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4 AN10 Abstracts 4 AN10 Abstracts IC1 IC4 On Dispersive Equations and Their Importance in Kinematics and Numerical Algebraic Geometry Mathematics Kinematics underlies applications ranging from the design Dispersive equations, like the Schr¨odinger equation for ex- and control of mechanical devices, especially robots, to the ample, have been used to model several wave phenom- biomechanical modelling of human motion. The major- ena with the distinct property that if no boundary con- ity of kinematic problems can be formulated as systems ditions are imposed then in time the wave spreads out of polynomial equations to be solved and so fall within spatially. In the last fifteen years this field has seen an the domain of algebraic geometry. While symbolic meth- incredible amount of new and sophisticated results proved ods from computer algebra have a role to play, numerical with the aid of mathematics coming from different fields: methods such as polynomial continuation that make strong Fourier analysis, differential and symplectic geometry, an- use of algebraic-geometric properties offer advantages in alytic number theory, and now also probability and a bit of efficiency and parallelizability. Although these methods, dynamical systems. In this talk it is my intention to present collectively called Numerical Algebraic Geometry, are ap- few simple, but still representative examples in which one plicable wherever polynomials arise, e.g., chemistry, biol- can see how these different kinds of mathematics are used ogy, statistics, and economics, this talk will concentrate on in this context. applications in mechanical engineering. A brief review of the main algorithms of the field will indicate their broad Gigliola Staffilani applicability. MIT [email protected] Charles Wampler General Motors Research Laboratories, Enterprise Systems Lab IC2 30500 Mound Road, Warren, MI 48090-9055, USA On the Role of Error and Uncertainty in the Nu- [email protected] merical Simulation of Complex Fluid Flows The failure of numerical simulation to predict physical real- IC5 ity is often a direct consequence of the compounding effects Cloaking and Transformation Optics of numerical error arising from finite-dimensional approx- imation and physical model uncertainty resulting from in- We describe recent theoretical and experimental progress exact knowledge and/or statistical representation. In this on making objects invisible to detection by electromagnetic topical lecture, we briefly review systematic theories for waves, acoustic waves and matter waves. For the case of quantifying numerical errors and restricted forms of model electromagnetic waves, Maxwell’s equations have transfor- uncertainty occurring in simulations of fluid flow. A goal mation laws that allow for design of electromagnetic ma- of this lecture is to elucidate both positive and negative terials that steer light around a hidden region, returning aspects of applying these theories to practical fluid flow it to its original path on the far side. Not only would ob- problems. Finite-element and finite-volume calculations of servers be unaware of the contents of the hidden region, subsonic and hypersonic fluid flow are presented to contrast they would not even be aware that something was being the differing roles of numerical error and model uncertainty hidden. The object, which would have no shadow, is said for these problems. to be cloaked. We recount some of the history of the sub- ject and discuss some of the mathematical issues involved. Timothy J. Barth NASA Ames Research Center [email protected] Gunther Uhlmann University of Washington Department of Mathematics IC3 [email protected] Communication Complexity of Algorithms Algorithms have two kinds of costs: arithmetic and com- IC6 munication, which means either moving data between lev- Algebraic Geometric Algorithms in Discrete Opti- els of a memory hierarchy (in the sequential case) or over a mization network connecting processors (in the parallel case). The costs of communication often dominate the cost of arith- It is common knowledge that the understanding of the ge- metic. Thus, minimizing communication is often highly de- ometry of convex bodies has helped speed up algorithms sirable. The main goal of this talk is to describe a general in discrete optimization. For example, cutting planes and method for deriving upper and lower bounds on the amount facet-description of polyhedra have been crucial in the suc- of data moved (also known as bandwidth) for a very gen- cess of branch-and-bound algorithms for mixed integer lin- eral class of algorithms, including most dense and sparse ear programming. Another example is how the ellipsoid linear algebra algorithms. The method involves combinato- method can be used to prove polynomiality