Posted August 8, 2008

Last modified August 12, 2008

Computational Mathematics Seminar

11:00 am - 12:00 pm 338 Johnston Hall
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.

Posted August 29, 2008

Computational Mathematics Seminar

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

Posted October 7, 2008

Last modified October 21, 2008

Computational Mathematics Seminar

11:00 am - 12:00 pm Johnston 338
Zhijun Wu, Iowa State University

Bioinformatics and Biocomputing

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

Posted November 11, 2008

Computational Mathematics Seminar

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

Posted March 16, 2009

Last modified April 13, 2009

Computational Mathematics Seminar

11:00 am - 12:00 pm
Luke Owens, Texas A&M University

Solving the Eikonal equation on adaptive triangular and tetrahedral meshes

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

Posted May 7, 2009

Computational Mathematics Seminar

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.

Posted August 20, 2009

Computational Mathematics Seminar

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.

Posted August 24, 2009

Last modified August 25, 2009

Computational Mathematics Seminar

3:10 pm - 4:00 pm 338 Johnston Hall
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.

Posted August 28, 2009

Computational Mathematics Seminar

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.

Posted September 8, 2009

Last modified October 15, 2009

Computational Mathematics Seminar

3:10 pm - 4:00 pm 338 Johnston Hall
Eun-Hee Park, Louisiana State University

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

Posted September 8, 2009

Last modified October 15, 2009

Computational Mathematics Seminar

3:10 pm - 4:00 pm 338 Johnston Hall
Hongchao Zhang, Louisiana State University

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

Posted February 10, 2010

Last modified February 17, 2010

Computational Mathematics Seminar

3:30 pm - 4:30 pm 338 Johnston Hall
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

Posted February 26, 2010

Last modified March 1, 2010

Computational Mathematics Seminar

3:40 pm - 4:30 pm 233 Lockett
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.

Posted March 30, 2010

Computational Mathematics Seminar

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

Introduction to CUDA

Posted March 21, 2010

Computational Mathematics Seminar

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

Posted April 8, 2010

Computational Mathematics Seminar

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

Posted June 5, 2011

Computational Mathematics Seminar

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

Posted July 27, 2011

Last modified August 20, 2011

Computational Mathematics Seminar

3:30 pm Johnston Hall 338Higher Order Estimates In Time For The Arbitrary Lagrangian Eulerian Formulation In Moving Domains

Speaker: Andrea Bonito

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

Posted July 15, 2011

Last modified August 20, 2011

Computational Mathematics Seminar

3:30 pm Johnston Hall 338A Diffuse Interface Model For Electrowetting

Speaker: Abner J. Salgado

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

Posted August 20, 2011

Last modified August 31, 2011

Computational Mathematics Seminar

3:30 pm - 4:30 pm Johnston 338
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.

Posted August 31, 2011

Last modified November 29, 2022

Computational Mathematics Seminar

3:30 pm - 4:30 pm Johnston 338
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 methods. 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

Posted September 12, 2011

Computational Mathematics Seminar

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.

Posted September 14, 2011

Computational Mathematics Seminar

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

Posted October 3, 2011

Computational Mathematics Seminar

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

Posted October 3, 2011

Last modified November 29, 2022

Computational Mathematics Seminar

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 solving 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 approximated 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

Posted November 1, 2011

Computational Mathematics Seminar

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)

Posted November 1, 2011

Last modified November 29, 2022

Computational Mathematics Seminar

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 elongated 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 )

Posted November 1, 2011

Computational Mathematics Seminar

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 )

Posted February 5, 2012

Computational Mathematics Seminar

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

Posted March 5, 2012

Computational Mathematics Seminar

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.

Posted December 5, 2011

Last modified March 5, 2012

Computational Mathematics Seminar

3:30 pm - 4:30 pm 338 Johnston Hall
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.

Posted March 6, 2012

Computational Mathematics Seminar

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.

Posted March 27, 2012

Computational Mathematics Seminar

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.

Posted October 1, 2012

Last modified October 9, 2012

Computational Mathematics Seminar

3:30 pm - 4:30 pm 338 Johnston Hall
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.

Posted September 8, 2012

Last modified November 29, 2022

Computational Mathematics Seminar

3:00 pm - 4:00 pm 338 Johnston Hall
Klaus Boehmer, Philipps-Universität 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, *symmetry breaking bifurcation*. Despite many simplifications 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.

Posted November 1, 2012

Computational Mathematics Seminar

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)

Posted November 1, 2012

Computational Mathematics Seminar

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.)

Posted October 9, 2012

Computational Mathematics Seminar

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.

Posted October 1, 2013

Computational Mathematics Seminar

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

Posted October 1, 2013

Last modified October 4, 2013

Computational Mathematics Seminar

3:30 pm - 4:30 pm Lockett 233
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

Posted November 12, 2013

Last modified November 20, 2013

Computational Mathematics Seminar

2:30 pm - 3:30 pm Lockett 233
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.

