Numerical Analysis Archives for Fall 2013 to Spring 2014
Optimal Control of a Free Boundary Problem with Surface Tension Effects
When: Tue, September 4, 2012 - 3:30pmWhere: Math 3206
Speaker: Patrick Sodre (Department of Mathematics, University of Maryland)
Abstract: 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 in 1991. This amounts to solving a second order system both in the bulk and on the interface. Our analysis provides 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 Frechet differentiable. We demonstrate how to slightly improve the regularity of the state variables and under this regime show existence of a control together with second order sufficient optimality conditions.
Tetrahedral $C^1$ Finite Elements
When: Tue, September 11, 2012 - 3:30pmWhere: Math 3206
Speaker: Noel Walkington (Department of Mathematical Sciences Carnegie Mellon University) - http://www.math.cmu.edu/~noelw/
Abstract: Historically the development of $C^1$ finite elements was motivated by the need to solve the plate and shell equations which are naturally posed in two dimensions. This resulted in a variety of two dimensional $C^1$ elements; however, constructing $C^1$ elements in three dimensions is more difficult. Currently there is a dearth of three dimensional $C^1$ elements in the finite element literature, so problems naturally posed in $H^2(\Omega)$ are often treated with mixed methods or non--conforming elements. Moreover, current tetrahedral $C^1$ elements require a basis for the planes perpendicular to each edge. Such a basis can not depend continuously upon the edge orientation, so global book-keeping is required which breaks the natural modularity inherent in traditional finite element codes.
This talk will review the $C^1$ elements currently available and the practical issues that arise when implementing them which motivated the development of a new tetrahedral element. Modification of classical interpolation arguments and transformation from a parent element for these "non--affine equivalent'" elements will be considered.
Two families of Discontinuous Galerkin Methods for the Monge-Ampere Equation
When: Tue, September 18, 2012 - 3:30pmWhere: Math 3206
Speaker: Prof. Michael Neilan (Department of Mathematics University of Pittsburgh)
Abstract: In this talk, I will discuss two approaches to construct and analyze discontinuous Galerkin (DG) methods for the Monge-Ampere equation, a fully nonlinear second order PDE. The motivating feature of the first approach is to construct consistent schemes such that the discrete inearization is stable. I will describe a methodology for constructing DG methods for the Monge-Ampere equation that inherit this trait and derive three examples which correspond to the SIPG, NIPG and IIPG methods. The second approach is based on the concept of a discrete Hessian recently introduced by Aguilera & Morin (2009), Huang et al. (2010) and Lakkis & Pryer (2011). Replacing the Hessian in the PDE by its discrete counterpart, we obtain convergence schemes that converge even when the exact solution possesses strong singularities.
Hybridizable discontinuous Galerkin surface finite element methods
When: Tue, September 25, 2012 - 3:30pmWhere: Math 3206
Speaker: Prof. Alan Demlow (Department of Mathematics University of Kentucky)
Abstract: While both discontinuous Galerkin and surface finite element methods have both become relatively standard parts of the computational toolbox for solving elliptic problems, little work has been done to develop and analyze surface DG methods. In this talk I will describe recent progress in development and analysis of surface hybridizable discontinuous Galerkin (surface HDG) methods. HDG methods are classes of DG methods in which the only globally coupled degrees of freedom occur on element interfaces. The talk will include discussion of relevant technical issues such as superconvergence properties and appropriate choices of element cornormals as well as connections with both mixed surface finite element methods and other types of discontinuous Galerkin methods.
Kinetic Adaptive Information Measure Closures of Continuum Equations
When: Tue, October 2, 2012 - 3:30pmWhere: Math 3206
Speaker: Prof. Sorin Mitran (Department of Mathematics University of North Carolina Chapel Hill)
Abstract: Continuum models of microscopically non-equilibrated states exhibit
state-dependent closure laws. A predictor-corrector computational
closure procedure is presented for non-equilibrated phenomena based
upon adaptive modification of the influence of continuum-level
constraints upon the probability distribution function describing the
microscopic states. The adaptive procedure results from learning
algorithms applied to previous microscopic time steps. Applications
are presented for both deterministic and stochastic microscopic
dynamics, and some interesting connections to optimal transport theory
are discussed.
