Computational Mathematics Seminar

Posted August 8, 2008

Last modified August 12, 2008

Jennifer Ryan, Delft University Of Technology

Local Smoothness-Increasing Accuracy-Conserving (SIAC) Filtering For Discontinuous Galerkin Methods

A detailed abstract can be found at www.cct.lsu.edu/events/talks/418 Refreshments will be served at 10:30.

Computational Mathematics Seminar

Posted August 29, 2008

2:30 pm - 3:30 pm 145 Coates Hall
Linda Petzold, UC Santa Barbara
Member, National Academy of Engineering

Multiscale Simulation Of Biochemical Systems

Linda Petzold (UC Santa Barbara) will visit LSU the end of next week. She will present an ITELS lecture on **Multiscale Simulation Of Biochemical Systems** on Friday, September 5 at 2:30 pm in Coates 145. Further information can be found at http://www.cct.lsu.edu/events/talks/420.

Among her many honors, let me mention that she is a member of the US National Academy of Engineering and a Fellow of the ASME and of the AAAS.

I am a co-host for her visit. Please contact me if you would like to meet with her.

Sue Brenner

Computational Mathematics Seminar

Posted October 7, 2008

Last modified October 21, 2008

Zhijun Wu, Iowa State University

Bioinformatics and Biocomputing

http://www.cct.lsu.edu/events/talks/430

Computational Mathematics Seminar

Posted November 11, 2008

11:00 am - 12:00 pm Johnston Hall 338
Robert M. Kirby, University of Utah

Visualization of High Order Finite Element Methods

http://www.cct.lsu.edu/events/talks/436 Refreshments at 10:30am

Computational Mathematics Seminar

Posted March 16, 2009

Last modified April 13, 2009

Luke Owens, Texas A&M University

Solving the Eikonal equation on adaptive triangular and tetrahedral meshes

http://www.cct.lsu.edu/events/talks/472

Computational Mathematics Seminar

Posted May 7, 2009

11:00 am - 12:00 pm 338 Johnston Hall
Jiangguo Liu, Colorado State University

The Enriched Galerkin (EG) Method For Local Conservation

Abstract: In this talk, we present a locally mass-conservative finite element method based on enriching the approximation space of the continuous Galerkin (CG) method with elementwise constant functions. The proposed method has a smaller number of degrees of freedom than the discontinuous Galerkin (DG) method. Numerical results on coupled flow and transport problems in porous media are provided to illustrate the advantages of this method. Optimal error estimates of the EG method and comparison with related post-processing methods will be discussed also. This is a joint work with Shuyu Sun at Clemson University. Refreshments will be served at 10:30.

Computational Mathematics Seminar

Posted August 20, 2009

3:00 pm - 3:50 pm 338 Johnston Hall
Michael Neilan, Louisiana State University

Numerical Methods for Fully Nonlinear Second Order PDEs and Applications

Fully nonlinear second order PDEs arise in many areas of science including optimal transport, meteorology, differential geometry, and optimal design. However, numerical methods for general fully nonlinear second order PDEs still remains a relatively untouched area. In this talk, I will introduce a new notion of solutions for these equations called moment solutions which are based on a constructive limiting process called the vanishing moment method. I will then present three finite element methods based on the vanishing moment method. Finally, I will demonstrate the effectiveness of the method with numerical examples.

Computational Mathematics Seminar

Posted August 24, 2009

Last modified August 25, 2009

Andrew Barker, Louisiana State University

Monolithically Coupled Scalable Parallel Algorithms For Simulation Of Fluid-structure Interaction

Simulation of fluid-structure interaction is a computationally difficult problem that is important in a variety of applications. Doing it well requires not only accurately modeling physics for the fluid and the structure, but also coupling them together in a stable and efficient manner, and developing scalable numerical methods for this highly nonlinear problem is a challenge. In this talk we describe and examine parallel, scalable techniques in the multilevel Newton-Krylov-Schwarz family for solving the fully implicit fluid-structure interaction system on dynamic unstructured moving finite element meshes in the arbitrary Lagrangian-Eulerian framework. Our emphasis is on tight monolithic coupling of the physical systems and the computational mesh, and on the parallel scalability of the method. We present applications of the method to the simulation of blood flow on vessel geometries derived from patient-specific clinical data.

Computational Mathematics Seminar

Posted August 28, 2009

3:10 pm - 4:00 pm 338 Johnston Hall
Xiaoliang Wan, Louisiana State University

Noise-induced Transition for the Kuramoto-Sivashinsky Equation

Noise-induced transition in the solutions of the Kuramoto-Sivashinsky equation is investigated using the minimum action method derived from the large deviation theory. This is then used as a starting point for exploring the configuration space of the Kuramoto-Sivashinsky equation. The particular example considered here is the transition between a stable fixed point and a stable traveling wave. Five saddle points, up to constants due to translational invariance, are identified based on the information given by the minimum action path (MAP). Heteroclinic orbits between the saddle points are identified. Relations between noise-induced transitions and the saddle points are examined.

Computational Mathematics Seminar

Posted September 8, 2009

Last modified October 15, 2009

Eun-Hee Park, Louisiana State University

A Domain Decomposition Method Based On Augmented Lagrangian With A Penalty Term

Computational Mathematics Seminar

Posted September 8, 2009

Last modified October 15, 2009

Hongchao Zhang, Louisiana State University

An Affine-scaling Method For Nonlinear Optimization With Continuous Knapsack Constraints

Computational Mathematics Seminar

Posted February 10, 2010

Last modified February 17, 2010

Joscha Gedicke, Humboldt University of Berlin

Optimal Convergence of the Adaptive Finite Element Method

Additional information at

http://www.cct.lsu.edu/events/talks/507

Computational Mathematics Seminar

Posted February 26, 2010

Last modified March 1, 2010

Jean-Marie Mirebeau, Laboratoire Jacques Louis Lions, Universite Pierre et Marie Curie

Optimally Adapted Finite Element Meshes

Abstract: Given a function f defined on a bounded domain and a number n>0, we study the properties of the triangulation Tn that minimizes the distance between f and its interpolation on the associated finite element space, over all triangulations of at most n elements. The error is studied in the Lp norm or W1p norm and we consider Lagrange finite elements of arbitrary polynomial order. We establish sharp asymptotic error estimates as n tends to infinity when the optimal anisotropic triangulation is used, and we illustrate these results with numerical experiments.

Computational Mathematics Seminar

Posted March 30, 2010

3:30 pm - 4:30 pm 338 Johnston Hall
Shiyuan Gu, Mathematics, LSU

Introduction to CUDA

Computational Mathematics Seminar

Posted March 21, 2010

10:00 am - 11:00 am 338 Johnston Hall
Juan Galvis, Texas A&M University

Wiener-Chaos finite element methods for the approximation of infinite-dimensional stochastic elliptic equations

Computational Mathematics Seminar

Posted April 8, 2010

1:00 pm - 2:00 pm 338 Johnston Hall
Luke Owens, Texas A&M University

An Algorithm For Surface Encoding And Reconstruction From 3D Point Cloud Data

http://www.cct.lsu.edu/events/talks/521

Computational Mathematics Seminar

Posted June 5, 2011

3:30 pm - 4:30 pm Lockett 233
Christopher Davis, UNCC

Meshless Boundary Particle Methods for Elliptic Problems

http://www.cct.lsu.edu/events/talks/569

Computational Mathematics Seminar

Posted July 27, 2011

Last modified August 20, 2011

Higher Order Estimates In Time For The Arbitrary Lagrangian Eulerian Formulation In Moving Domains

Speaker: Andrea Bonito

http://www.cct.lsu.edu/events/talks/576

Computational Mathematics Seminar

Posted July 15, 2011

Last modified August 20, 2011

A Diffuse Interface Model For Electrowetting

Speaker: Abner J. Salgado

http://www.cct.lsu.edu/events/talks/575

Computational Mathematics Seminar

Posted August 20, 2011

Last modified August 31, 2011

Computational Math Group

Research Summaries

Faculty members in the computational mathematics group (Bourdin, Brenner, Sung, Walker, Wan and Zhang) will give short presentations on their current research. Refreshments will be served at 3:00 pm.