Posted February 3, 2014

Computational Mathematics Seminar

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

Partition of Unity Methods for Fourth Order Problems

Posted February 8, 2014

Computational Mathematics Seminar

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

Posted February 19, 2014

Computational Mathematics Seminar

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

Numerical Methods for Transmission Eigenvalues

Posted February 11, 2014

Computational Mathematics Seminar

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

Posted February 23, 2014

Last modified November 29, 2022

Computational Mathematics Seminar

3:30 pm - 4:30 pm Lockett 233
Eun-Jae Park, Yonsei University, South Korea

Recent Progress in Hybrid Discontinuous Galerkin Methods

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 Baumann-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.

Posted August 25, 2014

Computational Mathematics Seminar

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

Posted September 2, 2014

Last modified November 29, 2022

Computational Mathematics Seminar

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.

Posted November 13, 2014

Computational Mathematics Seminar

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.

Posted January 19, 2015

Last modified March 2, 2021

Computational Mathematics Seminar

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

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.

Posted January 30, 2015

Computational Mathematics Seminar

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

Attempting to Reverse the Irreversible in Quantum Physics

Posted March 2, 2015

Last modified November 29, 2022

Computational Mathematics Seminar

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

A Fast Algorithm for Polyhedral Projection

In this talk, we discuss a very efficient algorithm for projecting a point onto a polyhedron. This algorithm solves the projection 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.

Posted January 26, 2015

Computational Mathematics Seminar

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.

Posted March 2, 2015

Last modified March 18, 2015

Computational Mathematics Seminar

3:30 pm - 4:30 pm 1034 Digital Media Center
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

Posted March 22, 2015

Computational Mathematics Seminar

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.

Posted March 2, 2015

Last modified March 18, 2015

Computational Mathematics Seminar

3:30 pm - 4:30 pm 1034 Digital Media Center
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.

Posted September 3, 2015

Computational Mathematics Seminar

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.

Posted September 3, 2015

Computational Mathematics Seminar

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.

Posted November 5, 2015

Computational Mathematics Seminar

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.

Posted January 21, 2016

Computational Mathematics Seminar

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.

Posted January 30, 2016

Last modified February 11, 2016

Computational Mathematics Seminar

3:30 pm - 4:30 pm Digital Media Center 1034
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.

Posted January 26, 2016

Last modified February 23, 2016

Computational Mathematics Seminar

9:00 am - 10:00 am Digital Media Center 1034 Originally scheduled for 3:30 pm, Tuesday, 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.

Posted March 7, 2016

Computational Mathematics Seminar

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.

Posted January 26, 2016

Last modified February 11, 2016

Computational Mathematics Seminar

3:30 pm - 4:30 pm Digital Media Center 1034
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).

Posted January 21, 2016

Computational Mathematics Seminar

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)

Posted September 14, 2016

Computational Mathematics Seminar

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.)

Posted September 14, 2016

Computational Mathematics Seminar

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.

Posted September 29, 2016

Last modified October 10, 2016

Computational Mathematics Seminar

3:30 pm - 4:30 pm 1034 Digital Media Center
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.

Posted October 18, 2016

Computational Mathematics Seminar

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.

Posted October 18, 2016

Computational Mathematics Seminar

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.

Posted October 31, 2016

Last modified November 27, 2016

Computational Mathematics Seminar

3:30 pm - 4:30 pm Digital Media Center 1034
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.

Posted February 2, 2017

Computational Mathematics Seminar

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.)

Posted April 3, 2017

Computational Mathematics Seminar

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.

Posted August 23, 2017

Last modified August 29, 2017

Computational Mathematics Seminar

3:30 pm - 4:30 pm 1034 Digital Media Center
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.

Posted August 22, 2017

Last modified September 26, 2017

Computational Mathematics Seminar

3:30 pm - 4:30 pm 1034 Digital Media Center
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.

Posted September 14, 2017

Last modified September 26, 2017

Computational Mathematics Seminar

4:15 pm - 5:00 pm 1034 Digital Media CenterComputational 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.

Posted August 22, 2017

Last modified November 29, 2022

Computational Mathematics Seminar

3:30 pm - 4:30 pm 1034 Digital Media Center
Xiang Xu, Old Dominion University

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

The Beris-Edwards 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-Edwards system that corresponds to the co-rotational case, and study the eigenvalue preservation property for the initial Q-tensor order parameter. Then we show that for the full system that relates to the non-rotational case, this property is not valid in general.

Posted August 22, 2017

Last modified September 26, 2017

Computational Mathematics Seminar

3:30 pm - 4:30 pm 1034 Digital Media Center
Joscha Gedicke, Universität 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.)

Posted August 23, 2017

Last modified October 17, 2017

Computational Mathematics Seminar

3:30 pm - 4:30 pm 1034 Digital Media Center
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.

Posted September 12, 2017

Last modified October 10, 2017

Computational Mathematics Seminar

3:30 pm - 4:30 pm 1034 Digital Media Center
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).

Posted January 16, 2018

Last modified November 29, 2022

Computational Mathematics Seminar

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

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.

Posted January 30, 2018

Last modified February 14, 2018

Computational Mathematics Seminar

3:30 pm - 4:30 pm 1034 Digital Media Center
Ellya Kawecki, Oxford University

A discontinuous Galerkin finite element method for Hamilton Jacobi Bellman equations on piecewise curved domains

Abstract: We introduce a discontinuous Galerkin finite element method (DGFEM) for Hamilton Jacobi Bellman equations on piecewise curved domains, and prove that the method is consistent, stable, and produces optimal convergence rates. Upon utilising a long standing result due to N. Krylov, we may characterise the Monge Ampere equation as a HJB equation; in two dimensions, this HJB equation can be characterised further as uniformly elliptic HJB equation, allowing for the application of the DGFEM.