Volume-of-fluid modeling of cellular deformation and adhesion
When: Tue, October 9, 2012 - 3:30pmWhere: Math 3206
Speaker: Prof. Damir B. Khismatullin (Department of Biomedical Engineering Tulane University New Orleans)
Abstract: The adhesion of circulating cells to the internal lining of
blood vessels (vascular endothelium) and their shear-induced
deformation play a key role in maintaining body homeostasis,
including protection of the body against invading pathogens or
defective cells and plugging vascular wounds by blood clots. We
have developed a Volume-of-Fluid algorithm for fully
three-dimensional simulation of the deformation and
receptor-mediated adhesion of living cells. In this talk, I
will present this algorithm, called Viscoelastic Cell Adhesion
Model (VECAM), and discuss its application to the problems of
leukocyte-endothelial cell adhesion and lateral migration of
leukocytes in a micro-channel.
Interpolatory Model Reduction: From Diffuse Optical Tomography to Energy Efficient Building Design
When: Tue, October 16, 2012 - 3:30pmWhere: Math 3206
Speaker: Prof. Serkan Gugercin (Department of Mathematics Virginia Tech Blacksburg, VA)
Abstract: Direct numerical simulation of dynamical systems is one of the few available means for accurate prediction or control of complex physical phenomena. However, the need for accuracy leads to ever greater detail in the modeling stage and hence to large-scale, complex dynamical systems, whose simulation may easily exceed available computational resources. This provides the main motivation for model reduction: The goal is to replace the original large-scale system with a lower dimensional one having as nearly as possible the same input/output response characteristics as the original. In recent years, interpolatory projection techniques for model reduction have emerged as powerful candidates as they can produce high-fidelity/optimal approximants without any dense matrix operations; requiring only sparse linear solves.
In this talk, we first introduce the main framework for interpolatory projection methods and then show how to obtain (locally) optimal reduced models via interpolation using the Iterative Rational Krylov Algorithm (IRKA). Effectiveness of IRKA is illustrated in an application from energy efficient building design. We then visit the nonlinear parameter inversion problem where we show how interpolatory methods reduce the evaluation cost of forward problems in diffuse optical tomography by approximating the cost functional and
associated Jacobian with little loss of accuracy while still significantly reducing the cost of the overall inversion problem.
Maximum principle and entropy consistency for numerical approximations of nonlinear hyperbolic conservation laws
When: Tue, October 23, 2012 - 3:30pmWhere: Math 3206
Speaker: Prof. Bojan Popov (Department of Mathematics Texas A&M University)
Abstract: The scalar theory for nonlinear hyperbolic conservation laws is well developed. Namely, maximum principle, entropy stability and convergence of viscosity approximations has been established a long time ago. However, in the case of second or higher order schemes, there are few limited convergence results. We will present our recent stability and convergence result for the second order Nessyahu-Tadmor scheme. The relation of this result with entropy viscosity and entropy stable schemes will be explained. In the case of the Euler system of gas dynamics, we will present a class of viscosity approximations and prove that they have an invariant domain property (the analogue of maximum principle for systems). Moreover, we will identify the subclass of these approximations which is consistent with all entropy inequalities. The connections with the related works of Lax, Hopf, Tadmor, and Harten et al. will be discussed. Some numerical results obtained with entropy viscosity schemes based on this first order approximation of the Euler system will be presented.
