All Seminars
| Title: Scalable Efficient Methods for Incompressible Fluid-dynamics in Engineering Problems |
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| Defense: Dissertation |
| Speaker: Umberto Villa of Emory University |
| Contact: Umberto Villa, uvilla@emory.edu |
| Date: 2012-11-12 at 4:30PM |
| Venue: MSC W301 |
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| Abstract: Accurate and effective methods for the numerical solution of incompressible fluid dynamics is an old but still important challenging problem, as more and more complex problems in engineering biology, ecology, medicine, sport are tackled with computational methods.\\ \\ In this dissertation defense, we investigate efficient solvers for two important models that governs the motion of a fluid, the incompressible Navier-Stokes and the Brinkman equations. The former describes the motion of an incompressible fluid in either an open or closed domain. The latter is used for describing the dynamics in a matrix of an inhomogeneous porous media, alternating bubbles and open channels.\\ \\ For the solution of the unsteady Navier-Stokes Equations, we move from the pressure correction algebraic factorization formerly proposed by Saleri, Veneziani (2005). These schemes feature an intrinsic hierarchical nature, such that an accurate approximation of the pressure Schur complement is obtained by computing intermediate low-order guesses. The difference between the pressure at two successive correction steps provides a natural a-posteriori estimator for time adaptivity with no additional computational cost. \\ \\ For the solution of the Brinkman Equations, we follow the approach presented in Mardal, Winther (2011) to precondition symmetric saddle point problems in a Hilbert settings. More specifically, we first present a novel mixed formulation of the Brinkman problem, with improved stability properties, in which we introduce the flow's vorticity as additional unknown. Based on stability analysis of the problem in the H(curl) - H(div) - L2 norms, we derive a scalable block diagonal preconditioner which is optimal in the constant coefficient case.\\ \\ Algorithms and preconditioners analysed in this thesis have been implemented in a parallel C++ code, using the finite element libraries LifeV and MFEM, and the linear algebra libraries Trilinos and HYPRE. We emphasize the performance of the proposed algorithms in solving problems of practical interest, involving complex geometries and realistic flow conditions. Numerical experiments in 2D and 3D confirm the effectiveness of our approach showing good efficiency and parallel scalability properties of the solvers proposed. |
| Title: Iterative Methods and Partitioning Techniques for Large Sparse Problems in Network Analysis |
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| Defense: Dissertation |
| Speaker: Verena Kuhlemann of Emory University |
| Contact: Verena Kuhlemann, vkuhlem@emory.edu |
| Date: 2012-11-09 at 1:00PM |
| Venue: MSC W303 |
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| Abstract: The analysis of networks is an important aspect in many fields. Here we consider three different topics: the numerical solution of Markov chains, ranking of genes, and parallel computations with large scale-free graphs.\\ \\ First, additive Schwarz methods are a class of domain decomposition methods that are suitable for the solution of large linear systems in serial as well as in parallel mode. We adapt the Restricted Additive Schwarz (RAS) method to the computation of the stationary probability distribution vector of large, sparse, irreducible stochastic matrices. The convergence properties are analyzed and extensive numerical experiments aimed at assessing the effect of varying the number of subdomains and the choice of overlap are discussed.\\ \\ Next, the ranking of genes plays an important role in the identification of key genes for a specific disease. A modification of the PageRank algorithm that is used to rank web pages is the GeneRank algorithm. We assessed the performance of additive Schwarz methods as well as a Chebyshev iteration for the solution of the GeneRank problem.\\ \\ Finally, many large networks are scale-free. That is, the degree distribution follows a power-law. Currently available graph partitioners are not efficient for such an irregular degree distribution. The lack of a good partitioning leads to excessive inter-processor communication requirements during every matrix-vector multiplication on parallel distributed-memory computers. We present an approach to alleviate this problem based on embedding the original irregular graph into a more regular one by disaggregating (splitting up) vertices in the original graph. Even though the latter graph is larger, we are able to decrease the communication requirements considerably and improve the performance of matrix-vector multiplication. |
| Title: On Folkman-type problems |
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| Seminar: Combinatorics |
| Speaker: Andrzej Dudek of Western Michigan University |
| Contact: Dwight Duffus, dwight@mathcs.emory.edu |
| Date: 2012-11-09 at 4:00PM |
| Venue: MSC W303 |
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| Abstract: A classical Ramsey theorem states that in any 2-coloring of the edges of a sufficiently large complete graph, one will always find a monochromatic complete subgraph. In 1970, Folkman extended this result showing that for any graph $G$ there exists a graph $H$ with the same clique number as $G$ such that any 2-coloring of the edges of $H$ yields a monochromatic copy of $G$. In this talk, we present some old and recent developments concerning Folkman-type results. |
