The GeoClaw software for tsunamis and other hazardous flows
Abstract: Many geophysical flows over topography can be modeled by two-dimensional depth-averaged fluid dynamics equations. The shallow water equations are the simplest example of this type, though it is often necessary to incorporate non-hydrostatic pressures, more complicated rheologies (e.g. for avalanches, landslides, or debris flows), or to use multi-layer models, e.g. for capturing internal waves or to model a landslide-induced tsunamis.
These equations are generally hyperbolic and can be modeled using high-resolution finite volume methods designed for such problems. However, several features of these flows lead to new algorithmic challenges, such as the fact that the depth goes to zero at the edge of the flow and that vastly differing spatial scales must often be modeled, making adaptive mesh refinement essential. I will discuss some of these algorithms and the GeoClaw software, a specialized version of Clawpack that is aimed at solving real-world geophysical flow problems over topography. In particular I will show results from some recent tsunamis and potential future events. For information about the software, see www.clawpack.org/geoclaw.
On detection of low emission sources in the presence of a large random background
Abstract: One of the missions of the Department of Homeland Security is to prevent smuggling of weapon-grade nuclear materials. It is expected that such materials, unlike those needed for a "dirty bomb," will have low emission rates and will be well shielded, so that very few gamma photons or neutrons could escape, and even less would be detected. An additional hurdle for the detection of illicit nuclear substances is the strong natural radiation background. In this talk I will discuss when detection of such sources is feasible and how Compton camera type detectors could be used for this purpose.
Multigrid methods for Maxwell’s equations
Abstract: In this work we study finite element methods for two-dimensional Maxwell’s equations and their solutions by multigrid algorithms. We first introduce two types of nonconforming finite element methods on graded meshes for a two-dimensional curl-curl and grad-div (CCGD) problem that appears in electromagnetics. The first method is based on a discretization using weakly continuous P1 vector fields. The second method uses discontinuous P1 vector fields. Optimal convergence rates in the energy norm and the L2 norm are established for both methods on graded meshes. Then we consider a class of symmetric discontinuous Galerkin methods for a model Poisson problem on graded meshes that share many techniques with the nonconforming methods for the CCGD problem. We establish the uniform convergence of W-cycle, V-cycle and F-cycle multigrid algorithms for the resulting discrete problems. Finally, we propose a new numerical approach for two-dimensional Maxwell’s equations that is based on the Hodge decomposition for divergence-free vector fields, and present multigrid results.
Hierarchical approximations, coarse-graining and fast lattice Monte Carlo simulations
Abstract: We shall discuss numerical analysis aspects of coarse-graining stochastic particle systems and the connection to acceleration of kinetic Monte Carlo simulations. Mathematical tools developed for error control in microscopic simulations using the coarse-grained stochastic processes and reconstruction of microscopic scales will be presented in connection with accelerating (kinetic) Monte Carlo simulations. On specific examples of lattice as well as off-lattice dynamics we demonstrate that computational implementation of constructed hierarchical algorithms results in significant speed up of simulations. The developed framework also leads to new parallel kinetic Monte Carlo algorithms that will be briefly described.
Simulating nonholonomic mechanics using variational integrators through Hamiltonization
Abstract: Although it is well known that nonholonomic mechanical systems are not Hamiltonian, recent research has uncovered a variety of techniques which allow one to express the reduced, constrained dynamics of certain classes of nonholonomic systems as Hamiltonian. In this talk I will discuss the application of these methods to develop alternative geometric integrators for nonholonomic systems with perhaps more eacuteciency than the known nonholonomic integrators. After showing how variational integrators theoretically preserve conserved mechanical quantities (such as momentum and energy), I will discuss how Hamiltonization can be used to apply these variational integrators to certain classes of nonholonomic systems. Finally, I will discuss some current research utilizing time reparameterizations.