<del>Error estimates DG methods for Stokes with minimal regularity</del> Seminar cancelled due to inclement weather
When: Tue, October 30, 2012 - 3:30pmWhere:
Speaker:
Abstract:
Analysis of Hybridizable Discontinuous Galerkin Methods Applied to the Wave Equation
When: Tue, November 6, 2012 - 3:30pmWhere: Math 3206
Speaker: Prof. Peter Monk (Department of Mathematical Sciences University of Delaware)
Abstract: Hybridizable Discontinuous Galerkin (HDG) methods are a class of discontinuous Galerkin methods that allow cell degrees of freedom to be condensed from the discrete linear problem and hence allow the discrete problem to be reduced to a linear system that
only involves unknowns on faces in the mesh. As a result hyrbridizable methods can be implemented more efficiently than other discontinuous Galerkin methods. In addition the HDG scheme of Cockburn, Dong and Guzman allows super-convergent postprocessing. I shall discuss the analysis of this method applied to two problems: the time harmonic Helmholtz equation and the time-dependent wave equation. In the latter case we use a
continuous time Galerkin method to discretize in time. In both cases optimal error estimates will be proved, and numerical results illustrating the convergence will be shown.
Model reduction for systems with high-dimensional parameter spaces
When: Tue, November 13, 2012 - 3:30pmWhere: Math 3206
Speaker: Prof. Karen Willcox (Department of Aeronautics & Astronautics Center for Computational Engineering Massachusetts Institute of Technology)
Abstract: Recent advances in projection-based model reduction methods for nonlinear and parametrically varying systems have opened up a broad new class of potential applications. Problems with large parameter dimension present a significant opportunity for model reduction to accelerate solution of large-scale systems with applications in optimization, inverse problems and uncertainty quantification. However, large parameter dimension also poses a significant challenge, since most model reduction methods rely on sampling the parameter space to
build the reduced-space basis. This talk highlights recent progress on model reduction for large-scale problems with many parameters. Our approaches use a goal-oriented philosophy combined with optimization methods to guide the selection of samples over the parameter space in an adaptive manner. We also show how reduced basis approximations of the state space can be extended to reduce the dimension of the parameter space. We demonstrate our methods in the context of applications in optimization, inverse problems and uncertainty quantification with a variety of engineering examples.
Error estimates DG methods for Stokes with minimal regularity (Note special date and time)
When: Mon, November 19, 2012 - 2:00pmWhere: Math 3206
Speaker: Prof. Johnny Guzman (Division of Applied Mathematics Brown University) -
Abstract: We discuss error estimation of discontinuous Galerkin methods ( DG) for Stokes problem assuming minimal regularity. We first assume that the right hand side belongs in H^{-1} intersect L1. We then modify the right hand side to consider DG methods for only H^{-1} data. We consider DG methods that have equal order polynomial for pressure and velocity and methods were the polynomial order for the pressure is one less than the velocity. We also consider two different ways of stabilizing the DG methods in the equal order case. This is
joint work with S. Badia (Barcelona), R. Codina (Barcelona), and T. Gudi (Bangalore).
Stochastic multiscale modeling of porous media flows
When: Tue, November 27, 2012 - 3:30pmWhere: Math 3206
Speaker: Prof. Ivan Yotov (Department of Mathematics University of Pittsburgh)
Abstract: We discuss a multiscale stochastic framework fo uncertainty quantification in flow in porous media. The governing equations are based on Darcy's law with stochastic permeability represented with a Karhunen-Loeve (KL) expansion. We consider a domain decomposition formulation with different KL expansions in different subdomains. This approach allows to model efficiently heterogeneous media with different rock types. The approximation is based on multiscale mortar mixed finite elements in the spatial domain coupled with stochastic collocation. We precompute a multiscale basis, which involves solving subdomain problems with for each realization of the local KL expansion. The basis is then used to solve the coarse scale mortar interface problem for each global KL realization. The resulting algorithm is orders of magnitude faster than a global stochastic collocation approach. Error analysis for the statistical moments of the pressure and the velocity is performed and experimentally verified with numerical simulations of single phase flow in porous media. If time permits, extensions to coupled Stokes-Darcy flows and contaminant transport will be presented.