Computational Mathematics Seminar

Posted August 31, 2011

Last modified September 12, 2011

Jiangguo Liu, Colorado State University

Mathematical Modeling For HIV-1 Gag Protein Trafficking And Assembly

The group-specific antigen (Gag) protein is encoded by human immunodeficiency virus type 1 (HIV-1), which causes the Acquired Immuno Deficiency Syndrome (AIDS). A better understanding of the mechanisms of trafficking and assembly of the HIV-1 Gag proteins inside the infected host cells will be undoubtedly helpful for developing new drugs for treatment of HIV. In this talk, we will present a mathematical model for HIV-1 Gag protein trafficking that accounts for both active transport on microtubules and diffusion in cytoplasm. The convection-diffusion equation can be efficiently solved using characteristic finite element menthods. Our in silico results are in good agreement with the in vitro experimental data for several cell lines. We shall also discuss math models for Gag multimerization inside cytoplasm and on cell membrane. The mechanism for kinesin-based viral egress will be examined to illustrate the stochastic features of protein trafficking. This is a joint work with Chaoping Chen, Roberto Munoz-Alicea, Simon Tavener at ColoState and Qing Nie at UC Irvine.

Additional information can be found at http://www.cct.lsu.edu/events/talks/577

Computational Mathematics Seminar

Posted September 12, 2011

3:30 pm - 4:30 pm Johnston 338
Yi Zhang, Department of Mathematics, LSU
Graduate Student

A Quadratic C0 Interior Penalty Method For The Displacement Obstacle Problem Of Clamped Plates

The displacement obstacle problem of clamped plates is an example of a fourth order variational inequality whose numerical analysis is more subtle than that of second order variational inequalities. In this talk we will introduce C0 interior penalty methods for this problem. Both error estimates and numerical results will be discussed. This is joint work with Susanne Brenner, Li-yeng Sung and Hongchao Zhang.

Computational Mathematics Seminar

Posted September 14, 2011

3:30 pm - 4:30 pm Johnston Hall 338
Mayank Tyagi, Louisiana State University

Fluid Flow Simulations of Diverse Petroleum Engineering Processes at the Rock Pores-, System Components- and Reservoir Field- Scales

http://www.cct.lsu.edu/events/talks/582

Computational Mathematics Seminar

Posted October 3, 2011

3:30 pm - 4:30 pm Johnston Hall 338
Georgios Veronis, Louisiana State University

Plasmonics For Controlling Light At The Nanoscale: Cavity And Slow-Light Enhanced Devices, And The Effect Of Disorder

http://www.cct.lsu.edu/events/talks/585

Computational Mathematics Seminar

Posted October 3, 2011

3:30 pm - 4:30 pm Johnston Hall 338
Jintao Cui, Institute for Mathematics and Its Applications

HDG Methods For The Vorticity-Velocity-Pressure Formulation Of The Stokes Problem

In this talk we discuss the hybridizable discontinuous Galerkin (HDG) method for solv- ing the vorticity-velocity-pressure formulation of the three-dimensional Stokes equations of incompressible fluid flow. The idea of the a priori error analysis consists in estimating a projection of the errors that is tailored to the very structure of the numerical traces of the method. We show that the approximated vorticity and pressure, which are polynomials of degree k, converge with order k + 1/2 in L2-norm for any k ≥ 0. Moreover, the approxi- mated velocity converges with order k + 1. This is joint work with Bernardo Cockburn from University of Minnesota. Further details at http://www.cct.lsu.edu/events/talks/579

Computational Mathematics Seminar

Posted November 1, 2011

3:30 pm - 4:30 pm 338 Johnston Hall
Xiaoliang Wan, Louisiana State University

Minimum Action Method And Dynamical Systems

In this work, we present an adaptive high-order minimum action method for dynamical systems perturbed by small noise. We use the hp finite element method to approximate the minimal action path and nonlinear conjugate gradient method to solve the optimization problem given by the Freidlin-Wentzell least action principle. The gradient of the discrete action functional is obtained through the functional derivative and the moving mesh technique is employed to enhance the approximation accuracy. Numerical examples are given to demonstrate the efficiency and accuracy of the proposed numerical method. We also discuss the application of the minimum action method to study the structure of the phase space and some open issues from the numerical point of view.

(Refreshments at 3pm. Further information at http://www.cct.lsu.edu/events/talks/594)

Computational Mathematics Seminar

Posted November 1, 2011

3:30 pm - 4:30 pm 338 Johnston Hall
Shawn Walker, LSU

Optimization Of Flapping Based Locomotion

Locomotion at the macro-scale is important in biology and industrial applications, such as for understanding the fundamentals of flight to enable design of artificial locomotors. We present an analysis of a fluid-structure interaction problem that models a rigid flapping body at intermediate Reynolds number (in 2-D). In particular, we have an energy estimate and a schur-complement method for solving the coupled system, which is valid for all mass densities of the body (even zero). We also describe an optimal control problem for the time-dependent actuation profile that drives the forward motion of the body. The actuation consists of a vertical velocity control attached to a pivot point of an ellongated rigid body, which is allowed to rotate and is affected by a torsional spring; the spring acts as an elastic recoil. We then solve the time-dependent, PDE-constrained optimization problem (with appropriate constraints). Optimization results for certain parameter variations (relative mass density, spring constant, etc) will be shown. This work is joint with Michael Shelley at NYU. (Refreshments at 3pm. Further details at http://www.cct.lsu.edu/events/talks/596 )

Computational Mathematics Seminar

Posted November 1, 2011

3:30 pm - 4:30 pm 338 Johnston Hall
Hongchao Zhang, Louisiana State University

An Adaptive Preconditioned Nonlinear Conjugate Gradient Method With Limited Memory

Nonlinear conjugate gradient methods are an important class of methods for solving large-scale unconstrained nonlinear optimization. However, their performance is often severely affected when the problem is very ill-conditioned. In the talk, efficient techniques for adaptively preconditioning the nonlinear conjugate method in the subspace spanned by a small number of previous searching directions will be discussed. The new method could take advantages of both nonlinear conjugate methods and limited-memory BFGS quasi-Newton methods, and achieves significant performance improvement compared with CG\_DESCENT conjugate gradient method and L-BFGS quasi-Newton method. (Refreshments at 3pm. Further details at http://www.cct.lsu.edu/events/talks/592 )

Computational Mathematics Seminar

Posted February 5, 2012

3:30 pm - 4:30 pm 338 Johnston Hall
Tchavdar Marinov, Southern University at New Orleans

Solitary Wave Solutions As Inverse Problem

A special numerical technique has been developed for identification of solitary wave solutions of Boussinesq and Korteweg--de Vries equations. Stationary localized waves are considered in the frame moving to the right. The original ill-posed problem is transferred into a problem of the unknown coefficient from over-posed boundary data in which the trivial solution is excluded. The Method of Variational Imbedding is used for solving the inverse problem. The generalized sixth order Boussinesq equation is considered for illustrations. http://www.cct.lsu.edu/events/talks/605

Computational Mathematics Seminar

Posted March 5, 2012

3:30 pm - 4:30 pm 338 Johnston Hall
Haijun Yu, Institute of Computational Mathematics, Chinese Academy of Sciences

Chebyshev Sparse Grid Method for High-dimensional PDEs

Sparse grid is a special discretization for high-dimensional problems.