| Title: Rotationally Symmetric Planes in Comparison Geometry |
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| Defense: Dissertation |
| Speaker: Eric Choi of Emory University |
| Contact: Eric Choi, echoi7@emory.edu |
| Date: 2012-11-08 at 12:00PM |
| Venue: MSC W502 |
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| Abstract: Global Riemannian geometry is concerned with relating geometric data such as curvature, volume, and radius to topological data. Cheeger-Gromoll showed that any noncompact complete manifold $M$ with nonnegative sectional curvature contains a boundaryless, totally convex, compact submanifold $S$, called a soul, such that $M$ is homeomorphic to the normal bundle over $S$. In the first part of our talk, we show that if $M$ is a rotationally symmetric plane $M_m,$ defined by metric $dr^2 + m^2(r) d\theta^2$, then the set of souls is a closed geometric ball centered at the origin, and if furthermore $M_m$ is a von Mangoldt plane, then the radius of this ball can be explicitly determined. We show that the set of critical points of infinity in $M_m$ is equal to this set of souls, and we make some additional observations on the set of critical points of infinity when $M_m$ is von Mangoldt with a point at which the sectional curvature is negative. We close out the first part of the talk with showing that the slope $m^{\prime}(r)$ of $M_m$ near infinity can be prescribed with any number in $(0, 1]$. In the second part of our talk, we extend results in a paper by Kondo-Tanaka in which the authors generalize the Toponogov Comparison Theorem such that an arbitrary noncompact manifold $M$ is compared with a rotationally symmetric plane $M_m$ satisfying certain conditions to establish that $M$ is topologically finite. We substitute one of the conditions for $M_m$ with a weaker condition and show that our method using this weaker condition enables us to draw further conclusions on the topology of $M$. We also completely remove one of the conditions required for the Sector Theorem, one of the principal results in the same paper. |
| Title: Modern Krylov Subspace Methods (and applications to Parabolic Control Problems) |
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| Seminar: Numerical Analysis and Scientific Computing |
| Speaker: Daniel B. Szyld of Temple University |
| Contact: Dr. Michele Benzi, benzi@mathcs.emory.edu |
| Date: 2012-11-08 at 4:00PM |
| Venue: MSC W201 |
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| Abstract: In many circumstances, a known good preconditioner is not easily computable. Instead, an approximation to it is available. This is the case, for example, when the preconditioner has an inverse associated with it, such as in Schur complements (e.g., in saddle point problems), or in the reduced Hessian in some control problems. The application of the preconditioner implies then an iterative solution of a linear system. In these cases, the question is: how accurately to solve the (inner) iteration? In our work on Inexact Krylov methods, we have shown that the inner iterations can be solved progressively less accurately, as the underlying Krylov method (e.g., GMRES) converges to the overall solution. Computable inner stopping criteria were developed to guarantee convergence of the overall method. We discuss these criteria, and illustrate its application to several problems. In particular, we apply these ideas to parabolic control problems, where the reduced Hessian contains two different inverses, and thus two inner iteration criteria are needed. Truncated methods are also discussed. |
| Title: Minimal Free Resolutions of the toppling ideal of a graph and its initial ideal |
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| Seminar: Algebra and Number Theory Seminar |
| Speaker: Madhusudan Manjunath of Georgia Tech |
| Contact: David Zureick-Brown, dzb@mathcs.emory.edu |
| Date: 2012-11-07 at 4:00PM |
| Venue: W306 |
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| Abstract: We describe minimal free resolutions of a lattice ideal associated with a graph and its initial ideal. These ideals are closely related to chip firing games and the Riemann-Roch theorem on graphs. Our motivations are twofold: describing information related to the Riemann-Roch theorem in terms of Betti numbers of the lattice ideal and the problem of explicit description of minimal free resolutions. This talk is based on joint work with Frank-Olaf Schreyer and John Wilmes. Analogous results were simultaneously and independently obtained by Fatemeh Mohammadi and Farbod Shokrieh. |
| Title: Numerical Analysis of Mixed Formulations for Bingham Fluids |
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| Defense: Dissertation |
| Speaker: Alexis Aposporidis of Emory University |
| Contact: Alexis Aposporidis, aapospo@emory.edu |
| Date: 2012-11-06 at 4:00PM |
| Venue: W304 |
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| Abstract: Visco-plastic materials have been attracting a great amount of attention among researchers in the study of fluid flow due to their widespread presence in various fields of science. However, the efficient solution of the nonlinear partial differential equations modelling their flow poses many challenges. From a mathematical point of view, the major difficulty associated with solving these equations is the presence of singularities in (a priori unknown) parts of the domain. This "irregularity" often reflects in a slow convergence of numerical solvers. In this presentation we consider an augmented formulation of the Bingham visco-plastic flow which is aimed at circumventing the singularity of the equations. We present a nonlinear solver based on this formulation and compare its performance to other common techniques for solving the Bingham flow, indicating superior convergence properties of the solver based on the augmented formulation. Upon linearization and discretization, a sequence of linear systems is obtained which are in general very large and sparse. For the efficient solution of these linear systems we present a nonlinear geometric multilevel technique which is used to precondition a flexible Krylov subspace method. We conclude by presenting some applications of this work to problems in hemodynamics. |
