Department of Mathematics

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

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

Wayne State University and the Beijing Center for Computational Science

**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.

- 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)*is sufficiently small, then the pollution error in the energy norm is bounded by^{p+1}*O(k(kh)*^{2p}*)*, which coincides with the phase error of the finite element method obtained by existent dispersion analyses. - For linear finite element method under certain mesh condition, we obtain the
*H*^{1}-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*H*^{1}-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

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.

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

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.

Monday, March 27, 2017

Katy Craig

*University of California, Santa Barbara
Host: A. J. Meir*

**Collective dynamics and optimal transport**

**Abstract**: For a range of physical and biological processes—from biological swarming to dynamics of granular media—the evolution of a large number of interacting agents can be described in terms of the competing effects of attraction and repulsion. But do solutions of the resulting partial differential equations exist? And if so, can we quantify their stability with respect to initial data and their convergence to equilibrium? Finally, how can we simulate them numerically?

The theory of optimal transport provides a unified framework in which to study these questions, allowing the evolution of the interacting agents to be reframed as movements to minimize an energy. While the case of convex energies is well understood, many energies of applied interest—including those coming from models of biological chemotaxis and social aggregation—fall outside the scope of the existing theory. In this talk, I will present new results that extend the well-posedness theory to a range of nonconvex energies and discuss applications of these results to a model of pedestrian crowds.

Wednesday, April 12, 2017

Jiguang Sun

*Michigan Tech
Host: Yunkai Zhou
*

**Innovative eigensolvers for large non-hermitian matrices**

**Abstract**: Recently, a non-classical eigenvalue solver, called RIM, was proposed to compute eigenvalues in a given region on the complex plane. Without solving any eigenvalue problem, it tests if a region contains eigenvalues using an approximate spectral projection. Regions that contain eigenvalues are subdivided and tested recursively until eigenvalues are isolated with a specified precision. This makes RIM an eigenvalue solver distinct from existing methods. Furthermore, it requires no a priori spectral information. In this talk, we discuss an improved version of RIM for non-Hermitian eigenvalue problems. Using Cayley transforms, the computation cost is reduced significantly also it inherits all the advantages of RIM. Numerical examples are presented and compared with 'eigs' in Matlab.