It was first introduced by S.A. Smolyak in 1960s for the integration

and interpolation of tensor product functions. During the 1990s, C.

Zenger et al. extended it to solve high-dimensional PDEs. The commonly

used bases are Fourier bases for periodic problems and linear finite

element bases for non-periodic problems. In this talk, we introduce

Chebyshev sparse grid method for solving non-periodic PDEs and apply it

to solve the electronic Schrodinger equation.

Computational Mathematics Seminar

Posted December 5, 2011

Last modified March 5, 2012

Guang Lin, Pacific Northwest National Laboratory

Uncertainty Quantification Algorithms and Applications for High Dimensional Stochastic PDE Systems

Experience suggests that uncertainties often play an important role in quantifying the performance of complex systems. Therefore, uncertainty needs to be treated as a core element in modeling, simulation and optimization of complex systems. In this talk, a new formulation for quantifying uncertainty in the context of subsurface flow and transport problem will be discussed. An integrated simulation framework will be presented that quantifies both numerical and modeling errors in an effort to establish "error bars" in CFD. In particular, stochastic formulations based on Galerkin and collocation versions of the generalized Polynomial Chaos (gPC), multi-output Gaussian process model, Multilevel Monte Carlo, scalable multigrid-based pre-conditioner for stochastic PDE, adaptive ANOVA decomposition, and some stochastic sensitivity analysis and Bayesian parameter estimation techniques will be discussed in some detail. Several specific examples on flow and transport in randomly heterogeneous porous media, Bayesian climate model parameter estimation will be presented to illustrate the main idea of our approach.

Computational Mathematics Seminar

Posted March 6, 2012

3:30 pm - 4:30 pm
Xiang Zhou, Brown University

The Study of Rare Events

Many methods for stochastic systems take into account only average behavior (or perhaps variance) of the model response. But this is often not enough as the performance is related to rare events with a small probability of occurring. In my talk, I will review the large deviation theory for analyzing rare events, introduce a minimum action method for small noise diffusion processes, and the recent importance sampling Monte Carlo method based on the large deviation. Throughout the talk, I will also stress the special features of noise-induced transition in non-gradient systems and how to understand subcritical instability in physics and fluid dynamics from perspective of noise-induced transition.

Computational Mathematics Seminar

Posted March 27, 2012

3:30 pm - 4:30 pm 338 Johnston Hall
Guido Kanschat, Texas A&M University

Discontinuous Galerkin Methods for Diffusion-Dominated Radiative Transfer Problems

Abstract: While discontinuous Galerkin (DG) methods had been developed and analyzed in the 1970s and 80s with applications in radiative transfer and neutron transport in mind, it was pointed out later in the nuclear engineering community, that the upwind DG discretization by Reed and Hill may fail to produce physically relevant approximations, if the scattering mean free path length is smaller than the mesh size. Mathematical analysis reveals, that in this case, convergence is only achieved in a continuous subspace of the finite element space. Furthermore, if boundary conditions are not chosen isotropically, convergence can only be expected in relatively weak topology. While the latter result is a property of the transport model, asymptotic analysis reveals, that the forcing into a continuous subspace can be avoided. By choosing a weighted upwinding, the conditions on the diffusion limit can be weakened. It has been known for long time, that the diffusion limit of radiative transfer is a diffusion equation; it turns out, that by choosing the stabilization carefully, the DG method can yield either the LDG method or the method by Ern and Guermond in its diffusion limit. We will close discussing solution techniques for the resulting discrete problems.

Refreshments at 3pm.

Computational Mathematics Seminar

Posted October 1, 2012

Last modified October 9, 2012

Duk-Soon Oh, Louisiana State University

A Balancing Domain Decomposition Method by Constraints for Raviart-Thomas Vector Fields

Balancing domain decomposition by constraints(BDDC) preconditioners consist of a coarse component, involving primal constraints across the interface between the subdomains, and local components related to the Schur complements of the local subdomain problems. A BDDC method for vector field problems discretized with Raviart-Thomas finite elements is introduced. The method is based on a new type of weighted averages developed to deal with more than one variable coefficient. Bounds on the condition number of the preconditioned linear system are also provided and the estimated condition number is quite insensitive to the values and jumps of the coefficients across the interface and has a polylogarithmic bound in terms of the number of degrees of freedom in the individual subdomains. Numerical experiments for 2D and 3D problems, which support the theory and show the effectiveness of our algorithm, are also presented.

Computational Mathematics Seminar

Posted September 8, 2012

Last modified October 21, 2012

Klaus Boehmer, Philipps-Universitaet Marburg

Dew Drops on Spider Webs: A Symmetry Breaking Bifurcation for a Parabolic Differential-Algebraic Equation

Lines of dew drops on spider webs are frequently observed on cold mornings. In this lecture we present a model explaining their generation. Although dew is supposed to condense somehow evenly along the thread, only lines of drops are observed along the spider thread. What are the reasons for this difference? We try to give an explanation by concentrating on some essential aspects only. This everyday observation is an example of one of the fascinating scenarios of nonlinear problems, {it symmetry breaking bifurcation}. Despite many simplications the model still provides very interesting mathematical challenges. In fact the necessary mathematical model and the corresponding numerical methods for this problem are so complicated that in its full complexity it has never been studied before. We analyze and numerically study symmetry breaking bifurcations for a free boundary problem of a degenerate parabolic differential-algebraic equation employing a combination of analytical and numerical tools.

Computational Mathematics Seminar

Posted November 1, 2012

3:30 pm - 4:30 pm 338 Johnston Hall
Harbir Antil, George Mason University

Optimal Control of a Free Boundary Problem with Surface Tension Effects

We consider a PDE constrained optimization problem governed by a free boundary problem. The state system is based on coupling the Laplace equation in the bulk with a Young-Laplace equation on the free boundary to account for surface tension, as proposed by P. Saavedra and L.R. Scott. This amounts to solving a second order system both in the bulk and on the interface. Our analysis hinges on a convex constraint on the control such that the state constraints are always satisfied. Using only first order regularity we show that the control to state operator is twice Fr'echet differentiable. We improve slightly the regularity of the state variables and exploit this to show existence of a control together with second order sufficient optimality conditions. Next we prove the optimal a priori error estimates for the control problem and present numerical examples. Finally, we give a novel analysis for a more practical model with Stokes equations in the bulk and slip boundary conditions on the free boundary interface. (Refreshments at 3pm)

Computational Mathematics Seminar

Posted November 1, 2012

3:30 pm - 4:30 pm 338 Johnston Hall
Shawn Walker, LSU

A New Mixed Formulation For a Sharp Interface Model of Stokes Flow and Moving Contact Lines

Two phase fluid flows on substrates (i.e. wetting phenomena) are important in many industrial processes, such as micro-fluidics and coating flows. These flows include additional physical effects that occur near moving (three-phase) contact lines. We present a new 2-D variational (saddle-point) formulation of a Stokesian fluid with surface tension that interacts with a rigid substrate. The model is derived by an Onsager type principle using shape differential calculus (at the sharp-interface, front-tracking level) and allows for moving contact lines and contact angle hysteresis through a variational inequality. We prove the well-posedness of the time semi-discrete and fully discrete (finite element) model and discuss error estimates (ongoing). Simulation movies will be presented to illustrate the method. We conclude with some discussion of a 3-D version of the problem as well as future work on optimal control of these types of flows. (Refreshments at 3pm.)