Posted February 14, 2018

Computational Mathematics Seminar

3:30 pm - 4:30 pm 1034 Digital Media Center
Yi Zhang, University of North Carolina at Greensboro

Numerical Approximations for a Singular Elliptic Variational Inequality

Abstract: The displacement obstacle problem of simply supported plates is an example of a fourth order variational inequality. As the bending rigidity tends to zero the problem degenerates to an elastic membrane obstacle problem which is a second order variational inequality. In this talk we will introduce C0 interior penalty methods for this singular perturbed problem with small parameter. Robust error estimates with respect to the parameter will be presented. We also discuss the convergence of numerical solutions to the unperturbed second order elliptic variational inequality. This is joint work with Susanne Brenner and Li-yeng Sung.

Posted January 30, 2018

Last modified February 14, 2018

Computational Mathematics Seminar

3:30 pm - 4:30 pm 1034 Digital Media Center
Liping Wang, Nanjing University of Aeronautics and Astronautics

A Joint Matrix Minimization Approach and the Applications in Collective Face Recognition and Seismic Wavefield Recovery

Abstract: Recently, image-set based face recognition and multi trace seismic wavefield recovery have attracted extensive attention in pattern recognition and geophysical community. Representation coding is one of popular ways for both face recognition and seismic wave reconstruction. Similar representative coding pattern among the group of samples is observed both in facial images and seismic signals. To take account of the collective correlation from a given set of testing samples as well as each individual, a matrix minimization model is presented to jointly representing all the testing samples over the coding bases simultaneously. A generalized matrix norms employed to measure the interrelation of the multiple samples and the entries of each one. For solving the involved matrix optimization problem, a unified algorithm is developed and the convergence analysis is accordingly demonstrated for the range of parameters p in (0,1]. Extensive experiments on public data of facial images and real-world seismic waves exhibit the efficient performance of the joint technique over the state-of-the-art methods in recognition or recovery accuracy and computational cost.

Posted March 5, 2018

Computational Mathematics Seminar

3:30 pm - 3:30 pm 1034 Digital Media Center
Jun-Hong Liang, Louisiana State University

Horizontal Dispersion of Buoyant Materials in the Ocean Surface Boundary Layer 1

Abstract: In this talk I will discuss our recent study that uses a large eddy simulation model for ocean surface gravity wave filtered incompressible Navier-Stokes equation to study how buoyant material spreads in the upper ocean. The results of the study will improve the prediction of the pathway of marine pollutants such as spilled oil and microplastics.

Posted March 5, 2018

Computational Mathematics Seminar

3:30 pm - 4:30 pm 1034 Digital Media Center
Shu Lu, Univeristy of North Carolina at Chapel Hill

Statistical inference for sample average approximation of constrained optimization and variational inequalities

Abstract: The sample average approximation is widely used as a substitute for the true expectation function in optimization and equilibrium problems. We study how to provide a confidence region or confidence intervals for the true solution, once the SAA solution is obtained. Our method is based on the asymptotic distribution of the SAA solution, and we handle polyhedral constraints by examining the nonsmooth structure of the asymptotic distribution.

Posted March 5, 2018

Last modified March 19, 2018

Computational Mathematics Seminar

3:30 pm - 4:30 pm 1034 Digital Media Center
Longfei Li, University of Louisiana at Lafayette

Overcoming the added-mass instability for coupling incompressible flows and elastic beams

Abstract: A new partitioned algorithm is described for solving fluid-structure interaction (FSI) problems coupling incompressible flows with elastic structures undergoing finite deformations. The new algorithm, referred to as the Added-Mass Partitioned (AMP) scheme, overcomes the added-mass instability that has for decades plagued partitioned FSI simulations of incompressible flows coupled to light structures. Within a Finite-Difference framework, the AMP scheme achieves fully second-order accuracy and remains stable, without sub-time-step iterations, even for very light structures when added-mass effects are strong. The stability and accuracy of the AMP scheme is validated through mode analysis and numerical experiments. Aiming to extend the AMP scheme to an Finite-Element framework, we also develop an accurate and efficient Finite-Element Method for solving the incompressible Navier-Stokes Equations with high-order accuracy up-to the boundary.

Posted August 20, 2018

Computational Mathematics Seminar

3:30 pm - 4:30 pm 1034 Digital Media Center
Jianchao Bai, Xian Jiaotong University

Deterministic and Stochastic ADMM for Structured Convex Optimization

Abstract: The Alternating Direction Method of Multipliers (ADMM) has a long history, but its algorithmic idea can be still used to design new algorithms for the application examples involving big-data. In this talk, we show our recent work about two kinds of deterministic ADMMs and a family of stochastic ADMM for solving structured convex optimization. We also present the convergence complexity of these ADMM-type algorithms. Several further questions are discussed finally. (Refreshments at 3pm.)