| Title: Mining and Using Contextual Information from Large-Scale Web Search Logs |
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| Seminar: Computer Science |
| Speaker: Paul Bennet of Microsoft |
| Contact: Eugene Atichtein, eugene@mathcs.emory.edu |
| Date: 2012-11-05 at 4:00PM |
| Venue: MSC W303 |
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| Abstract: Information retrieval has made significant progress in returning relevant results for a single query. However, much search activity is conducted within a much richer context of a current task focus, recent search activities as well as longer-term preferences. For example, our ability to accurately interpret the current query can be informed by knowledge of the web pages a searcher was viewing when initiating the search or recent actions of the searcher such as queries issued, results clicked, and pages viewed. We develop a framework that enables representation of a broad variety of context including the searcher's long-term interests, recent activity, current focus, and other user characteristics. We then demonstrate how that can be used to improve the quality of search results. We describe recent progress on three key challenges in this domain: mining contextual signals from large scale logs; understanding and modeling the combination of short-term and long-term behavior; and learning a more robust model that mitigates the risk of applying the contextual model when a simpler model would suffice.\\ \\ This talk will present joint work with Filip Radlinski, Lidan Wang, Ryen White, Kevyn Collins-Thompson, Wei Chu, Susan Dumais, Peter Bailey, Emine Yilmaz, Fedor Borisyuk, and Xiaoyuan Cui.\\ \\ Bio:\\ \\ Paul Bennett is a Researcher in the Context, Learning and User Experience for Search (CLUES) group at Microsoft Research where he works on using machine learning technology to improve information access and retrieval. His recent research has focused on classification-enhanced and contextual information retrieval, pairwise preferences, human computation, and text classification while his previous work focused primarily on ensemble methods, active learning, and obtaining reliable probability estimates, but also extended to machine translation, recommender systems, and knowledge bases. He completed his dissertation on combining text classifiers using reliability indicators in 2006 at Carnegie Mellon where he was advised by Profs. Jaime Carbonell and John Lafferty. |
| Title: 3D Floorplanning and Tree Representations |
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| Seminar: Combinatorics |
| Speaker: Paul Horn of Harvard University |
| Contact: Dwight Duffus, dwight@mathcs.emory.edu |
| Date: 2012-11-02 at 4:00PM |
| Venue: MSC W303 |
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| Abstract: A (2D) mosaic floorplan is a partitioning of a square into $n$ rectangles. Schemes which can be used to represent and count these floorplans, under a suitable topological equivalence, have applications in chip design. A 3D mosaic floorplan is a partitioning of a box into $n$ smaller boxes. Modern chip construction techniques, allowing chips with multiple layers, have fueled the desire to understand the number of 3D mosaic floorplans and how to represent them. For the 2D case, the number of unlabeled floorplans with $n$ rectangles, up to topological equivalence, is exponential in $n$. For many classes of 3D floorplans the number of floorplans is also exponential. Notable among these are 'slicing' floorplans, where to obtain an $n$ box floorplan one starts with an $n-1$ box floorplan and divides a box in two. In this talk, we discuss some of the difficulties with more general 3D floorplans, even with only two layers. In particular, if $F_n$ denotes the number of 3D floorplans with two levels, we show that $\log F_n \sim \frac{1}{3}n \log n$, and hence $F_n$ grows superexponentially in $n$. The upper bound comes from a representation scheme using trees, while the lower bound comes from a random construction. |
| Title: On computations of Shanks and Schmid |
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| Seminar: Algebra and Number Theory |
| Speaker: Robert Osburn of University College Dublin |
| Contact: David Zureick-Brown, dzb@mathcs.emory.edu |
| Date: 2012-10-30 at 4:00PM |
| Venue: W306 |
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| Abstract: In 1966, Shanks and Schmid investigated the asymptotic behavior of the number of positive integers less than or equal to $x$ which are represented by the quadratic form $X^2+nY^2$, n greater than or equal to 1. Based on some numerical computations, they observed that the constant occurring in the main term appears to be the largest for $n=2$. In this talk, we discuss a proof of the fact that this constant is actually unbounded as one runs through fundamental discriminants with a fixed number of distinct prime divisors. This is joint work with David Brink and Pieter Moree (MPIM). |