Computational Mathematics Seminar

Posted October 9, 2012

3:30 pm - 4:30 pm 338 Johnston Hall
Brittany Froese, University of Texas at Austin

Numerical Solution of the Optimal Transportation Problem Via Viscosity Solutions of the Monge-Ampere Equation

Despite the importance of optimal transportation in both theoretical and applied mathematics, the computation of solutions remains an extremely challenging problem. We describe a numerical method for the widely-studied case when the cost is quadratic and mass is being mapped onto a convex set. The solution is obtained by solving the Monge-Ampere equation, a fully nonlinear elliptic partial differential equation (PDE), coupled to a non-standard implicit boundary condition. First, we describe a variational formulation of the PDE operator, which enables us to construct a monotone finite difference discretisation. This is used as the foundation of a more accurate, almost-monotone discretisation. Next, we re-express the transport condition as a Hamilton-Jacobi equation on the boundary. We construct an upwind discretization of this equation that only requires data inside the domain. Using the theory of viscosity solutions, we prove convergence of the resulting method. A range of challenging computational examples demonstrate the effectiveness and efficiency of this method.

Computational Mathematics Seminar

Posted October 1, 2013

3:30 pm - 4:30 pm Lockett 233
Hyea Hyun Kim, Kyung Hee University, South Korea

A Staggered Discontinuous Galerkin Method for the Stokes System and its Fast Solvers by Domain Decomposition Methods

https://www.cct.lsu.edu/lectures/staggered-discontinuous-galerkin-method-stokes-system-and-its-fast-solvers-domain-decomposi

Computational Mathematics Seminar

Posted October 1, 2013

Last modified October 4, 2013

Michael Friedlander, University of British Columbia

Gauge Optimization, Duality, and Applications

Gauge functions significantly generalize the notion of a norm, and gauge optimization is the class of problems for finding the element of a convex set that is minimal with respect to a gauge. These conceptually simple problems appear in a remarkable array of applications. Their gauge structure allows for a special kind of duality framework that may lead to new algorithmic approaches. I will illustrate these ideas with applications in signal processing and machine learning.

https://www.cct.lsu.edu/lectures/gauge-optimization-duality-and-applications

Computational Mathematics Seminar

Posted November 12, 2013

Last modified November 20, 2013

Neela Nataraj, Indian Institute of Technology Bombay

A C0 interior penalty method for an optimal control problem governed by the biharmonic operator

Abstract: In the recent past, C0 interior penalty methods have been attractive for solving the fourth order problems. In this talk, a C0 interior penalty method is proposed and analyzed for distributed optimal control problems governed by the biharmonic operator. The state equation is discretized using continuous piecewise quadratic finite elements while piecewise constant approximations are used for discretizing the control variable. *A priori *and *a posteriori *error estimates are derived for both the state and control variables under minimal regularity assumptions. Theoretical results are demonstrated by numerical experiments. The * a posteriori *error estimators are useful in adaptive finite element approximation and the numerical results indicate that the sharp error estimators work efficiently in guiding the mesh refinement and saving the computational effort substantially.

Computational Mathematics Seminar

Posted February 3, 2014

3:15 pm - 4:15 pm Lockett 233
Christopher Davis, LSU

Partition of Unity Methods for Fourth Order Problems

Computational Mathematics Seminar

Posted February 8, 2014

3:30 pm - 4:30 pm Lockett 233
Peter Minev, University of Alberta

A Fast Parallel Algorithm for Direct Simulation of Particulate Flows Using Conforming Grids

Computational Mathematics Seminar

Posted February 19, 2014

3:30 pm - 4:30 pm Lockett 233
Jiguang Sun, Michigan Technological University

Numerical Methods for Transmission Eigenvalues

Computational Mathematics Seminar

Posted February 11, 2014

3:30 pm - 4:30 pm Lockett 233
Jichun Li, University of Nevada Las Vegas

Mathematical study and finite element modeling of invisibility cloaks with metamaterials

Computational Mathematics Seminar

Posted February 23, 2014

Last modified April 21, 2014

Eun-Jae Park, Yonsei University, South Korea

Recent Progress in Hybrid Discontinuous Galerkin Methods

Abstract: A new family of hybrid discontinuous Galerkin methods is studied for second-order elliptic equations. Our approach is composed of generating PDE-adapted local basis and solving a global matrix system arising from a flux continuity equation. Our method can be viewed as a hybridizable discontinuous Galerkin method using a Bauman-Oden type local solver. A priori and a posteriori error estimates are derived and applications to the Stokes equations and Convection-Diffusion equations are discussed. Numerical results are presented for various examples.

Computational Mathematics Seminar

Posted August 25, 2014

3:30 pm - 4:30 pm 1008B Digital Media Center
Kamana Porwal, Louisiana State University

A Posteriori Error Estimates of Discontinuous Galerkin Methods for Elliptic Obstacle Problems

Computational Mathematics Seminar

Posted September 2, 2014

3:30 pm - 4:20 pm DMC, Room 1034Discrete ABP estimate and rates of convergence of linear elliptic PDEs in non-divergence form

We design a finite element method (FEM) for linear elliptic equations in

non-divergence form, which hinges on an integro-differential

approximation of the PDE. We show the FEM satisfies the discrete maximum

principle (DMP) provided that the mesh is weakly acute. Thanks to the

DMP and consistency property of the FEM, we establish convergence of the

numerical solution to the viscosity solution.

We derive a discrete Alexandroff-Bakelman-Pucci (ABP) estimate for

finite element methods. Its proof relies on a geometric interpretation

of Alexandroff estimate and control of the measure of the

sub-differential of piecewise linear functions in terms of jumps, and

thus of the discrete PDE. The discrete ABP estimate leads to optimal

rates of convergence for the finite element method under suitable

regularity assumptions on the solution and coefficient matrix.

Computational Mathematics Seminar

Posted November 13, 2014

3:30 pm - 4:30 pm 1034 Digital Media Center
Li-yeng Sung, Louisiana State University

Multigrid Methods for Saddle Point Problems

In this talk we will present a general framework for the design and analysis of multigrid methods for saddle point problems arising from mixed finite element discretizations of elliptic boundary value problems. These multigrid methods are uniformly convergent in the energy norm on general polyhedral domains where the elliptic boundary value problems in general do not have full elliptic regularity. Applications to Stokes, Lam\'e, Darcy and related nonsymmetric systems will be discussed. This is joint work with Susanne Brenner, Hengguang Li and Duk-Soon Oh.