Robust numerical solution of singularly perturbed problems
Abstract: Singularly perturbed differential equations are usually posed with a small positive (perturbation) parameter multiplying the highest derivative. Their solutions typically exhibit boundary or interior layers. In recent years much effort has been directed towards constructing and analysing so-called "parameter robust" methods. Such methods should yield solutions whose accuracy does not depend on the perturbation parameter, and should resolve any layers present.
In this talk I will survey some of these methods, and the mathematics behind them, with particular emphasis on finite differences for coupled systems.
Discontinuous Galerkin approximation for the Vlasov-Poisson system
Abstract: One of the simplest model problems in the kinetic theory of plasma--physics is the Vlasov-Poisson (VP) system with periodic boundary conditions. Such system describes the evolution of a plasma of charged particles (electrons and ions) under the effects of the transport and self-consistent electric field. In this talk, we construct a new family of semi-discrete numerical schemes for the approximation of the Vlasov-Poisson system. The methods are based on the coupling of discontinuous Galerkin (DG) approximation to the Vlasov equation and several finite element (conforming, non-conforming and mixed) approximations for the Poisson problem. We present the error analysis in the case of smooth solutions. The issue of energy conservation is also analyzed for some of the methods. If time allows, I will also comment on the issue of approximating non-smooth solutions of the VP system. The talk is based on joint works with J.A. Carrillo (Universidad Autonoma de Barcelona) and C-W. Shu (Brown University).
The Poincaré lemma and the computation of domain integrals in BEM
Abstract: An effective technique to evaluate domain integrals appearing in a boundary element method (BEM) has been developed . The proposed approach first converts a domain integral with continuous or weakly-singular integrand into an equivalent boundary integral. The resulting surface integral is then calculated via standard quadrature rules commonly used for boundary elements. This transformation of a domain integral into a boundary counterpart is accomplished through a systematic and rigorous generalization of the fundamental theorem of calculus to higher dimension. Moreover, it is shown that the higher-dimensional version of the first fundamental theorem of calculus corresponds to the classical Poincaré lemma.
Employed together with the singular treatment of surface integrals that is well established in the literature [2, 3, 4], the proposed method can be utilized to ectively solve boundary-value problems involving non-homogeneous source terms by way of a collocation or a Galerkin BEM without partitioning the problem domain into volume cells. Several key features of this study, including the extension of the fundamental theorem of calculus to higher dimension, are highlighted. In addition, numerical examples dealing with mixed boundary-value problems for the Poisson equation on representative test geometries are carried out successfully to validate the proposed method.
 S. Nintcheu Fata, Treatment of domain integrals in boundary element methods, Appl. Numer. Math., doi:10.1016/j.apnum.2010.07.003, 2010.
 S. Nintcheu Fata, Explicit expressions for 3D boundary integrals in potential theory, Int. J. Num. Meth. Eng., 78(1), pp. 32—47, 2009.
 S. Nintcheu Fata, L. J. Gray, Semi-analytic integration of hypersingular Galerkin BIEs for three-dimensional potential problems, J. Comput. Appl. Math., 231(2), pp. 561—576, 2009.
 S. Nintcheu Fata, Semi-analytic treatment of nearly-singular Galerkin surface integrals, Appl. Numer. Math., 60(10), pp. 974—993, 2010.
This research was supported by the Office of Advanced Scientific Computing Research, U.S. Department of Energy, under contract DE-AC05-00OR22725 with UT-Battelle, LLC.
December 7, 2010
Guido Kanschat (Department of Mathematics, Texas A & M University)
Discontinuous Galerkin methods for radiative transfer: Some old results and some new results
Abstract: In the 1970s, discontinuous Galerkin (DG) methods were invented as a means to solve neutron/radiation transport problems. Their convergence analysis had been developed by the early 1980s. Between 1989 and 2000 several publications in nuclear engineering suggested, that the method does not converge in scattering dominated regimes. In this presentation, we will review these seemingly contradicting results. A first robustness result requires that the DG finite element space contains an approximating continuous space. Since this result is not sufficient for applications, we use the information contained in the analysis to devise a new DG method, which will converge to the correct solution independent of the model parameters.
