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Title: Regularization by Krylov-Tikhonov methods
Seminar: Numerical Analysis and Scientific Computing
Speaker: Silvia Gazzola of University of Padova
Contact: James Nagy, nagy@mathcs.emory.edu
Date: 2014-10-31 at 12:00AM
Venue: W301
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Abstract:
Krylov subspace methods have always played a central role in the iterative regularization of large-scale linear discrete ill-posed problems, which arise in a variety of scientific and engineering applications; we are particularly interested in image deblurring and denoising issues. In addition to a purely iterative approach to regularization, some "hybrid" Krylov-Tikhonov methods have also been derived, which merge an iterative and a variational (Tikhonov-like) approach to regularization. The purpose of this talk is to survey some classical Krylov and Krylov-Tikhonov methods, and to present some original ones, comparing their performance on some meaningful test problems. Particular emphasis will be posed on the strategies to be employed to set the regularization parameters and matrices in the Krylov-Tikhonov framework.
Title: High Performance Spatial Query Processing for Large Scale Spatial Data Warehousing
Defense: Dissertation
Speaker: Ablimit Aji of Emory University
Contact: James Lu, jlu@mathcs.emory.edu
Date: 2014-10-31 at 3:00PM
Venue: W303
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Abstract:
Support of high performance queries on large volumes of spatial data have become important in many application domains, including geowspatial problems in numerous fields, location based services, geo-social networks, and emerging scientific applications that are increasingly data- and compute-intensive. There are two major challenges for managing and querying massive spatial data: the explosion of spatial data, and the high computational complexity of spatial queries due to the multi-dimensional nature of spatial analytics. High performance computing capabilities are fundamental to efficiently handling of massive spatial datasets. MapReduce based computing model provides a highly scalable, reliable, elastic and cost effective framework for processing massive data on a cluster or cloud environment. While the MapReduce model fits nicely with large scale problems through data partitioning, spatial queries and analytics are intrinsically complex to fit into the MapReduce model easily. Meanwhile, hybrid systems combining CPUs and GPUs are becoming commonly available in commodity clusters, but the computing capacity of such systems is often underutilized. Providing new spatial querying and analytical methods to run on such architecture requires us to answer several fundamental research questions that are of practical importance. The goal of my dissertation is to create a framework with new systematic methods to support high performance spatial queries for spatial big data on MapReduce and CPU-GPU hybrid platforms, driven by real-world use cases. Towards that end, we have researched multi-level parallelism methods of spatial queries running on these platforms. Specifically, we have conducted following studies: 1) create new spatial data processing methods and pipelines with spatial partition level parallelism through a simple programming model MapReduce and propose multi-level indexing methods to accelerate spatial data processing, 2) develop two critical components to enable data parallelism: effective and scalable spatial partitioning in MapReduce (pre-processing), and query normalization methods for partition effect, 3) integrate GPU-based spatial operations into MapReduce pipelines 4) investigate optimization methods for data skew mitigation, and CPU/GPU resource coordination in MapReduce, and 5) support declarative spatial queries for workload composition, and create a query translator to automatically translate the queries into MapReduce programs. Consequently, we have developed Hadoop-GISb a MapReduce based high performance spatial querying system for spatial data warehousing. The system supports multiple types of spatial queries on MapReduce through spatial partitioning, implicit parallel spatial query execution on MapReduce, and effective methods for amending query results through handling bound- ary objects. Hadoop-GIS utilizes global partition indexing and customizable on demand local spatial indexing to achieve efficient query processing. Hadoop-GIS is integrated into Hive to support declarative spatial queries with an integrated architecture. The systems and developed approaches are released as an open source software package for use.
Title: The genus of a division algebra
Seminar: Algebra
Speaker: Igor Rapinchuk of Harvard
Contact: David Zureick-Brown, dzb@mathcs.emory.edu
Date: 2014-10-28 at 4:00PM
Venue: W306
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Abstract:
In this talk, I will address the following problem. Suppose D and D' are central division algebras over a field K. What can be said about D and D' if they have the same maximal subfields? I will discuss various motivations for this question and recent results. I will also mention some generalizations to arbitrary absolutely almost simple algebraic groups. This is joint work with V. Chernousov and A. Rapinchuk.
Title: Embeddings of maximal tori in classical groups and explicit Brauer
Seminar: Algebra
Speaker: Eva Bayer of EPFL
Contact: David Zureick-Brown, dzb@mathcs.emory.edu
Date: 2014-10-28 at 5:00PM
Venue: W306
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Abstract:
This is a joint work with Parimala and Ting–Yu Lee. Embeddings of maximal tori into classical groups over global fields of characteristic $\neq$ 2 are the subject matter of several recent papers (for instance by Prasad and Rapinchuk, Fiori, Lee), with special attention to the Hasse principle. The aim of this talk is to describe a complete criterion for the Hasse principle to hold, and to give necessary and sufficient conditions for a classical group to contain a maximal torus of a given type. The embedding problem will be described in terms of embeddings of \'etale algebras with involution into central simple algebras with involution.