Computational Mathematics Seminar

Posted January 19, 2015

3:30 pm - 4:30 pm DMC 1034
Natasha Sharma, University of Texas El Paso

An Adaptive DG-θ Method with Residual-type Error Estimates for Nonlinear Parabolic Problems

Abstract: In this talk, we propose and analyze a fully discretized adaptive Discontinuous Galerkin-θ (DG-θ) method for nonlinear parabolic problems with the space discretized by the DG finite elements and the time discretization realized by the popular θ-time stepping scheme. The a posteriori error analysis is based on the residual-type estimator derived by Verfurth for conforming approximations in space and θ-scheme in time. This DG-θ estimator will enable us to then realize the adaptive algorithm for local mesh refinement. The desirable properties of reliability and efficiency of the estimator will be then be discussed and finally, we will present numerical results to illustrate the performance of this method.

Computational Mathematics Seminar

Posted January 30, 2015

3:30 pm - 4:30 pm Lockett 233
Mark Wilde, LSU Department of Physics/CCT

Attempting to Reverse the Irreversible in Quantum Physics

Computational Mathematics Seminar

Posted March 2, 2015

3:30 pm - 4:30 pm 1034 Digital Media Center
Hongchao Zhang, Louisiana State University

A Fast Algorithm for Polyhedral Projection

Abstract: In this talk, we discuss a very efficient algorithm for projecting a point onto a polyhedron. This algorithm solves the projeciton problem through its dual and fully exploits the sparsity. The SpaRSA (Sparse Reconstruction by Separable Approximation) is used to approximately identify active constraints in the polyhedron, and the Dual Active Set Algorithm (DASA) is used to compute a high precision solution. Some interesting convergence properties and very promising numerical results compared with the state-of-the-art software IPOPT and CPLEX will be discussed in this talk.

Computational Mathematics Seminar

Posted January 26, 2015

3:30 pm - 4:30 pm 1034 Digital Media Center
Yi Zhang, University of Tennessee

Finite Element Methods for the Stochastic Allen-Cahn Equation with Gradient-type Multiplicative Noises

Abstract: In this talk, we study two fully discrete finite element methods for the stochastic Allen-Cahn equation with a gradient-type multiplicative noise that is white in time and correlated in space. The sharp interface limit of this stochastic equation formally approximates a stochastic mean curvature flow. Strong convergence with rates are established for both fully discrete methods. The key ingredients are bounds for arbitrary moments and Holder estimates in the L2 and H1 norms for the strong solution of the stochastic equation. Numerical results are presented to gauge the performance of the proposed fully discrete methods and to study the interplay of the geometric evolution and gradient type noises. This is the joint work with Xiaobing Feng and Yukun Li.

Computational Mathematics Seminar

Posted March 2, 2015

Last modified March 18, 2015

Clint Whaley, Louisiana State University

Automated Empirical Computational Optimization in ATLAS and iFKO

Abstract: This talk will overview empirical tuning, and highlight its importance for computational scientists / applied mathematicians of all types. Clint Whaley will present the two main empirical tuning projects that he maintains as part of his empirical tuning research, ATLAS and iFKO. Both of these research projects involve large software frameworks designed to be used by computational scientists. ATLAS provides dense linear algebra routines designed for direct for use by mathematicians, engineers, and industry, and is already used by hundreds-of-thousands worldwide. iFKO is a computational-oriented compiler framework, which is currently targeted for computational groups with significant tuning expertise. div

Computational Mathematics Seminar

Posted March 22, 2015

3:30 pm - 4:30 pm 1034 Digital Media Center
Bin Zheng, Pacific Northwest National Laboratory

Fast Multilevel Solvers for Discrete Fourth Order Parabolic Problems

Abstract: In this work, we study fast iterative solvers for the solution of fourth order parabolic equations discretized by mixed finite element method. We propose to use consistent mass matrix in the discretization and use lumped mass matrix to construct efficient preconditioners. We provide eigenvalue analysis for the preconditioned system and estimate the convergence rate of the preconditioned GMRes method. Furthermore, we show these preconditioners only need to be solved inexactly by optimal multigrid algorithms. We also investigate the performance of multigrid algorithms with either collective smoothers or distributive smoothers when solving the preconditioner systems. Our numerical examples indicate the proposed preconditioners are very efficient and robust with respect to both discretization parameters and diffusion coefficients.

Computational Mathematics Seminar

Posted March 2, 2015

Last modified March 18, 2015

Xin (Shane) Li, Louisiana State University

Partial Geometric Mapping for Data Reassembly and Reconstruction

Abstract: To compute geometric mapping is to establish a bijective correspondence between two 3D objects/regions or images. Effective mapping computation could facilitate pattern discovery, similarity detection, and deformation tracking/prediction in geometric data analysis. I will discuss the partial geometric mapping problem between two objects and among multiple objects, which has many practical applications in data reassembly and reconstruction. A few algorithms recently developed in our group will be explained and their applications in computational forensics and medical imaging will be demonstrated.

Computational Mathematics Seminar

Posted September 3, 2015

3:30 pm - 4:30 pm 1034 Digital Media Center
Amanda Diegel, Louisiana State University

Numerical Analysis of Convex Splitting Schemes for Cahn-Hilliard and Coupled Cahn- Hilliard-Fluid-Flow Equations

Abstract: In this talk, we investigate numerical schemes for the Cahn-Hilliard equation and the Cahn-Hilliard equation coupled with a Darcy-Stokes flow. Considered independently, the Cahn-Hilliard equation is a model for spinodal decomposition and domain coarsening. When coupled with a Darcy-Stokes flow, the resulting system describes the flow of a very viscous block copolymer fluid. Challenges in creating numerical schemes for these equations arise due to the nonlinear nature and high derivative order of the Cahn-Hilliard equation. Further challenges arise during the coupling process as the coupling terms tend to be nonlinear as well. The numerical schemes which will be presented preserve the energy dissipative structure of the Cahn-Hilliard equation while maintaining unique solvability and optimal error bounds.

Computational Mathematics Seminar

Posted September 3, 2015

3:30 pm - 4:30 pm 1034 Digital Media Center
Christopher Davis, Tennessee Tech University

A Two Level Additive Schwarz Preconditioner for a Partition of Unity Method

Abstract: The partition of unity finite element method is a type of finite element method that enables one to construct smooth approximation functions at low cost. Investigation into the conditioning of partition of unity methods is an active field or research. In this talk, we discuss the use of two level additive Schwarz preconditioners for a partition of unity method. The numerical algorithm will be presented and analyzed. Numerical examples will be given to demonstrate the effectiveness of the method. This is joint work with Susanne C. Brenner and Li-yeng Sung.

Computational Mathematics Seminar

Posted November 5, 2015

3:30 pm - 4:30 pm 1034 Digital Media Center
Hongchao Zhang, Louisiana State University

Inexact Alternating Direction Algorithm for Separable Convex Optimization

Abstract: We introduce inexact alternating direction algorithms with variable stepsize for solving separable convex optimization. These algorithms generate the Bregman Operator Splitting Algorithm with Variable Stepsize (BOSVS) to the multiblock case and allow to solve the convex subproblems to an adaptive accuracy. Global convergence and some preliminary numerical results will be discussed.