January 25, 2011
Pawel Konieczny (2nd year Postdoc)
On dispersive effect of the Coriolis force for the stationary Navier-Stokes equations
Abstract: The dispersive effect of the Coriolis force for the stationary and nonstationary Navier-Stokes equations is investigated. Existence of a unique stationary solution is shown for arbitrary large external force provided the Coriolis force is large enough. In addition to the stationary case, counterparts of several classical results for the non-stationary Navier-Stokes problem have been proven. The analysis is carried out in a new framework of the Fourier-Besov spaces.
February 1, 2011
Aycil Cesmelioglu (1st year Postdoc)
On the coupling of surface/subsurface flow with transport
Abstract: The coupling of porous media flow with free flow arises in many applications an example of which is groundwater contamination through rivers. The free flow is characterized by the Navier-Stokes equations whereas the porous media flow is described by the Darcy equations. Beavers-Joseph-Saffman interface condition is prescribed at the interface separating two regions. A transport equation for the contaminant concentration is fully coupled to the flow problem via the velocity field and the viscosity. First, we discuss the existence result to the related weak formulation of the full coupling problem. Second, we analyze numerical schemes based on classical finite element methods and discontinuous Galerkin methods for the special case where the coupling is only one-way, that is, the velocity field from the Navier-Stokes/Darcy problem is an input for the transport equation. Numerical solutions for non homogeneous porous media are also presented.
February 15, 2011
Hengguang Li (1st year Postdoc)
FEMs and MG methods for axisymmetric problems
Abstract: We shall discuss finite element and multigrid techniques solving the axisymmetric Poisson's equation and the azimuthal Stokes problem on polygonal domains with possible singular solutions. In particular, we construct stable interpolation operators and establish the well-posedness and regularity in some weighted Sobolev space, which in turn, leads to special finite element spaces to approximate the solutions in the optimal rate. With a careful formulation, we also obtain uniform convergence of the MG methods. These estimates can also be used to show the stability of the Taylor-Hood elements for the axisymmetric Stokes problem and to precondition the indefinite system from the axisymmetric Stokes equations.
February 22, 2011
Weifeng (Frederick) Qiu (1st year Postdoc)
An hp DPG method for linear elasticity with symmetric stresses abstract.pdf
Joint work with Jamie Bramwell 3, Leszek Demkowicz 2, and Jay Gopalakrishnan 1.
Abstract: In this research, we present two Discontinuous Petrov-Galerkin (DPG) finite element methods for linear elasticity. For the first method, we consider asymmetric test tensors for the constitutive equation and compute infinitessimal rotations, while in the second method we only use symmetric test tensors and therefore have fewer unknowns. We define optimal test functions which are shown to deliver the best approximation error if an optimal global test norm is used. To make the method practical, we show a localizable test norm is equivalent to the global optimal norm. The majority of this proof is the verification that the inf-sup condition holds for our DPG formulations using the localizable test space norm. From DPG theory, this proves our methods are quasi-optimal with constants independent of the mesh. We can then use results from approximation theory to show h and p convergence for both methods.
Since the quasi-optimal test space norm is localizable, we have implemented practical finite element codes that show h and p convergence of both methods at optimal rates. Additionally, the DPG framework provides an a priori error estimator determined by a local auxilliary variational problems. We use this estimator as the basis for various 'greedy' adaptive schemes. We test our adaptive algorithm using a manufactured smooth solution as well as a singular solution L-shape domain problem and observe adaptive h and hp convergence.
The principal contributions of this research are proving p convergence for the dual-mixed elasticity system, particularly without the need for a discrete exact sequence or commuting diagram, as well as a practical adaptive 2D elasticity code with a priori error estimation. We will present an overview of the theoretical DPG framework, the convergence proofs for both methods, and the numerical results for both a singular and smooth solution.