Capturing Community Behavior in Very Large Networks using MapReduce
When: Tue, December 4, 2012 - 3:30pmWhere: Math 3206
Speaker: Dr. Tamara G. Kolda (Sandia National Labs, Livermore, CA)
Abstract: http://www.math.umd.edu/~abnersg/golda.pdf
Application of Finite Element Exterior Calculus (FEEC) to Flow Problems
When: Tue, December 11, 2012 - 3:30pmWhere: Math 3206
Speaker: Prof. Long Chen (University of California at Irvine ) -
Abstract: FEEC is a recent advance in the mathematics of finite element methods that employs differential complexes to construct stable numerical schemes for several important types of application problems. In this talk, FEEC is applied to flow problems including Stokes equations and Biot model. More precisely we shall use H(div) elements to obtain point-wise divergence free velocity
approximation to the Stokes equations and use H(curl) element for the Biot model. These discretization is solver-friendly in the sense that efficient multigrid solvers can be developed based on the underling exact sequence. For more general finite element discretizations, our discretization can be served as a preconditioner in the framework of Fast Auxiliary Space Preconditioning (FASP) method to speed up the simulation.
Local Structure Analysis and Defect Characterization in Atomic-Resolution Images
When: Sun, January 13, 2013 - 10:00amWhere: Math 3206
Speaker: Matt Elsey (Courant Institute, NYU) - http://cims.nyu.edu/~melsey/
Abstract: Recent advances allow us to capture enormous atomic-resolution images, but techniques for efficiently analyzing these images are lacking. To address this shortcoming, we propose a variational method which yields a tensor field describing the local crystal strain at each point. Local values of this field describe the crystal orientation and elastic distortion, while the curl of the field locates and characterizes crystal defects and grain boundaries. The proposed energy functional has a simple L2-L1 structure which permits minimization via a split Bregman iteration, and GPU parallelization results in short computing times. Joint work with Benedikt Wirth.
Quadrature by Expansion: A New Method for the Evaluation of Layer Potentials
When: Tue, January 29, 2013 - 3:30pmWhere: Math 3206
Speaker: Andreas Kloeckner (Courant Institute of Mathematical Sciences, New York University) - http://mathema.tician.de/aboutme/
Abstract: Integral equation methods for the solution of partial differential
equations, when coupled with suitable fast algorithms, yield
geometrically flexible, asymptotically optimal and well-conditioned
schemes in either interior or exterior domains. The practical
application of these methods, however, requires the accurate evaluation
of boundary integrals with singular, weakly singular or nearly singular
kernels. I will be presenting a new systematic, high-order approach that
works for any singularity (including hypersingular kernels), based only
on the assumption that the field induced by the integral operator is
locally smooth when restricted to either the interior or the
exterior. Discontinuities in the field across the boundary are
permitted. The scheme, denoted QBX (quadrature by expansion), is easy to
implement and compatible with fast hierarchical algorithms such as the
fast multipole method. I will also present accuracy tests for a variety
of integral operators in two dimensions on smooth and corner
domains. (with L. Greengard, A. Barnett, M. O'Neil)
Heat transmission problems across irregular fractal interfaces
When: Tue, February 5, 2013 - 3:30pmWhere: Math 3206
Speaker: Haodong Liang (Department of Mathematics, Worcester Polytechnic ) -
Abstract:In recent years, there has been a growing interest in studying irregular interfaces that model certain diffusion phenomena across disordered and wild media in natural and industrial processes. These phenoiena are usually modeled by an elliptic or a parabolic boundary value problem with a transmission condition on the interface of order zero, one or two. From the geometrical point of view, irregular interfaces of fractal type are of great interest for these applications, since many irregular media are believed to exhibit fractal properties, such as large surfaces confined in small volumes. More specifically, we consider a two-dimensional second order heat transmission problem across a highly conductive pre-fractal interface of Koch mixture type. This research is motivated by the previous work done by Vacca, Wasyk and Lancia-Cefalo-DellAcqua in the case of Koch curves with fixed fractal contraction factors in (2; 4). We expect to obtain analogous analytic and numeric results by considering more irregular fractal boundaries, which are constructed by alternating mappings with different contraction parameters in each iteration.