Title: On the Number of B_h Sets of a Given Size
Seminar: Combinatorics
Speaker: Domingos Dellamonica of Sao Paulo
Contact: Vojtech Rodl, rodl@mathcs.emory.edu
Date: 2014-10-27 at 4:00PM
Venue: W303
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Abstract:
For an integer h bigger or equal to 2, a $B_h$ set is a set of integers with the property that every collection containing h of its elements yield a unique sum (and repetitions are allowed). For $h = 2$, such sets are also called Sidon sets. In this talk we will describe our recent results on estimating $F(n, s, h)$, which we define as the number of $B_h$ sets of cardinality s containing integers from $[n] = {1, 2, ..., n}$. It is not hard to see that for $s > n^(1/h)$, we have $F(n, s, h) = 0$. Indeed, in this case there are more h-sums than possible outcomes for the sums. On the other hand, there are constructions of B-h sets having cardinality $c.n^(1/h)$, (with c depending on h only) hence we shall estimate the behavior of $F(n, s, h)$ for s up to $O( n^(1/h))$. Our counting shows the existence of a surprising threshold function $T(n)$: for values of $s << T(n)$, the B-h sets are abundant while for $s >> T(n)$ the B-h sets become very rare. More precisely, we show that $T(n) ~ n^{(1 + o(1))/(2h - 1)}$ and establish fairly precise estimates of $F(n, s, h)$ for the entire range of s.
Title: Intermittence and Entropy-Equivalent Measures in Brain Function and Behavior
Colloquium: Ergodic Theory and Dynamical Systems
Speaker: Arnold Mandell of UCSD School of Medicine
Contact: Vaidy Sunderam, vss@emory.edu
Date: 2014-10-23 at 4:30PM
Venue: W201
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Abstract:
TBA
Title: Tropical Independence and Algebraic Curves
Seminar: Algebra
Speaker: David Jensen of University of Kentucky
Contact: David Zureick-Brown, dzb@mathcs.emory.edu
Date: 2014-10-21 at 4:00PM
Venue: W306
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Abstract:
Many questions about an algebraic curve concern the ranks of linear maps between linear series on the curve. Recent years have witnessed the development of a new, combinatorial approach to studying such questions via tropical geometry. We will discuss the basics of this theory, and how it can be used to gain new insight into the geometry of general curves. If time permits, we will discuss joint work with Sam Payne in which we use these techniques to provide a new proof of the Gieseker-Petri Theorem.
Title: Recent Advances and Open Problems in the Degree/Diameter Problem
Seminar: Combinatorics
Speaker: Mirka Miller of Department of Mathematics, University of West Bohemia, Pilsen, Czech Republic; and School of Mathematical and Physical Sciences, University of Newcastle, Australia
Contact: Vojtech Rodl, rodl@mathcs.emory.edu
Date: 2014-10-20 at 4:00PM
Venue: W302
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Abstract:
The degree/diameter problem is to find the largest possible graphs in terms of the number of vertices, given the maximum degree and diameter. The directed version is similar except that instead of the maximum degree constraint we are given the maximum out-degree. For both the undirected and directed versions of the problem there are general upper bounds on the number of vertices, called the Moore bounds. On the other hand, the best current lower bounds are obtained from constructions of ever larger graphs (given maximum degree and diameter).\\ \\ In this talk we will give an overview of the undirected, directed and mixed versions (allowing both undirected edges and directed arcs in the graph) of the problem and the most recent advances.\\ \\ The talk concludes with several open problems.
Title: Recent developments in rationality of cubic fourfolds
Seminar: Algebra
Speaker: Nick Addington of Duke
Contact: David Zureick-Brown, dzb@mathcs.emory.edu
Date: 2014-10-07 at 4:00PM
Venue: W306
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Abstract:
Which cubic hypersurfaces in $P^5$ are rational? Certainly some, but conjecturally not all. The problem has classical roots, but it has seemed tractable since the 70s, when Clemens and Griffiths gave a beautiful solution to the analogous problem for cubic threefolds using Hodge theory. Hassett worked on adapting their argument to cubic fourfolds in the late 90s; Kuznetsov tried again around 2008, using derived categories in place of Hodge theory; Galkin and Shinder have gotten further this year using the Grothendieck ring of varieties. While none of these attempts has succeeded at solving the problem, each has brought new insights. My own contribution has been to explain how these three would-be rationality criteria are interrelated. In the course of the talk I'll pose an elementary number theory puzzle that might get a student a paper.
Title: Some Old and New Results on an Elimination Game
Seminar: Numerical Analysis and Scientific Computing
Speaker: Esmond G. Ng of Lawrence Berkeley National Laboratory
Contact: James Nagy, nagy@mathcs.emory.edu
Date: 2014-10-06 at 12:00AM
Venue: W301
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Abstract:
Sparse matrix problems arise at the heart of many large-scale scientific and engineering applications. State-of-the-art algorithms for solving these problems involve not only numerical techniques, but may also require knowledge of data structures, graph theory, algorithm design, complexity analysis, and computer architectures. This is particularly true for factorization-based algorithms, such as those for solving sparse systems of linear equations, in which the nonzero entries of a matrix are eliminated according to certain prescribed rules. However, the order in which the nonzero entries are eliminated can have a dramatic effect on the overall performance of the solution process. This is referred to as the ordering problem, which is often posed as an elimination game on a graph. In this talk, an overview of the elimination game will be presented. In particular, previously known results and some recently work will be described.