Computational Mathematics Seminar

Posted January 21, 2016

3:30 pm - 4:30 pm Digital Media Center 1034
Jennifer Ryan, University of East Anglia, UK

Exploiting Approximation Properties in the Discontinuous Galerkin Scheme for Improved Trouble Cell Indication

Abstract:
In this talk, we present a generalized discussion of discontinuous Galerkin methods concentrating on a basic concept: exploiting the existing approximation properties. The discontinuous Galerkin method uses a piecewise polynomial approximation to the variational form of a PDE. It uses polynomials up to degree k for a k+1 order accurate scheme. Using this formulation, we concentrate on nonlinear hyperbolic equations and specifically discuss how to obtain better discontinuity detection during time integration by rewriting the approximation using a multi-wavelet decomposition. We demonstrate that this multi-wavelet expansion allows for more accurate detection of discontinuity locations. One advantage of using the multi-wavelet expansion is that it allows us to specifically relate the jumps in the DG solution and its derivatives to the multi-wavelet coefficients. This is joint work with Thea Vuik, TU Delft.

Computational Mathematics Seminar

Posted January 30, 2016

Last modified February 11, 2016

Gang Bao, Zhejiang University

Inverse Problems for PDEs: Analysis, Computation, and Applications

Abstract: Inverse problems for PDEs arise in diverse areas of industrial and military applications, such as nondestructive testing, seismic imaging, submarine detections, near-field and nano optical imaging, and medical imaging. A model problem in wave propagation is concerned with a plane wave incident on a medium enclosed by a bounded domain. Given the incident field, the direct problem is to determine the scattered field for the known scatterer. The inverse problem is to determine the scatterer from the boundary measurements of near field currents densities. Although this is a classical problem in mathematical physics, mathematical issues and numerical solution of the inverse problems remain to be challenging since the problems are nonlinear, large-scale, and most of all ill-posed! The severe ill-posedness has thus far limited in many ways the scope of inverse problem methods in practical applications. In this talk, the speaker will first introduce inverse problems for PDEs and discuss the state of the arts of the inverse problems. Our recent progress in mathematical analysis and computational studies of the inverse boundary value problems will be reported. Several classes of inverse problems will be studied, including inverse medium problems, inverse source problems, inverse obstacle problems, and inverse waveguide problems. A novel stable continuation approach based on the uncertainty principle will be presented. By using multi-frequency or multi-spatial frequency boundary data, our approach is shown to overcome the ill-posedness for the inverse problems. New stability results and techniques for the inverse problems will be presented. Related topics will be highlighted.

Computational Mathematics Seminar

Posted January 26, 2016

Last modified February 23, 2016

Alexandre Madureira, Laboratorio Nacional de Computacao Cientifica, Brazil

Hybrid Finite Element Methods for Multiscale Problems

Abstract: In this talk we discuss the use of hybrid methods for multiscale partial differential equations, in particular concerning the development of a hybrid scheme to solve the linear elasticity system. The unknowns are the displacements and the boundary tractions at each element. Starting from a primal hybrid formulation, the method has a domain decomposition flavor, and the displacements can be discontinuous, with continuous tractions. A decomposition of the primal space allows the reformulation of the continuous problem as a coupled system of elementwise equations, and a global mixed system posed on the mesh skeleton. The scheme is embarrassingly parallel, where the local problems are solved independently. We shall discuss the connections between this and some other methods.

Computational Mathematics Seminar

Posted March 7, 2016

3:30 pm Digital Media Center 1034
Mayank Tyagi, Mechanical Engineering Department, Louisiana State University.

Insights into Complex Wellbore Construction Processes and Completions Performance using Computation Fluid Dynamics (CFD) Simulations

Multiphysics CFD simulations on HPC platforms provide a great opportunity to learn about the complex processes during drilling and completions operations of oil & gas wells. Several computational fluid dynamics (CFD) models with different features are presented for cuttings transport, cement placement, and production through completions in this presentation. All simulation cases are both verified and validated against available experimental data for their corresponding physics. In order to get accurate flow predictions while optimizing computational resources requirements, unsteady shear stress transport (SST) k-ω turbulence model is used to model turbulence closure while solving Reynolds-averaged Navier-Stokes (RANS) equations using unstructured finite volume method (FVM) for discretization. Discrete phase is modeled with discrete element method (DEM) by including particle-particle and particle-fluid interactions with two-way coupling in Eulerian-Lagrangian simulations. Volume of Fluid (VOF) model is used to model displacement of different fluid types with non-Newtonian fluid rheology for cement placement applications. Specifically, during the drilling of highly deviated wellbores, the cuttings transport becomes difficult due to the rolling/sliding transport of the cuttings due to settling around the lower side of the annular region between wellbore and drillpipe. Inefficient cuttings transport may lead to several critical problems such as stuck pipe, increased torque and drag, damaged material and poor quality of cementing jobs. Increasing mud flowrates and improving mud properties for a proper wellbore cleaning is usually limited due to the hydraulic and mechanical thresholds for wellbore formation integrity. Further, understanding of cement placement process remains a critical step in achieving zonal isolation between casings and hydrocarbon bearing formations in all types of well construction operation. Lastly, a gravel-packed completion is modeled to showcase the capabilities of CFD simulations by gaining new insights into modeling and representation of high-rate producer wells in reservoir simulators.

Computational Mathematics Seminar

Posted January 26, 2016

Last modified February 11, 2016

Haomin Zhou, Georgia Tech University

Stochastic Differential Equations and Optimal Control with Constraints

Abstract: We design a new stochastic differential equation (SDE) based algorithm to efficiently compute the solutions of a class of infinite dimensional optimal control problems with constraints on both state and control variables. The main ideas include two parts. 1) Use junctions to separate paths into segments on which no constraint changes from active to in-active, or vice versa. In this way, we transfer the original infinite dimensional optimal control problems into finite dimensional optimizations. 2) Employ the intermittent diffusion (ID), a SDE based global optimization strategy, to compute the solutions efficiently. It can find the global optimal solution in our numerical experiments. We illustrate the performance of this algorithm by several shortest path problems, the frogger problem and generalized Nash equilibrium examples. This talk is based on joint work with Shui-Nee Chow (Math, Georgia Tech), Magnus Egerstedt (ECE, Georgia Tech). Wuchen Li (Math, Georgia Tech), and Jun Lu (Wells Fargo).

Computational Mathematics Seminar

Posted January 21, 2016

3:30 pm - 4:30 pm Digital Media Center 1034
Francisco Javier Sayas, University of Delaware

Hybridizable Discontinuous Galerkin Methods for Elastodynamics

Abstract: In this talk I will present some preliminary results on the use of an Hybridizable Discontinuous Galerkin method for the simulation of elastic waves. I will show how the Qiu and Shi choice of spaces and stabilization parameters for an HDG scheme applied to quasi-static elasticity also apply for time harmonic elastic waves, providing a superconvergent method. I will next discuss a conservation of energy property that holds in the transient case when the elasticity equations are semidiscretized in space with the same HDG strategy. This work is a collaboration with Allan Hungria (University of Delaware) and Daniele Prada (Indiana University Purdue University at Indianapolis)

Computational Mathematics Seminar

Posted September 14, 2016

3:30 pm - 4:30 pm 1034 Digital Media Center
Susanne Brenner, Louisiana State University

Computational Mathematics

Abstract. This is a talk for a general audience. We will first take
a look at computational instruments and mathematical algorithms
from ancient times to the twenty-first century. We will then
discuss the role of mathematics in computing and present some
real life examples of computational mathematics in action.
Finally, we will provide some information on career opportunities.