Our focus is on the numerical approximation of the solutions, when the number of fractal iteration is fixed. Firstly, we consider the semi-discrete problem by only discretizing the space variable. With the Galerkin method, we would like to obtain an optimal rate of convergence when the refined mesh size goes to zero, as in the classical case when the solution has H2- regularity. It requires to build up suitable refined meshes in the vicinity of the reentrant corners in the domain. This technique goes back to the early work of Grisvard on finite element approximations to Dirichlet problems in polygonal domains. We are concerned to develop a suitable mesh refinement algorithm that could be adapted to our situation, by taking into account of the self-similarity of fractals. Secondly, the fully-discrete problem is obtained by applying a finite difference scheme, so-called $\theta$-method, on the time variable. We finally show results on error estimates, mesh refinements and numerical simulations.
A finite element method for PDEs posed on surfaces
When: Tue, February 12, 2013 - 3:30pmWhere: Math 3206
Speaker: Maxim Olshanskii (Department of Mathematics, University of Houston) - http://www.math.uh.edu/~molshan/
Abstract: Partial differential equations posed on surfaces arise in mathematical models for many natural phenomena:
diffusion along grain boundaries, lipid interactions in biomembranes, pattern formation, and transport of surfactants on multiphase flow interfaces to mention a few. Numerical methods for solving PDEs posed on manifolds
recently received considerable attention. In this talk we review some existing approaches and focus on a new Eulerian finite element method for the discretization of elliptic and parabolic partial differential equations on surfaces. The method uses traces of volume finite element space functions
on a surface to discretize equations posed on the surface. The approach is particularly suitable for problems in which the surface is given implicitly by a level set function, may evolve, and in which there is a coupling with a problem in a fixed outer domain. We present an error analysis, a posteriori estimates and adaptivity, stabilization for transport dominated surface equations, and discuss algebraic properties of the method.
Fully Nonlinear Second Order PDEs and Their Numerical Solutions
When: Tue, February 19, 2013 - 3:30pmWhere: Math 3206
Speaker: Xiaobing Feng (Department of Mathematics, University of Tennessee ) - http://www.math.utk.edu/~xfeng/
Abstract: In this talk, I shall first review some recent developments (and attempts) in numerical methods for fully nonlinear second order PDEs such as the Monge-Ampere type equations and Bellman equations. I shall then present some latest advances on developing finite difference and discontinuous Galerkin (DG) methods for those PDEs. The focus of this talk is to present a newly developed framework for constructing finite difference and DG methods which can reliably approximate viscosity solutions of fully nonlinear second order PDEs. The connection between this new framework with the well-known finite difference and DG framework for fully nonlinear first order Hamilton-Jacobi equations will also be discussed.
Fast Spectral-Galerkin Methods for High-Dimensional PDEs and Applications to the electronic Schrodinger equation
When: Tue, February 26, 2013 - 3:30pmWhere: Math 3206
Speaker: Jie Shen (Department of Mathematics, Purdue University) - http://www.math.purdue.edu/~shen/
Abstract: Many scientific, engineering and financial applications require solving high-dimensional PDEs. However, traditional tensor product based algorithms suffer from the so called "curse of dimensionality".
We shall construct a new sparse spectral method for high-dimensional problems, and present, in particular, rigorous error estimates as well as efficient numerical algorithms for elliptic equations in both bounded and unbounded domains.
As an application, we shall use the proposed sparse spectral method to solve the N-particle electronic Schrodinger equation.
