Department of Mathematics

Research Colloquium

Fall 2016/Spring2017


Clements Hall 126
Wednesday: 3:30--4:30
Tue & Thu: 3:45--4:45
Refreshments are served 15 minutes before the talk.

 



Wednesday, October 19, 2016

Dr. Stephen Davis
Engineering Sciences and Applied Mathematics
Northwestern University
Host: V. Ajaev

Dynamics of Clean Foams

Abstract: A foam is a two-phase mixture of liquid and gas bubbles in which the liquid fraction is small, say, below 15%. Because the bubbles are crowded, they have polygonal shapes composed of thin films and Plateau borders. The curved borders induce film thinning due to capillarity. When the films have thinned sufficiently, the films rupture causing adjacent bubbles to coalesce. There is then a cascade of successive ruptures taking an initial coherent pattern to a strongly disordered state. Unlike foams with surfactant, clean foams possess no stable static configurations.
The talk will present a numerical simulation of 2D foams using network analysis in which films are represented by lines and borders by nodes. Thin film asymptotics are used to describe 1D flow in films, Newton’s laws are used to describe nodal motions, and a new criterion is used to determine when film rupture due to van der Waals attractions. The simulation is then used to extract statistical data regarding the evolution of the foam.

 



Wednesday, October 26, 2016

Dr. Jesse Chan
Department of Computational and Applied Mathematics
RiceUniversity
Host: T. Hagstrom

Efficient time-domain DG methods for wave propagation

Abstract: High order time-domain discontinuous Galerkin methods can yield accurate numerical models of wave propagation on complex geometries due to their low numerical dispersion and dissipation. Additionally, the computationally intensive nature of time-explicit nodal discontinuous Galerkin methods is well-suited to implementation on Graphics Processing Units (GPUs). However, the efficiency of GPU-accelerated DG methods suffers when high orders of approximation or high order accurate approximations of heterogeneous media are used. We discuss how to address both issues using properties of Bernstein-Bezier polynomial bases and "weighted-adjusted" inner products.

 



Wednesday, November 2, 2016

Dr. Zhimin Zhang
Department of Mathematics
Wayne State University and the Beijing Center for Computational Science
Host: T. Hagstrom

 

Numerical analysis of the Galerkin and weak Galerkin method for the Helmholtz equation with high wave number

Abstract: We study convergence property of the weak Galerkin method of fixed degree p and supercovergence property of the linear finite element method for the Helmholtz problem with large wave number.

  1. Using a modified duality argument, we improve the existing error estimates of the WG method, in particular, the error estimates with explicit dependence on the wave number k are derived, it is shown that if k(kh)p+1 is sufficiently small, then the pollution error in the energy norm is bounded by O(k(kh)2p), which coincides with the phase error of the finite element method obtained by existent dispersion analyses.
  2. For linear finite element method under certain mesh condition, we obtain the H1-error estimate with explicit dependence on the wave number k and show that the error between the finite element solution and the linear interpolation of the exact solution is superconvergent in the H1-seminorm, although the pollution error still exists. We proved a similar result for the recovered gradient by polynomial preserving recovery (PPR) and found that the PPR can only improve the interpolation error and has no effect on the pollution error. Furthermore, we estimated the error between the finite element gradient and recovered gradient and discovered that the pollution error is canceled between these two quantities. Finally, we apply the Richardson extrapolation to the recovered gradient and demonstrate numerically that PPR combined with the Richardson extrapolation can reduce the interpolation and pollution errors simultaneously, and therefore, leads to an asymptotically exact a posteriori error estimator.

All theoretical findings are verified by numerical tests.

 


 

Wednesday, November 9, 2016

Dr. Christopher Donahue
Postdoctoral Scholar, The Gladstone Institutes
UCSF School of Medicine
Host: A. Barreiro

Pathway-specific coding of actions and outcomes in the basal ganglia

Abstract: Neural circuits of the basal ganglia are critical for adaptive motor control and have been implicated in both neurological and psychiatric disorders. Two parallel basal ganglia circuits have been described: the direct and indirect pathways. Selective stimulation of the direct pathway increases movement, whereas stimulation of the indirect pathway suppresses movement. While the role of these circuits in motor control have been well characterized, their role in higher cognitive function is less understood. To address this, we trained mice to perform a value-based dynamic foraging task which required animals to learn by integrating their recent history of actions and outcomes. We imaged pathway-specific single-cell activity through the use of a head-mounted fluorescence microscope while animals performed the task. We found that the direct (but not the indirect) pathway neurons systematically combined action and outcome information in a manner that could potentially bias decisions towards options that maximize reward.



Wednesday, November 30, 2016 

Dr. Haizhao Yang 
Department of Mathematics
Duke University

Host: Y. Zhou

 

A Cubic Scaling Algorithm For Excited States Calculations In pp-RPA

Abstract: In the investigation of excitation problems, the particle-particle random phase approximation (pp-RPA) has been shown to be capable of describing double, Rydberg, and charge transfer excitations, for which the conventional time-dependent density functional theory (TDDFT) might not be suitable. However, the application of the pp-RPA to large molecules is still limited because of its expensive scaling. We introduce a cubic scaling algorithm to calculate the lowest few excitations in the pp-RPA framework.