(Refreshments will be served at 3pm in 1034 DMC.)

Computational Mathematics Seminar

Posted September 14, 2016

3:30 pm - 4:30 pm 1034 Digital Media Center
Runchang Lin, Texas A&M International University

A discontinuous Galerkin least-squares finite element method for reaction-diffusion problems with singular perturbation

Abstract: A discontinuous Galerkin least-squares finite element method is proposed to solve reaction-diffusion equations with singular perturbations. This method produces solutions without numerical oscillations when uniform meshes are used, where neither special treatments nor manually adjusted parameters are required. This method can be applied to linear and nonlinear reaction-diffusion problems with strong reactions. Numerical examples are provided to demonstrate the efficiency of the method.

Computational Mathematics Seminar

Posted September 29, 2016

Last modified October 10, 2016

Xiaoliang Wan, Louisiana State University

On Small Random Perturbations of Elliptic Problems

Abstract: The large deviation principle (LDP) plays an important role for studying rare events induced by small random noise. One challenging task of applying the LDP is to minimize the rate functional numerically, especially when a spatially extended system is considered. Many numerical issues arise depending on the properties of the system and the noise. In this talk we discuss the regularization for the spatial covariance operator using Poisson's equation perturbed by small random forcing. The Euler-Lagrange (E-L) equation suggests that it is critical to approximate a nonlocal biharmonic-like operator, which is ill-posed due to the inverse of the covariance operator. We first study the properties of the nonlocal biharmonic-like operator and then consider the Lavrentiev regularization. The convergence of the approximated minimizer is established in terms of Gamma-convergence. Furthermore, we construct an LDP-based importance sampling estimator, and provide a sufficient condition for such an estimator to be asymptotically efficient. The effect of the regularization parameter on the importance sampling estimator is studied numerically.

Computational Mathematics Seminar

Posted October 18, 2016

3:30 pm - 4:30 pm Digital Media Center 1034
Huan Lei, Pacific Northwest National Laboratory

Quantifying Quasi-equilibrium and Non-equilibrium Properties for Complex Multiphysics Systems

Abstract: We propose a data-driven method to quantify quasi-equilibrium and non-equilibrium properties for complex physical systems with high dimensional stochastic space based on generalize polynomial chaos (gPC) expansion and Mori-Zwanzig projection method. For quasi-equilibrium properties, we demonstrate that sparse grid method suffers instability problem due to the high-dimensionality. Alternatively, we propose a numerical method to enhance the sparsity by defining a set of collective variables within active subspace, yielding more accurate surrogate model recovered by compressive sensing method. Moreover, non-equilibrium properties further depends on the non-local memory term representing the high-dimensional unresolved states. We propose a data-driven method based on appropriate parameterization to compute the memory kernel of the generalized Langevin Equation (GLE) by merely using trajectory data. The approximated kernel formulation satisfies the second fluctuation-dissipation conditions naturally with invariant measure. The proposed method enables us to characterize transition properties such as reaction rate where Markovian approximation shows limitation.

Computational Mathematics Seminar

Posted October 18, 2016

3:30 pm - 4:30 pm Digital Media Center 1034
Xiao Wang, Chinese Academy of Sciences

Inexact Proximal Stochastic Gradient Method for Convex Composite Optimization

Abstract: We study an inexact proximal stochastic gradient (IPSG) method for convex composite optimization, whose objective function is a summation of an average of a large number of smooth convex functions and a convex, but possibly nonsmooth, function. The variance reduction technique is incorporated in the method to reduce the stochastic gradient variance. The main feature of this IPSG algorithm is to allow solving the proximal subproblems inexactly while still keeping the global convergence with desirable complexity bounds. Different accuracy criteria are proposed for solving the subproblem, under which the global convergence and the component gradient complexity bounds are derived for the both cases when the objective function is strongly convex or generally convex. Preliminary numerical experiment shows the overall efficiency of the IPSG algorithm.

Computational Mathematics Seminar

Posted October 31, 2016

Last modified November 27, 2016

Yi Zhang, University of Notre Dame

Error Analysis of C0 Interior Penalty Methods for An Elliptic State-Constrained Optimal Control Problem

Abstract:

We study C0 interior penalty methods for an elliptic optimal control problem with pointwise state constraints on two and three dimensional convex polyhedral domains. The approximation of the optimal state is solved by a fourth order variational inequality and the approximation of the optimal control is computed by a post-processing procedure. To circumvent the difficulty caused by the low regularity of the optimal solutions, we carried out an a priori error analysis based on the complementarity form of the variational inequality. Furthermore, we develop an a posteriori analysis using a residual based error estimator. Numerical experiments are provided to gauge the performance of the proposed methods. This is joint work with Susanne Brenner and Li-yeng Sung.

Computational Mathematics Seminar

Posted February 2, 2017

3:30 pm - 4:20 pm Lockett 233 (Note: *different* date and room for the Comp. Math Seminar)
David Shirokoff, New Jersey Institute of Technology

Approximate global minimizers to pairwise interaction problems through a convex/non-convex energy decomposition: with applications to self-assembly

Abstract: A wide range of particle systems are modeled through energetically driven interactions, governed by an underlying non-convex and often non-local energy. Although numerically finding and verifying local minima to these energies is relatively straight-forward, the computation and verification of global minimizers is much more difficult. Here computing the global minimum is important as it characterizes the most likely self-assembled arrangement of particles (in the presence of low thermal noise) and plays a role in computing the material phase diagram. In this talk I will examine a general class of model functionals: those arising in non-local pairwise interaction problems. I will present a new approach for computing approximate global minimizers based on a convex/non-convex splitting of the energy functional that arises from a convex relaxation. The approach provides a sufficient condition for global minimizers that may in some cases be used to show that lattices are exact, and also be used to estimate the optimality of any candidate minimizer. Physically, the approach identifies the emergence of new length scales seen in the collective behavior of interacting particles. (This is a joint Applied Analysis/Computational Mathematics Seminar.)

Computational Mathematics Seminar

Posted April 3, 2017

3:30 pm 1034 Digital Media Center (CCT)
Yangyang Xu, University of Alabama

Primal-dual methods for affinely constrained problems

Optimization has been applied in many areas including engineering, statistics, finance, and data sciences. Modern applications often have rich structure information. Traditional methods like projected subgradient and the augmented Lagrangian can be used, but they do not utilize structures of the problems and thus are not so efficient. This talk will focus on convex optimization problems with affine constraints. The first part assumes two-block structure on the problem and presents the alternating direction method of multipliers (ADMM) and its accelerated variant. With strong convexity on one block variable, the ADMM can be accelerated from O(1/k) rate to O(1/k^2). Numerical results will be given to demonstrate the improved speed. In the second part, I will present a novel primal-dual block update method for a multi-block (at least three blocks) problem. Existing works have shown that directly extending two-block ADMM to multi-block problems may diverge. To guarantee convergence, either strong assumptions are made or updating order of the blocks has to be changed. Our method uses a simple randomization technique on choosing block variables, and it enjoys O(1/k) ergodic convergence rate and also global convergence in probability. In addition, by choosing a few blocks every time and using Jacobi-type update, the method enables parallel computing with guaranteed convergence. Numerical experiments will be shown to demonstrate its efficiency compared to other methods.