A PDE Approach to Fractional Diffusion: A Priori and A Posteriori Error Analyses
When: Tue, March 5, 2013 - 3:30pmWhere: Math 3206
Speaker: Enrique Otarola (Department of Mathematics, University of Maryland) - http://www2.math.umd.edu/~eotarol1/
Abstract: We study PDE solution techniques for problems involving fractional powers of symmetric coercive elliptic operators in a bounded domain with Dirichlet boundary conditions. These operators can be realized as the Dirichlet to Neumann map of a degenerate/singular elliptic problem posed on a semi-infinite cylinder, which we analyze in the framework of weighted Sobolev spaces. Motivated by the rapid decay of the solution of this problem, we propose a truncation that is suitable for numerical approximation. We discretize this truncation using first order tensor product finite elements. We derive a priori error estimates in weighted Sobolev spaces, which exhibit optimal regularity but suboptimal order for quasi-uniform meshes and quasi-optimal order-regularity for anisotropic meshes. As a first step towards adaptivity, we also present a computable a posteriori error estimator which relies on the solution of small discrete problems on stars. The estimator exhibits built-in flux equilibration and is equivalent to the energy error up to data oscillation. We design a simple adaptive strategy, which reduces error and data oscillation, and present numerical experiments to illustrate the a priori and a posteriori error estimates.
THE MICROPOLAR NAVIER-STOKES EQUATIONS: NUMERICAL ANALYSIS AND PROSPECTIVE APPLICATIONS FOR FERROFLUIDS
When: Tue, March 12, 2013 - 3:30pmWhere: Math 3206
Speaker: Ignacio Tomas (Department of Mathematics, University of Maryland) -
Abstract: The Micropolar Navier-Stokes equations (MNSE) are a system of partial differential equations that describes the motion of a fluid where the microconstituents have both translational and rotational degrees of freedom. It encompasses the laws of conservation mass, linear and angular momenta. We first outline the derivation of the MNSE from continuum mechanics and their extension to ferrofluids along with some prospective applications. Then, we propose and analyze a first order semi-implicit fully-discrete scheme for the unsteady MNSE, which decouples the computation of the linear and angular velocities. The scheme is unconditionally stable and delivers optimal convergence rates under assumptions analogous to those used for the classical Navier-Stokes equations. Finally, we point the main challenges behind the devise of stable and consistent numerical schemes for the equations of Ferrofluids.
Finding directions of variability: A dimension reduction strategy for multivariate response surfaces with applications in uncertainty quantification
When: Tue, April 2, 2013 - 11:00amWhere: 4172 AV Williams (NOTE SPECIAL TIME AND PLACE)
Speaker: Paul Constantine (Department of Mechanical Engineering, Stanford University) -
Abstract: As computational power increases, scientists and engineers
increasingly rely on simulations of complex models to test hypotheses
and inform analyses. Inputs to these models, such as boundary
conditions and material properties, are often underspecified due to a
lack of data, which leads to uncertainty in the model outputs.
Uncertainty quantification (UQ) attempts to endow simulation outputs
with measures of confidence given uncertainty in the model inputs.
Monte Carlo methods are the workhorse of UQ; model inputs are sampled
from a distribution, and the corresponding model outputs are treated
as a data set for statistical analyses. However, the slow convergence
of Monte Carlo approximations coupled with the time intensive model
evaluations has led many to employ response surfaces; once the
response surface has been trained with a few expensive simulations,
its inexpensive predictions can be used in place of the full model.
Unfortunately, most response surfaces suffer from the curse of
dimensionality, i.e., the work required to create an accurate
approximation increases exponentially as the number of inputs
increases.
Many practical models with high dimensional inputs vary primarily
along only a few directions in the space of inputs. I will describe a
method for detecting and exploiting these primary directions of
variability to construct a response surface on a low dimensional
linear subspace of the full input space; detection is accomplished
through analysis of the gradient of the model output with respect to
the inputs, and the subspace is defined by a projection. I will show
error bounds for the low dimensional approximation that motivate
computational heuristics for building a Kriging response surface on
the subspace. As a demonstration, I will apply the method to a
nonlinear heat transfer problem on a turbine blade, where a 250
parameter model for the heat flux represents uncertain transition to
turbulence of the flow field. I will also discuss the range of
existing applications of the method and the future research
challenges.