Posted September 17, 2018

Computational Mathematics Seminar

3:30 pm - 4:30 pm 1034 Digital Media Center
Jose Garay, Louisiana State University

Asynchronous Optimized Schwarz Methods for Partial Differential Equations in Rectangular Domains

https://www.cct.lsu.edu/lectures/asynchronous-optimized-schwarz-methods-partial-differential-equations-rectangular-domains

Posted August 19, 2018

Last modified September 2, 2018

Computational Mathematics Seminar

3:30 pm - 4:30 pm 1034 Digital Media Center
Minah Oh, James Madison University

The Hodge Laplacian on Axisymmetric Domains

Abstract: An axisymmetric problem is a problem defined on a three-dimensional (3D) axisymmetric domain, and it appears in numerous applications. An axisymmetric problem can be reduced to a sequence of two-dimensional (2D) problems by using cylindrical coordinates and a Fourier series decomposition. A discrete problem corresponding to the 2D problem is significantly smaller than that corresponding to the 3D one, so such dimension reduction is an attractive feature considering computation time. Due to the Jacobian arising from change of variables, however, the resulting 2D problems are posed in weighted function spaces where the weight function is the radial component r. Furthermore, formulas of the grad, curl, and div operators resulting from the so-called Fourier finite element methods are quite different from the standard ones, and it is well-known that these operators do not map the standard polynomial spaces into the next one. In this talk, I will present stability and convergence results of the mixed formulations arising from the axisymmetric Hodge Laplacian by using a relatively new family of finite element spaces that forms an exact sequence and that satisfies the abstract Hilbert space framework developed by Arnold, Falk, and Winther.

Posted September 5, 2018

Last modified October 18, 2018

Computational Mathematics Seminar

3:30 pm - 4:30 pm 1034 Digital Media Center
Shawn Walker, LSU

A Numerical Scheme for the Generalized Ericksen Model of Liquid Crystals With Applications to Virus DNA Packing

Abstract: We consider the generalized Ericksen model of liquid crystals, which is an energy with 8 independent ``elastic'' constants that depends on two order parameters n (director) and s (variable degree of orientation). In addition, we present a new finite element discretization for this energy, that can handle the degenerate elliptic part without regularization, is stable and it Gamma-converges to the continuous energy. Moreover, it does not require the mesh to be weakly acute (which was an important assumption in our previous work). A minimization scheme for computing discrete minimizers will also be discussed. Furthermore, we include other effects such as weak anchoring (normal and tangential), as well as fully coupled electro-statics with flexo-electric and order-electric effects. We also present several simulations (in 2-D and 3-D) illustrating the effects of the different elastic constants and electric field parameters. At the end of the talk, we discuss a problem about the packing of DNA inside viral capsids. We show how the generalized Ericksen model can be used to simulate the packing of DNA inside viral capsids, and to estimate packing pressures inside the capsid. This part is joint with Carme Calderer (UMN), Dmitry Golovaty (U. Akron).

Posted September 5, 2018

Last modified October 18, 2018

Computational Mathematics Seminar

3:30 pm - 4:30 pm 1034 Digital Media Center
Andrew Gillette, University of Arizona

Serendipity Finite Element Methods in Theory and Practice

Abstract: Serendipity finite element methods present a promising computational advantage over traditional tensor product finite elements: a significant reduction in degrees of freedom without sacrificing the order of accuracy in the computed solution. The theory of serendipity methods dates back to the 1970s but has seen a resurgence of interest in recent years within the context of finite element exterior calculus and the Periodic Table of the Finite Elements. In this talk, I will review modern perspectives on the family of serendipity elements and present the accompanying family of ``trimmed serendipity'' elements from my recent work. On the practical side, I will also discuss developments on the construction of basis functions for serendipity-type elements and their use on non-affinely mapped mesh element geometries. This is joint work with Tyler Kloefkorn and Victoria Sanders.

Posted December 2, 2018

Computational Mathematics Seminar

3:30 pm - 4:30 pm 1034 Digital Media Center
Xin Wang, University of Maryland

Semidefinite Optimization for Quantum Information Processing

Abstract: In this talk, I will show how to apply semidefinite optimization to study two basic lines of quantum information processing: entanglement manipulation and communication over quantum channels. Novel mathematical tools improve our understanding of the structure of quantum entanglement and the limits of information processing with quantum systems. In the first part, I will discuss the fundamental features of quantum entanglement and develop quantitative approaches to better exploit the power of entanglement. I will introduce a computable and additive entanglement measure to quantify the amount of entanglement, which also plays an important role as the improved semidefinite programming (SDP) upper bound of distillable entanglement. Notably, I will demonstrate the irreversibility of asymptotic entanglement manipulation under positive-partial-transpose-preserving quantum operations, resolving a long-standing open problem in quantum information. In the second part, I will develop a framework of semidefinite programs to evaluate the classical and quantum communication capabilities of quantum channels in both the non-asymptotic and asymptotic regimes, which can be applied as benchmarks for near-term quantum codes. In particular, I will discuss the first general SDP strong converse bound on the classical capacity of an arbitrary quantum channel and give in particular the best known upper bound on the classical capacity of the amplitude damping channel. I will further establish a finite resource analysis of classical communication over basic channels such as the quantum erasure channel.