Wednesday  December 7, 2016

Travis Thompson
Department of Computational and Applied Mathematics
Rice University
Host: K. Hedrick

A DG approach to poroelasticity, with applications to edema formation in tissue

Abstract: Edema is a generalized clinical condition referring to the accumulation of excess fluid between cells, or in cavities; in the intestine the result is a loss of contractility by disrupting peristaltic activity.  The canonical clinical model of edema used in the medical literature is an ordinary differential equation representing a simple balance of lymphatic and vascular fluid exchange based on the Drake Lane and Starling Landis equations.  These ODE models rely on an tissue bed structure hypothesis that is too simplistic to represent the layered, heterogeneous physiology of the intestine; furthermore, they neglect the mechanical response of the tissue itself.

Recently, an approach for modeling the combined fluid / mechanical interaction based on a simplified, mixed form of Biot's equations of poroelasticity has been put forth (Young & Riviere); the model was discretized using the symmetric interior penalty discontinuous Galerkin method with piecewise linear elements. Clinical experiments conducted at the UT Health Sciences center in 2008 provided a means for verification; no error analysis was conducted.

In this talk I will present the full mixed form of the Biot model, discretization with the (non-conforming) discontinuous Galerkin SIPG / NIPG approaches, a-priori error estimates, implementation in 2D and 3D, numerical convergence results, and robustness to locking.  I will then introduce a perturbed model, based on the generalization, suited for use in the context of the heterogeneous physiology of the intestine.  I will share preliminary physiological computations, and ongoing work of both a clinical and mathematical nature.



Friday, January 20, 2017

Anotida Madzvamuse
University of Sussex
Host: A. J. Meir

A robust and efficient adaptive multigrid solver for the optimal control of phase field formulations of geometric evolution laws

Abstract: In this talk, I will present a novel solution strategy to efficiently and accurately compute approximate solutions to semilinear optimal control problems, focusing on the optimal control of phase field formulations of geometric evolution laws.  The optimal control of geometric evolution laws arises in a number of applications in fields including material science, image processing, tumour growth and cell motility.  Despite this, many open problems remain in the analysis and approximation of such problems.  In the current work we focus on a phase field formulation of the optimal control problem, hence exploiting the well developed mathematical theory for the optimal control of semilinear parabolic partial differential equations.  Approximation of the resulting optimal control problem is computationally challenging, requiring massive amounts of computational time and memory storage.  The main focus of this work is to propose, derive, implement and test an efficient solution method for such problems.  The solver for the discretised partial differential equations is based upon a geometric multigrid method incorporating advanced techniques to deal with the nonlinearities in the problem and utilising adaptive mesh refinement.  An in-house two-grid solution strategy for the forward and adjoint problems, that significantly reduces memory requirements and CPU time, is proposed and investigated computationally.  Furthermore, parallelisation as well as an adaptive-step gradient update for the control are employed to further improve efficiency.  Along with a detailed description of our proposed solution method together with its implementation we present a number of computational results that demonstrate and evaluate our algorithms with respect to accuracy and efficiency.  A highlight of the present work is simulation results on the optimal control of phase field formulations of geometric evolution laws in 3-D which would be computationally infeasible without the solution strategies proposed in the present work. 


Thursday, February 23, 2017

Margot Gerritsen
Director, Institute for Computational & Mathematical Engineering
Senior Associate Dean, School of Earth, Energy and Environmental Sciences
Stanford University
Host: A. Aceves

Distinguished SMU/UTD SIAM Student Chapter Speaker

Fun numerical problems in subsurface flow modeling

Abstract: Subsurface flow modeling is a mecca for the numerical analyst and computational engineer. For the last 15 years, our research group has had a terrific time exploring problems in this field.  We are particularly interested in highly nonlinear problems, such as those governing thermal recovery processes.  In this talk, I will discuss one such process, in-situ combustion, and the associated challenges around upscaling of kinetics and strong coupling between flow and transport.



Thursday, March 2, 2017

Dr. Vakhtang Poutkaradze
Department of Mathematical and Statistical Sciences
University of Alberta
Host: A. Aceves

Exact geometric approach to the discretization of fluid-structure interactions and the dynamics of tubes conveying fluid

Abstract: Variational integrators for numerical simulations of Lagrangian systems have the advantage of conserving the momenta up to machine precision, independent of the time step. While the theory of variational integrators for mechanical systems is well developed, there are obstacles in direct applications of these integrators to systems involving fluid-structure interactions. In this talk, we derive a variational integrator for a particular type of fluid-structure interactions, namely, simulating the dynamics of a bendable tube conveying ideal fluid that can change its cross-section (collapsible tube). We start by deriving a fully three-dimensional, geometrically exact theory for flexible tubes conveying fluid. Our approach is based on the symmetry-reduced, exact geometric description for elastic rods, coupled with the fluid transport and subject to the volume conservation constraint for the fluid. Using these methods, we obtain the fully three dimensional equations of motion. We then proceed to the linear stability analysis and show that our theory introduces important corrections to previously derived results, both in the consistency at all wavelength and in the effects arising from the dynamical change of the cross-section. Based on this theory, we derive a variational discretization of the dynamics based on the appropriate discretization of the fluid’s back-to-labels map, coupled with a variational discretization of elastic part of the Lagrangian. We also show the results of simulations of the system with the discretization using a very small number of spatial points demonstrating a non-trivial and interesting behavior.  Time permitting, we shall also discuss some fully nonlinear solutions and the results of experiments.