Another challenge that arises in UQ is managing and analyzing the
large volumes of data (e.g., tens to hundreds of terabytes) generated
by many runs of a highly resolved physical simulation. In the second
part of the talk, I will discuss recent activities employing
Hadoop/MapReduce to compute the singular value decomposition of a
simulation data set whose elements depend on space, time, and and
model parameters; the computed decomposition is used to build reduced
order models for the physical simulation.
Discretizing a model of fracture in a poro-elastic medium
When: Tue, April 2, 2013 - 3:30pmWhere: Math 3206
Speaker: Vivette Girault (Emeritus Professor, Laboratoire Jacques-Louis Lions, Universite Pierre et Marie Curie, Paris, France ) -
Abstract: Discretizing a fracture in a medium is a difficult process because the width of the fracture is very small compared to that of the medium. To avoid the difficulty inherent to this large difference in scale, we resent a nonlinear fracture model in a poro-elastic medium where the fracture is described as a curve or surface according to the dimension, the width of the crack being included into the equation of flow in the fracture. I shall present the model, discuss some mathematical issues related first to a poro-elastic system and next to its coupling with the fracture model, propose some discrete schemes and algorithms for a linearized version of the problem, and present a numerical experiment.
This work is common with M. Wheeler, M. Mear, B. Ganis, and G. Singh (Center for Subsurface Modeling, ICES, U. of Texas at Austin).
Stability and error estimates in the maximum norm for finite-element discretizations of the Stokes and Navier-Stokes problems
When: Wed, April 3, 2013 - 3:15pmWhere: Math 3206 (This is an Applied Math Colloquium)
Speaker: Vivette Girault (Emeritus Professor, Laboratoire Jacques-Louis Lions, Universite Pierre et Marie Curie, Paris, France ) -
Abstract: The derivation in the energy norm of uniform stability estimates for the finite element discretization of the Stokes problem is now a fairly standard procedure, i.e., the discrete pressure and components of the velocity gradient are readily measured in $L^2(\Omega)$. This follows easily from the variational formulation provided the discrete pressure and velocity satisfy a uniform inf-sup condition.
But deriving uniform stability estimates in $L^\infty(\Omega)$ is much more complex because variational formulations do not lend themselves to maximum norms. I shall present here the main ideas of a proof that relies on weighted $L^2$ estimates for regularized Green's functions associated with the Stokes problem and on a weighted inf-sup condition. The domain $\Omega$ is a convex polygon or polyhedron according to the dimension. The triangulation is shape-regular and quasi-uniform. The finite element spaces satisfy a super-approximation property, which is shown to be valid for most commonly used stable finite-element spaces. Extending this result to error estimates and to the solution of the steady incompressible Navier-Stokes problem is straightforward.
This work is common with R. Nochetto (U. of Maryland) and R. Scott (U. of Chicago).
A priori error estimates for parabolic optimal control problems with point controls
When: Tue, April 16, 2013 - 3:30pmWhere: Math 3206
Speaker: Dmitriy Leykekhman (Department of Mathematics, University of Connecticut) - http://www.math.uconn.edu/~leykekhman/
Abstract: We consider a parabolic optimal control problem with point controls in space, but variable in time, in two space dimensions. We use the standard continuous piecewise linear approximation in space and the first order discontinuous Galerkin method in time to approximate the problem numerically. Despite low regularity of the state equation, we obtain almost optimal $h^2+k$ convergence rate for the control in $L^2$ norm. The main
ingredients of the analysis are the new global and local fully discrete a priori error estimates in $L^2([0,T]; L^\infty(\Omega))$ norms for the parabolic problems.
Tree Approximation in High Dimensions
When: Tue, April 23, 2013 - 3:30pmWhere: Math 3206
Speaker: Peter Binev (Department of Mathematics, University of South Carolina) -
Abstract: Trees are often used to describe the process of receiving adaptive partitions. For problems in which the complexity of the algorithms is important, the results in tree approximation are therefore much more appropriate than the ones for the general N-term approximation. The results about near-best approximation on trees are critical in establishing convergence rates in many problems, in particular for adaptive finite element methods. In this talk we consider the specific obstacles that appear when applying adaptive partitioning and tree approximation in cases in which the spatial dimension is high. As a particular example we consider the approximation of a high dimensional function known through sample points drown according to an unknown probability measure and the quality of fit is evaluated via a norm with respect to this measure (that often is essentially low dimensional). The paradigm of sparse occupancy trees is the main tool to process the data and to find the approximation. It allows scalable algorithms and on-line data assimilation.
