-equilibrium gases, since it is beyond the applicability of ordinary fluid dynamics. In the present talk, a brief introduction to kinetic theory is given first. Then, its application to some basic problems, such as the approach to equilibrium of a gas and a simple model of the Crookes radiometer, is discussed.




RLM 12.176

(Joint with Math Physics Seminar)

Peter Kuchment

(Texas A&M)
Wannier function frames in presence of non-trivial Bloch bundles

Let L be a Schroedinger operator with periodic magnetic and electric potentials, a Maxwell operator in a periodic medium, or an arbitrary self-adjoint elliptic linear partial differential operator in R^n with coefficients periodic with respect to a lattice G. Let also S be a finite part of its spectrum separated by gaps from the rest of the spectrum. We consider the old question of existence of a finite set of exponentially decaying Wannier functions such that their G-shifts span the whole spectral subspace corresponding to S in some "nice" manner. These sets of functions are extensively used for computations in solid state physics, photonic crystals, etc. However, it is known (Thouless, 1984) that a topological obstruction sometimes exists to finding exponentially decaying Wannier functions that form an orthonormal basis of the spectral subspace. This obstruction has the form of non-triviality of certain finite dimensional (with the dimension equal to the number of spectral bands in S) analytic vector bundle. We show that, in spite of this obstacle, it is always possible to find a finite number of exponentially decaying composite Wannier functions such that their G-shifts form a 1-tight (Parseval) frame in the spectral subspace. This appears to be the best one can do when the topological obstruction is present. The number of the functions required coincides with the number of spectral bands if and only if the bundle is trivial, in which case an orthonormal basis of exponentially decaying composite Wannier functions is known to exist. An estimate on this number is provided. In particular, in physical dimensions only one additional function (and its shifts) is needed.

Brittany Froese
(Mathematics, UT Austin)
Fast sweeping methods for hyperbolic systems of conservation laws

Fast sweeping methods have become a useful tool for computing the solutions of static Hamilton-Jacobi equations. By adapting the main idea behind these methods, we describe a new approach for computing steady state solutions to systems of conservation laws. By exploiting the flow of information along characteristics, these fast sweeping methods can compute solutions very efficiently. Furthermore, the methods capture shocks sharply by directly imposing the Rankine-Hugoniot shock conditions. We present numerics for several one- and two-dimensional examples to illustrate the use and advantages
of this approach.



Jingwei Hu
(ICES, UT Austin)

Fast algorithms for the quantum Boltzmann collision operator

The quantum Boltzmann equation describes the non-equilibrium dynamics of a quantum system consisting of bosons or fermions. The most prominent feature of the equation is a high-dimensional integral operator modeling particle collisions, whose nonlinear and nonlocal structure poses a great challenge for numerical simulation. I will introduce two fast algorithms for the quantum Boltzmann collision operator. The first one is a quadrature based solver specifically designed for the collision operator in reduced energy space. Compared to cubic complexity of direct evaluation, our algorithm runs in only linear complexity (optimal up to a logarithmic factor). The second one accelerates the computation of the full phase space collision operator. It is a spectral algorithm based on a separated expansion of the collision kernel. Numerical examples are presented to illustrate the efficiency and accuracy of proposed algorithms. This is joint work with Lexing Ying.



Bryan Quaife
Adaptive time stepping for vesicle suspensions

Vesicles are deformable and inextensible capsules, filled with and submerged in a viscous fluid, whose dynamics are governed by hydrodynamic and elastic forces. The evolution equations are a system
of integro-differential-algebraic equations resulting from a boundary integral formulation for the Stokes equations coupled with a balance of forces across the interface of the vesicle. In order to resolve
complicated vesicle flows, we have recently introduced a high-order adaptive time integrator. The time step size is chosen by requiring that an estimate of the local truncation error is less than a given tolerance. Most estimates require multiple numerical solutions, but vesicle suspensions have a natural estimate that requires only one numerical solution. To achieve high-order accuracy, we use a spectral deferred correction method which is naturally compatible with adaptive time stepping.



Elena Cherkaev
(University of Utah)

Inverse problem for the structure of composite materials

The talk discusses inverse homogenization problem which is a problem of deriving information about the microgeometry of a two-component composite media from given effective properties. The approach is based on reconstruction of the spectral measure of a self-adjoint operator that depends on the geometry of composite. Stieltjes analytic representation of the effective property relates the n-point correlation functions of the microstructure to the moments  of the spectral measure, which contains all information about the microgeometry. I show that the problem of identification of the spectral function from effective measurements known in an interval of frequency, has a unique solution. In particular, the volume fractions of materials in the composite and an inclusion separation parameter, as well as the spectral gaps at the ends of the spectral interval, can be uniquely recovered. I will discuss reconstruction of microstructural parameters from electromagnetic and viscoelastic effective measurements, application to coupling of different effective properties, and show an extension to nonlinear composites.