Posted February 14, 2019

Last modified March 15, 2019

Computational Mathematics Seminar

3:30 pm - 4:30 pm 1034 Digital Media Center
Hongbo Dong, Washington State University

On structured sparsity learning with affine sparsity constraints

Abstract: We introduce a new constraint system, namely affine sparsity constraints (ASC), as a general optimization framework for structured sparse variable selection in statistical learning. Such a system arises when there are nontrivial logical conditions on the sparsity of certain unknown model parameters to be estimated. One classical nontrivial logical condition is the heredity principle in regression models, where interaction terms of predictor variables can be introduced into the model only if the corresponding linear terms already exist in the model. Formally, extending a cardinality constraint, an ASC system is defined by a system of linear inequalities of binary indicators, which represent nonzero patterns of unknown parameters in estimation. We study some fundamental properties of such a system, including set closedness and set convergence of approximations, by using tools in polyhedral theory and variational analysis. We will also study conditions under which optimization with ASC can be reduced to integer programs or mathematical programming with complementarity conditions (MPCC), where algorithms and efficient implementation already exist. Finally, we will focus on the problem of regression with heredity principle, with our previous results, we derive nonconvex penalty formulations that are direct extensions of convex penalties proposed in the literature for this problem.

Posted March 6, 2019

Last modified March 15, 2019

Computational Mathematics Seminar

3:30 pm - 4:30 pm 1034 Digital Media Center
Xiaoliang Wan, Louisiana State University

Coupling the reduced-order model and the generative model for an importance sampling estimator

Abstract: In this work, we develop an importance sampling estimator by coupling the reduced-order model and the generative model in a problem setting of uncertainty quantification. The target is to estimate the probability that the quantity of interest (QoI) in a complex system is beyond a given threshold. To avoid the prohibitive cost of sampling a large scale system, the reduced-order model is usually considered for a trade-off between efficiency and accuracy. However, the Monte Carlo estimator given by the reduced-order model is biased due to the error from dimension reduction. To correct the bias, we still need to sample the fine model. An effective technique to reduce the variance reduction is importance sampling, where we employ the generative model to estimate the distribution of the data from the reduced-order model and use it for the change of measure in the importance sampling estimator. To compensate the approximation errors of the reduced-order model, more data that induce a slightly smaller QoI than the threshold need to be included into the training set. Although the amount of these data can be controlled by a posterior error estimate, redundant data, which may outnumber the effective data, will be kept due to the epistemic uncertainty. To deal with this issue, we introduce a weighted empirical distribution to process the data from the reduced-order model. The generative model is then trained by minimizing the cross entropy between it and the weighted empirical distribution. We also introduce a penalty term into the objective function to deal with the overfitting for more robustness. Numerical results are presented to demonstrate the effectiveness of the proposed methodology.

Posted February 14, 2019

Last modified March 15, 2019

Computational Mathematics Seminar

3:30 pm - 4:30 pm 1034 Digital Media Center
Winnifried Wollner, Technische Universität Darmstadt

PDE constrained optimization with pointwise constraints on the derivative of the state

Abstract: In many processes modeled by partial differential equations (PDE) the, pointwise, size of the gradient is a key quantity. Prominent examples for this are damage or plasticity models. In the optimization of such processes pointwise constraints on the gradient are natural. The numerical analysis of these problems is complicated by the fact, that the natural topology coming from the PDE is too weak for handling the bounds on the gradient. Within this talk, we will discuss existence of solutions to such problems as well as their approximability by finite elements with particular emphasis on non-smooth domains.

Posted March 6, 2019

Last modified November 29, 2022

Computational Mathematics Seminar

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

A Revisit of Gradient Descent Method for Nonlinear Optimization

In this talk, we will discuss some recent advances of the gradient methods developed in nonlinear optimization, including steepest descent methods, Barzilai-Borwein type methods, optimal gradient methods, quasi-Newton methods and conjugate gradient methods. Our focus will be the convergence properties of these methods as well as their practical performances.

Posted October 8, 2019

Computational Mathematics Seminar

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

A Nonmonotone Smoothing Newton Algorithm for Weighted Complementarity Problem

Abstract: The weighted complementarity problem, often denoted by WCP, significantly extends the general complementarity problem and can be used for modeling a larger class of problems from science and engineering. In this talk, by introducing a one-parametric class of smoothing functions, we will introduce a smoothing Newton algorithm with nonmonotone line search to solve WCP. We will discuss the global convergence as well as local superlinear or quadratic convergence of this algorithm under assumptions weaker than assuming the nonsingularity of the Jacobian. Some promising numerical results will be also reported.

Posted September 9, 2019

Last modified October 13, 2019

Computational Mathematics Seminar

3:30 pm - 4:30 pm 1034 Digital Media Center
Jose Garay, Louisiana State University

Localized Orthogonal Decomposition Method with Additive Schwarz for the Solution of Multiscale Elliptic Problems

Abstract: The solution of elliptic Partial Differential Equations (PDEs) with multiscale diffusion coefficients using regular Finite Element methods (FEM) typically requires a very fine mesh to resolve the small scales, which might be unfeasible. The use of generalized finite elements such as in the method of Localized Orthogonal Decomposition (LOD) requires a coarser mesh to obtain an approximation of the solution with similar accuracy. We present a solver for multiscale elliptic PDEs based on a variant of the LOD method. The resulting multiscale linear system is solved by using a two-level additive Schwarz preconditioner. We provide an analysis of the condition number of the preconditioned system as well as the numerical results which validate our theoretical results.