Large-scale sparse optimization
When: Tue, April 30, 2013 - 3:30pmWhere: Math 3206
Speaker: Wotao Yin (Department of Computational and Applied Mathematics, Rice University) - http://www.caam.rice.edu/~wy1/
Abstract: Sparse optimization has found interesting applications in many data-processing areas such as machine learning, signal processing, compressive sensing, medical imaging, finance, etc. This talk reviews existing sparse optimization algorithms and introduces ones tailored for very large scale sparse optimization problems with terabytes of data through distributed and/or decentralized computation. The talk introduces ideas that enable the state-of-the-art sequential algorithms such as the fast iterative soft-thresholding algorithm (FISTA), alternating direction method of multipliers (ADMM), linear Bregman method (LBreg), and greedy block coordinate-descent method (GBCD) for parallel and even decentralized processing of data. Besides the typical complexity analysis, we analyze the increased iterations and communication overhead due to parallelization or decentralization. Numerical results are presented to demonstrate the scalability of the parallel codes for handling tera-scale problems.
Short bio: Wotao Yin is an associate professor with Rice University, the Department of Computational and Applied Mathematics. His research interests lie in computational optimization and its applications in image processing, machine learning, medical imaging, and other inverse problems. He received his B.S. in mathematics from Nanjing University in 2001, and then M.S. and Ph.D. in operations research from Columbia University in 2003 and 2006, respectively. Since 2006, he has been with Rice University. He won NSF CAREER award in 2008 and Alfred P. Sloan Research Fellowship in 2009. His recent work has been in optimization algorithms for large-scale and distributed signal processing and machine learning problems.
Discrete total variation flows without regularization and applications
When: Tue, May 7, 2013 - 3:30pmWhere: Math 3206
Speaker: Abner J. Salgado (Department of Mathematics, University of Maryland) - www.math.umd.edu/~abnersg
Abstract: We propose and analyze an algorithm for the solution of the L2-subgradient flow of the total variation (TV) functional. The algorithm involves no regularization, thus the numerical solution preserves the main features that motivates the use of this type of energy both in imaging and materials sciences. We derive L2 error estimates under minimal regularity assumptions, and introduce a TV-diminishing interpolation operator which yields improved error bounds. We also propose an iterative scheme for the solution of the ensuing discrete problems and analyze it. This methodology extends to the TV functional augmented with a strictly convex functional, such as a p-Laplacian term. We discuss several numerical experiments which illustrate the power and potentials of the method.
Finite Element Approximations of Fractional Power of Self-Adjoint Elliptic Operators on Bounded Domains
When: Tue, May 14, 2013 - 3:30pmWhere: Math 3206 (This is a special seminar during finals week)
Speaker: Andrea Bonito (Department of Mathematics, Texas A&M University) - http://www.math.tamu.edu/~bonito/
Abstract: Taking advantage of the spectral properties of elliptic self-adjoint operators,
we deduce a representation formula for their fractional powers.
We show that fractional powers reduce to a singular integral over the positive real numbers of a perturbation of the original operator.
Then, we derive a novel numerical algorithm based on this new representation.
A quadrature formula approximating the one dimensional singular integral is proposed while standard finite element methods are advocated for the space discretization.
The quadrature points are distributed adequatly to capture the (known) singularity and the multiple but independent resulting elliptic problems are approximated using standard finite elements.
The algorithm turns out to be adequate for parallel implementations and scales perfectly.
We finally discuss optimal a-priori L2-error estimates in terms of the number of degree of freedoms used for the space discretization and the number of quadrature points.
This is joint work with J.E. Pasciak.