results in com- rial analysis of computation graphs, in particular, of their binatorial optimization. For the future, the importance expansion properties. This is joint work with Grey Ballard, of algebra and geometry in optimization is even greater James Demmel, and Oded Schwartz. since applications now demand non-linearity constraints together with discrete variables. In the past 5 years two Olga Holtz beautiful algebraic geometric algorithms on polyhedra have University of California, Berkeley been used to prove unexpected new results on the compu- Technische Universitat Berlin tation of integer programs with non-linearly objective func- [email protected] tions. The first is Barvinok’s algorithm for polytopes, and the second is Graver’s bases method on polyhedral cones. I will describe these two algorithms and explain why we AN10 Abstracts 5 can now prove theorems that were beyond our reach be- understanding of the options available to reduce energy fore. I will also describe attempts to turn these two al- use in buildings and will highlight the role of mathematics gorithms into practical computation, not just theoretical and particularly computational science in delivering low results. This is a nice story collecting results contained energy buildings to the market in cost-effective ways. A in several papers, joint work with various subsets of the systems approach to designing and operating low energy following people: R. Hemmecke, M. Koeppe, S. Onn, U. buildings will be described and the opportunities to de- Rothblum, and R. Weismantel. velop and deploy methods for uncertainty quantification, dynamical systems and embedded systems will be high- Jesus De Loera lighted. University of California, Davis [email protected] Clas Jacobson United Technologies Research Center IC7 [email protected] Scalable Tensor Factorizations with Incomplete Data IC10 Incomplete data is ubiquitous in biomedical signal pro- Semidefinite Programming: Algorithms and Appli- cessing, network traffic analysis, bibliometrics, social net- cations work analysis, chemometrics, computer vision, communi- cation networks, etc. We explain how factor analysis can The last two decades have seen dramatic advances in the identify the underlying latent structure even when signif- theory and practice of semidefinite programming (SDP), icant portions of data are missing and the remainder is stimulated in part by the realization that SDP is an ex- contaminated with noise. Though much of what we will tremely powerful modeling tool. Applications of SDP in- say is also applicable to matrix factorizations, we focus clude signal processing, relaxations of combinatorial and on the CANDECOMP/PARAFAC (CP) tensor decompo- polynomial optimization problems, covariance matrix es- sition. We demonstrate that it is possible to factorize in- timation, and sensor network localization. The first part complete tensors that have an underlying low-rank struc- of the talk will describe several applications, such as ma- ture. Further, there are approaches that can scale to sparse trix completion, that have attracted recent interest in large-scale data, e.g., 1000 x 1000 x 1000 with 99.5% miss- large scale SDP. The second part will focus on algorithms ing data. Real-world applicability is demonstrated in two for SDP. First, we present interior-point methods (IPMs) examples: EEG (electroencephalogram) applications and for medium scale SDP, follow by inexact IPMs (with lin- network traffic data. ear systems solved by iterative solvers) for large scale SDP and discuss their inherent limitations. Finally, we Tamara G. Kolda present recent algorithmic advances for large scale SDP Sandia National Laboratories and demonstrate that semismooth Newton-CG augmented [email protected] Lagrangian methods can be very efficient. Kim-Chuan Toh IC8 National University of Singapore Factorization-based Sparse Solvers and Precondi- Department of Mathematics tioners [email protected] Efficient solution of large-scale, ill-conditioned and highly- indefinite algebraic equations often relies on high quality IP0 preconditioners together with iterative solvers. Because W. T. and Idalia Reid Prize in Mathematics Lec- of their robustness, factorization-based algorithms often ture: William T. and Idalia Reid: His Mathematics play a significant role in developing scalable solvers. We and Her Mathematical Family present our recent work of using state-of-the-art sparse fac- torization techniques to build domain-decomposition type The W. T. and Idelia Reid Prize in Mathematics was es- direct/iterative hybrid solvers and efficient incomplete fac- tablished in 1993 by SIAM and is funded by an endowment torization preconditioners. In addition to algorithmic prin- from the late Mrs. Idalia Reid to honor the memory
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