1 Professor, Mathematics, University of Florida
2 Professor, Institute for Computational Engineering and Sciences,
3 Graduate Research Assistant, Institute for Computational Engineering and Sciences, University of Texas at Austin
March 22, 2011
David Yu Mao (1st year Postdoc)
Reconstruction of binary functions and shapes from incomplete frequency information
Abstract: Binary functions are a class of important functions that appears in many applications, e.g. image segmentation, bar code recognition, shape detection and so on. In this research we proved that under certain conditions the binary function can be reconstructed from very limited frequency information by using only simple linear programming. Numerical results and applications will be discussed.
On the use of the finite-fault solution for tsunami generation problems
Abstract: We present a new approach to describe accurately the generation of a tsunami wave due to an underwater earthquake. The main goal of this work is two-fold. First of all, we propose a simple and computationally inexpensive model for the description of the sea bed displacement during an underwater earthquake, based on the finite fault solution for the slip distribution under some assumptions on the dynamics of the rupturing process. Once the bottom motion is reconstructed, we study waves induced on the free surface of the ocean. For this purpose we consider three different models approximating the Euler equations of the water wave theory. Namely, we use the linearized Euler equations (we are in fact solving the Cauchy-Poisson problem), a Boussinesq system and a novel weakly nonlinear model. An intercomparison of these approaches is performed. The developments of the present study are illustrated on the July 27, 2006 Java event, where an underwater earthquake of magnitude 7.7 generated a tsunami that inundated the southern coast of Java.
Data Analysis and Uncertainty Quantification of Inverse Problems
Abstract: We present exploratory data analysis methods to assess inversion estimates using simple examples based on classic l2- and l1-regularization. These methods can be used to reveal the presence of systematic errors such as bias and discretization effects, or to validate assumptions made on the statistical model used in the analysis. The methods include: bound for randomized trace estimators, confidence intervals and bounds for the bias, resampling methods for model validation, and construction of training sets of functions with controlled local regularity.
April 26, 2011
Maria Pia Gualdani (Department of Mathematics, The University of Texas at Austin)
A factorization method for non-symmetric linear operator: enlargement of the functional space while preserving hypo-coercivity
Abstract: We present a factorization method for non-symmetric linear operators: the method allows to enlarge functional spaces while preserving spectral properties for the considered operators. In particular, spectral gap and related convergence towards equilibrium follow easily by hypo-coercivity and resolvent estimates. Applications of this theory on several kinetic equations will be presented.
Smoothness of Nonlinear Subdivision Curves
Abstract: Subdivision Curves were discovered by Georges de Rham in 1947. He noticed that the sequence of polygonal lines obtained by a "corner cutting" (or "subdivision") process converges to a C1 curve. B-splines are obtained by a generalization of this procedure, which depends on the affine structure of Euclidean space. A "linear subdivision curve" is the limiting curve obtained by such a linear subdivision process. The theory of such curves, is well-understood. Because subdivision curves also support multiresolution structure, various wavelet based compression algorithms apply to them. Consequently, subdivision curves offer a way to compress data related to curves in Euclidean space.
But not all data live in Euclidean space (rotations, rigid motions, deformation tensors are examples), and the quantity of such data is proliferating. In "Multiscale representations of manifold-valued data", Rahman, Rori, Stodden, Donoho, and Schroder, therefore, considered manifold-valued subdivision curves, and their multiresolution structure. These curves are all based on corresponding linear subdivision curves, and Rahman-et-al. conjectured that they enjoy the same regularity properties.
In my talk, I will review linear subdivision curves, explain the generalization to manifold-valued curves, and close with some surprising results concerning their regularity properties.
The talk is based on joint work with Gang Xie and Thomas Yu.
An introduction to DPG methods
Abstract: We will discuss a new class of discontinuous Petrov-Galerkin (DPG) methods that achieves stability by automatically finding stable pairs of trial and test spaces in a Petrov-Galerkin framework. The key is that the use of discontinuous finite element spaces (as in DG methods) allows local computation of the test spaces that guarantee stability. (This is joint work with L. Demkowicz.)
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