Posted September 9, 2019

Last modified October 13, 2019

Computational Mathematics Seminar

3:30 pm - 4:30 pm 1034 Digital Media Center
Yakui Huang, Hebei University of Technology

On the Asymptotic Convergence and Acceleration of Gradient Methods

Abstract: We consider the asymptotic behavior of a family of gradient methods, which include the steepest descent and minimal gradient methods as special instances. It is proved that each method in the family will asymptotically zigzag between two directions. Asymptotic convergence results of the objective value, gradient norm, and stepsize are presented as well. To accelerate the family of gradient methods, we further exploit spectral properties of stepsizes to break the zigzagging pattern. In particular, a new stepsize is derived by imposing finite termination on minimizing two dimensional strictly convex quadratic function. It is shown that, for the general quadratic function, the proposed stepsize asymptotically converges to the reciprocal of the largest eigenvalue of the Hessian. Furthermore, based on this spectral property, we propose a periodic gradient method by incorporating the Barzilai-Borwein method. Numerical comparisons with some recent successful gradient methods show that our new method is very promising.

Posted January 7, 2020

Computational Mathematics Seminar

3:30 pm - 4:20 pm Digital Media Center 1034
Giordano Tierra-Chica, University of North Texas

Numerical Schemes for Mixtures of Isotropic and Nematic Flows Taking into Account Anchoring and Stretching Effects

The study of interfacial dynamics between two different components has become the key role to understand the behavior of many interesting systems. Indeed, two-phase flows composed of fluids exhibiting different microscopic structures are an important class of engineering materials. The dynamics of these flows are determined by the coupling among three different length scales: microscopic inside each component, mesoscopic interfacial morphology and macroscopic hydrodynamics. Moreover, in the case of complex fluids composed by the mixture between isotropic (newtonian fluid) and nematic (liquid crystal) flows, its interfaces exhibit novel dynamics due to anchoring effects of the liquid crystal molecules on the interface.

In this talk I will introduce a PDE system to model mixtures composed by isotropic fluids and nematic liquid crystals, taking into account viscous, mixing, nematic, stretching and anchoring effects and reformulating the corresponding stress tensors in order to derive a dissipative energy law. Then, I will present new linear unconditionally energy-stable splitting schemes that allows us to split the computation of the three pairs of unknowns (velocity-pressure, phase field-chemical potential and director vector-equilibrium) in three different steps. The fact of being able to decouple the computations in different linear sub-steps maintaining the discrete energy law is crucial to carry out relevant numerical experiments under a feasible computational cost and assuring the accuracy of the computed results.

Finally, I will present several numerical simulations in order to show the efficiency of the proposed numerical schemes, the influence of the shape of the nematic molecules (stretching effects) in the dynamics and the importance of the interfacial interactions (anchoring effects) in the equilibrium configurations achieved by the system.

(Refreshments at 3:00PM in the Computational Math Area of LDMC)

Posted February 15, 2020

Computational Mathematics Seminar

3:30 pm - 4:30 pm Digital Media Center 1034
Li Wang, University of Texas at Arlington

Probabilistic Semi-supervised Learning via Sparse Graph Structure Learning

Abstract: We present a probabilistic semi-supervised learning (SSL) framework based on sparse graph structure learning. Different from existing SSL methods with either a predefined weighted graph heuristically constructed from the input data or a learned graph based on the locally linear embedding assumption, the proposed SSL model is capable of learning a sparse weighted graph from the unlabeled high-dimensional data and a small amount of labeled data, as well as dealing with the noise of the input data. Our representation of the weighted graph is indirectly derived from a unified model of density estimation and pairwise distance preservation in terms of various distance measurements, where latent embeddings are assumed to be random variables following an unknown density function to be learned and pairwise distances are then calculated as the expectations over the density for the model robustness to the data noise. Moreover, the labeled data based on the same distance representations is leveraged to guide the estimated density for better class separation and sparse graph structure learning. A simple inference approach for the embeddings of unlabeled data based on point estimation and kernel representation is presented. Extensive experiments on various data sets show the promising results in the setting of SSL compared with many existing methods, and significant improvements on small amounts of labeled data. div

Posted September 6, 2022

Computational Mathematics Seminar

3:30 pm DMC 1034
Casey Cavanaugh, Louisiana State University

Structure-Preserving Discretizations for Partial Differential Equations

Models arising from partial differential equations (PDEs) often include physical laws such as conservation of energy or source-free flows, and theoretical properties such as the maximum principle. To accurately capture these types of features in numerical simulation, we consider structure-preserving discretization methods which guarantee that continuous level properties are satisfied exactly on the discrete level. This talk will focus on building connections between two known structure-preserving methods: the mimetic finite-difference (MFD) method, where discrete differential operators "mimic" their continuous level counterparts, and a mixed finite-element method (FEM) based on finite-element exterior calculus. First, we examine MFD discretizations for two PDE models: Maxwell's equations, describing the coupling between electric and magnetic fields, and convection-dominated diffusion equations, which are typically challenging to solve due to numerical oscillations from shocks and boundary layers. Then, by exploiting the connections between MFD and FEM, we demonstrate how a FE framework can provide the MFD method with supplementary theory such as well-posedness, stability, error estimates, and multigrid solvers.