Computational Mathematics Seminar

Posted August 23, 2017

Last modified August 29, 2017

Jingyong Tang, Xinyang Normal University, China

Strong convergence properties of a modified nonmonotone smoothing algorithm for the SCCP

Abstract: The symmetric cone complementarity problem (denoted by SCCP) provides a simple unified framework for various existing complementarity problems and has wide applications. Smoothing algorithms have been successfully applied to solve the SCCP, which in general have the global and local superlinear/quadratic convergence if the solution set of the SCCP is nonempty and bounded. We propose a new nonmonotone smoothing algorithm for solving the SCCP and prove that the algorithm is globally and locally superlinearly/quadratically convergent if the solution set of the SCCP is only nonempty, without requiring its boundedness. This convergence result is stronger than those obtained by most smoothing-type algorithms. Finally, some numerical results are reported.

Computational Mathematics Seminar

Posted August 22, 2017

Last modified September 26, 2017

Yangyang Xu, Rensselaer Polytechnic Institute

Primal-dual methods for affinely constrained problems

Abstract: Optimization has been applied in many areas including engineering, statistics, finance, and data sciences. Modern applications

often have rich structure information. Traditional methods like projected subgradient and the augmented Lagrangian can be used, but they do not utilize structures of the problems and thus are not so efficient. This talk will focus on convex optimization problems with affine constraints. The first part assumes two-block structure on the problem and presents the alternating direction method of multipliers (ADMM) and its accelerated variant. With strong convexity on one block variable, the ADMM can be accelerated from O(1/k) rate to O(1/k^2). Numerical results will be given to demonstrate the improved speed. In the second part, I will present a novel primal-dual block update method for a multi-block (at least three blocks) problem. Existing works have shown that directly extending two-block ADMM to multi-block problems may diverge. To guarantee convergence, either strong assumptions are made or updating order of the blocks has to be changed. Our method uses a simple randomization technique on choosing block variables, and it enjoys O(1/k) ergodic convergence rate and also global convergence in probability. In addition, by choosing a few blocks every time and using Jacobi-type update, the method enables parallel computing with guaranteed convergence. Numerical experiments will be shown to demonstrate its efficiency compared to other methods.

Computational Mathematics Seminar

Posted September 14, 2017

Last modified September 26, 2017

Computational Mathematics Presentations

In this event for a general audience, we will share information on the education and research opportunities in computational mathematics at LSU. There will be a presentation on the Concentration in Computational Mathematics and several faculty members will talk about their current research. All are welcome.

Computational Mathematics Seminar

Posted August 22, 2017

Last modified September 14, 2017

Xiang Xu, Old Dominion University

Eigenvalue preservation for the Beris-Edwards system modeling nematic liquid crystals

Abstract: The Beris-Edward equations are a hydrodynamic system modeling nematic liquid crystals in the setting of Q-tensor order parameter. Mathematically speaking it is the incompressible Navier-Stokes equations coupled with a Q-tensor equation of parabolic type. In this talk we first consider the simplified Beris-Edward system that corresponds to the co-rotational case, and study the eigenvalue preservation property for the initial Q-tensororder parameter. Then we show that for the full system that relates to the non-rotational case, this property is not valid in general.

Computational Mathematics Seminar

Posted August 22, 2017

Last modified September 26, 2017

Joscha Gedicke, Universitaet Wien

Numerical homogenization of heterogeneous fractional Laplacians

Abstract: In this talk, we develop a numerical multiscale method to solve the fractional Laplacian with a heterogeneous diffusion coefficient. The fractional Laplacian is a non-local operator in its standard form, however the Caffarelli-Silvestre extension allows for a localization of the equation. This adds a complexity of an extra spacial dimension and a singular/degenerate coefficient depending on the fractional order. Using a sub-grid correction method, we correct the basis functions in a natural weighted Sobolev space and show that these corrections are able to be truncated to design a computationally efficient scheme with optimal convergence rates. We further show that we can obtain a greater rate of convergence for sufficient smooth forces, and utilizing a global projection on the critical boundary. We present some numerical examples, utilizing our projective quasi-interpolation in dimension 2+1 for analytic and heterogeneous cases to demonstrate the rates and effectiveness of the method. (This is joint work with Donald L. Brown and Daniel Peterseim.)

Computational Mathematics Seminar

Posted August 23, 2017

Last modified October 17, 2017

Daniele Venturi, University of California Santa Cruz

Data-driven closures for kinetic equations

Abstract: In this talk, I will address the problem of constructing data-driven closures for reduced-order kinetic equations. Such equations arise, e.g., when we coarse-grain high-dimensional systems of stochastic ODEs and PDEs. I will first review the basic theory that allows us to transform such systems into conservation laws for probability density functions (PDFs). Subsequently, I will introduce coarse-grained PDF models, and describe how we can use data, e.g., sample trajectories of the ODE/PDE system, to estimate the unclosed terms in the reduced-order PDF equation. I will also discuss a new paradigm to measure the information content of data which, in particular, allows us to infer whether a certain data set is sufficient to compute accurate closure approximations or not. Throughout the lecture I will provide numerical examples and applications to prototype stochastic systems such as Lorenz-96, Kraichnan-Orszag and Kuramoto-Sivashinsky equations.

Computational Mathematics Seminar

Posted September 12, 2017

Last modified October 10, 2017

Shawn Walker, LSU

A Finite Element Scheme for a Phase Field Model of Nematic Liquid Crystal Droplets

Abstract: We present a phase field model for nematic liquid crystal droplets. Our model couples the Cahn-Hilliard equation to Ericksen's one constant model for liquid crystals with variable degree of orientation. We present a special discretization of the liquid crystal energy that can handle the degenerate elliptic part without regularization. In addition, our discretization uses a mass lumping technique in order to handle the unit length constraint. Discrete minimizers are computed via a discrete gradient flow. We prove that our discrete energy Gamma-converges to the continuous energy and our gradient flow scheme is monotone energy decreasing. Numerical simulations will be shown in 2-D to illustrate the method. This work is joint with Amanda Diegel (post-doc at LSU). Near the end of the talk, I will discuss 3-D simulations of the Ericksen model coupled to the Allen-Cahn equations (with a mass constraint). This work is joint with REU 2017 students (E. Seal and A. Morvant).

Computational Mathematics Seminar

Posted January 16, 2018

3:30 pm - 4:30 pm 1034 Digital Media Center
Amanda Diegel, Louisiana State University

The Cahn-Hilliard Equation, a Robust Solver, and a Phase Field Model for Liquid Crystal Droplets

Abstract: We begin with an introduction to the Cahn-Hilliard equation and some motivations for the use of phase field models. We will then go on to describe a first order finite element method for the Cahn-Hilliard equation and the development of a robust solver for that method. The key ingredient of the solver is a preconditioned minimal residual algorithm (with a multigrid preconditioner) whose performance is independent of the spatial mesh size and the time step size for a given interfacial width parameter. In the second part of the talk, we present a novel finite element method for a phase field model of nematic liquid crystal droplets. The model considers a free energy comprised of three components: the Ericksen''s energy for liquid crystals, the Cahn-Hilliard energy for phase separation, and an anisotropic weak anchoring energy that enforces a boundary condition along the interface between the droplet and surrounding substance. We present the key properties of the finite element method for this model including energy stability and convergence and conclude with a few numerical experiments.