Posted September 21, 2022

Computational Mathematics Seminar

3:30 pm - 4:30 pm Digital Media Center 1034
Frederic Marazzato, Louisiana State University

Homogenized Origami Surfaces

Origami folds have found a large range of applications in Engineering as, for instance, solar panels for satellites, or the folding of airbags for optimal deployment or metamaterials. A homogenization process turning origami folds into smooth surfaces, developed in [Nassar et al, 2017], is first discussed. Then, its application to two specific folds is presented alongside the PDEs characterizing the associated smooth surfaces. The talk will then focus on the PDEs describing Miura surfaces by studying existence and uniqueness of solutions and by proposing a numerical method to approximate them. Finally, some numerical examples are presented. div

Posted October 2, 2022

Computational Mathematics Seminar

3:30 pm - 4:30 pm Zoom
Zi Yang, University of California, Santa Barbara

The Multi-Objective Polynomial Optimization

The multi-objective optimization is to optimize several objective functions over a common feasible set. Since the objectives usually do not share a common optimizer, people often consider (weakly) Pareto points. This paper studies multi-objective optimization problems that are given by polynomial functions. First, we study the convex geometry for (weakly) Pareto values and give a convex representation for them. Linear scalarization problems (LSPs) and Chebyshev scalarization problems (CSPs) are typical approaches for getting (weakly) Pareto points. For LSPs, we show how to use tight relaxations to solve them, how to detect existence or nonexistence of proper weights. For CSPs, we show how to solve them by moment relaxations. Moreover, we show how to check if a given point is a (weakly) Pareto point or not and how to detect existence or nonexistence of (weakly) Pareto points. We also study how to detect unboundedness of polynomial optimization, which is used to detect nonexistence of proper weights or (weakly) Pareto points. ZOOM: Meeting ID958 6951 8026 SecurityPasscode BRENNER https://lsu.zoom.us/j/95869518026?pwd=T2U3R0J1WGdMUFlKNEVhbkJndXZQZz09

Posted November 11, 2022

Computational Mathematics Seminar

3:30 pm - 4:30 pm Zoom
Tongtong Li, Dartmouth College

Improving Numerical Accuracy for the Viscous-plastic Formulation of Sea Ice

Accurate modeling of sea ice dynamics is critical for predicting environmental variables, which in turn is important in applications such as navigating ice breaker ships, and has led to extensive research in both modeling and simulating sea ice dynamics. The most widely accepted model is the one based on the viscous-plastic formulation introduced by Hibler, which is intrinsically difficult to solve numerically due to highly nonlinear features. In particular, sea ice simulations often significantly differ from satellite observations. In this study we focus on improving the numerical accuracy of the viscous-plastic sea ice model. We explore the convergence properties for various numerical solutions of the sea ice model and in particular examine the poor convergence seen in existing numerical methods. To address these issues, we demonstrate that using higher order methods for solving conservation laws, such as the weighted essentially non-oscillatory (WENO) schemes, is critical for numerically solving viscous-plastic formulations whenever the solution is not smooth. Moreover, WENO yields higher order convergence for smooth solutions than standard central differencing does. Our numerical examples verify this, and in particular by using WENO, we are able to resolve the discontinuities in the sharp features of sea ice covers. We also propose an approach utilizing the idea of phase field method to develop a potential function method which naturally incorporates the physical restrictions of ice thickness and ice concentration in transport equations. Our approach results in modified transport equations with extra forcing terms coming from potential energy function, and has the advantage of not requiring any post-processing procedure that might introduce discontinuities and thus ruin the solution behavior. Zoom: Meeting ID958 6951 8026 SecurityPasscode BRENNER https://lsu.zoom.us/j/95869518026?pwd=T2U3R0J1WGdMUFlKNEVhbkJndXZQZz09

Posted November 11, 2022

Last modified November 22, 2022

Computational Mathematics Seminar

3:30 pm - 4:30 pm DMC 1034
Jose Garay, Louisiana State University

DD-LOD: A Localized Orthogonal Decomposition Method for Elliptic Problems with Rough Coefficients Based on Domain Decomposition Techniques

The solution of multi-scale elliptic problems with non-separable scales and high contrast in the coefficients by standard Finite Element Methods (FEM) is typically prohibitively expensive since it requires the resolution of all characteristic lengths to produce an accurate solution. Numerical homogenization methods such as Localized Orthogonal Decomposition (LOD) methods provide access to feasible and reliable simulations of such multi-scale problems. These methods are based on the idea of a generalized finite element space whose basis functions are obtained by modifying standard coarse standard FEM basis functions to incorporate relevant microscopic information in a computationally feasible procedure. Using this enhanced basis one can solve a much smaller problem to produce an approximate solution whose accuracy is comparable to the solution obtained by the expensive standard FEM. We present a variant of the LOD method that utilizes domain decomposition techniques and its applications in the solution of elliptic partial differential equations with rough coefficients as well as elliptic optimal control problems with rough coefficients with and without control constraints.

Posted January 17, 2023

Last modified January 20, 2023

Computational Mathematics Seminar

3:30 pm - 4:20 pm LDMC: room 1034
Daniel Massatt, Louisiana State University

Convergence of the Planewave Approximations for Quantum Incommensurate Systems

Incommensurate structures arise from stacking single layers of low-dimensional materials on top of one another with misalignment such as an in-plane twist in orientation. While these structures are of significant physical interest, they pose many theoretical challenges due to the loss of periodicity. In this paper, we characterize the density of states of Schrödinger operators in the weak sense for the incommensurate system and develop novel numerical methods to approximate them. In particular, we (i) justify the thermodynamic limit of the density of states in the real space formulation; and (ii) propose efficient numerical schemes to evaluate the density of states based on planewave approximations and reciprocal space sampling. We present both rigorous analysis and numerical simulations to support the reliability and efficiency of our numerical algorithms.

Posted April 10, 2023

Computational Mathematics Seminar

3:30 pm - 4:30 pm Zoom
Jia-Jie Zhu, Weierstrass Institute for Applied Analysis and Stochastics

Distributionally Robust Optimization in Kernel and Unbalanced Transport Geometry

This distribution shift in machine learning (ML) can happen as a consequence of causal confounding, unfairness due to data biases, and adversarial attacks. In such cases, recent optimizers adopt robustification strategies derived from distributionally robust optimization (DRO). For example, one of the most interesting directions of DRO is the adoption of the optimal transport distance, the Wasserstein distance. While the Wasserstein DRO literature has exploded, it is restricted to the pure transport regime, often similar to existing regularization techniques already used by machine learners. To make matters worse, many state-of-the-art Wasserstein DRO methods based place severe limitations on the learning models and scalability, making them inapplicable to modern ML tasks. With those limitations in mind, I will introduce mathematical tools beyond the Wasserstein DRO using unbalanced optimal transport and kernel geometry. I will also discuss ML applications such as robust learning under distribution shift. Zoom link: https://lsu.zoom.us/j/6653973295

Posted January 12, 2023

Last modified April 20, 2023

Computational Mathematics Seminar

3:30 pm - 4:20 pm LDMC: room 1034
Matthias Maier, Department of Mathematics Texas A&M University

Structure-preserving finite-element schemes for the Euler-Poisson equations

We discuss structure-preserving numerical discretizations for the repulsive and attractive Euler-Poisson equations. The scheme is fully discrete and structure preserving in the sense that it maintains a discrete energy law, as well as hyperbolic invariant domain properties, such as positivity of the density and a minimum principle of the specific entropy. We discuss the underlying algebraic discretization technique based on collocation and convex limiting that maintain hyperbolic invariants and a discrete energy law, and discuss recovery strategies to maintain the discrete Gauss law.

Posted August 22, 2023

Last modified September 5, 2023

Computational Mathematics Seminar

3:30 pm - 4:30 pm Digital Media Center 1034 Originally scheduled for 3:30 pm, Tuesday, September 5, 2023
Xiaoliang Wan, Louisiana State University

Adaptive Sampling for the Neural Network Approximation of PDEs

Abstract: Deep learning-based numerical methods are being actively investigated for the approximation of PDEs from different perspectives including numerical analysis, algorithm development and applications. One common key component of these learning-based approximation methods is the training set, which consists of random samples in the computation domain. These random samples define a discrete optimization problem for the optimal neural network approximate solution. In this talk, we pay particular attention to the training set and demonstrate that adaptive sampling can improve significantly the accuracy of the neural network approximation especially for low-regularity and high-dimensional problems. https://www.cct.lsu.edu/lectures/adaptive-sampling-neural-network-approximation-pdes

Posted August 23, 2023

Computational Mathematics Seminar

3:30 pm - 4:30 pm Digital Media Center 1034
Zequn Zheng, Louisiana State University

Generating Polynomial Method for Non-symmetric Tensor Decomposition

Abstract: Tensors or multidimensional arrays are higher order generalizations of matrices. They are natural structures for expressing data that have inherent higher order structures. Tensor decompositions play an important role in learning those hidden structures. There exist both optimization-based methods and algebraic methods for the tensor decomposition problem, optimization-based methods regard the tensor decomposition problem as a nonconvex optimization problem and apply optimization methods to solve it. Hence, they usually suffer from local minimum and may not be able to find a satisfactory tensor decomposition. Algebraic methods usually require the tensor rank to be not too large and the running time is not so satisfying for large tensors. In this talk, we present a novel algorithm to find the tensor decompositions utilizing generating polynomials. Under some conditions on the tensor's rank, we prove that the exact tensor decomposition can be found by our algorithm. Numerical examples successfully demonstrate the robustness and efficiency of our algorithm. https://www.cct.lsu.edu/lectures/generating-polynomial-method-non-symmetric-tensor-decomposition

Posted September 28, 2023

Computational Mathematics Seminar

3:30 pm - 4:30 pm DMC 1034
Sara Pollock, University of Florida

TBA