Computational Mathematics and Applications
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Thu, 26/04/2007 14:00 |
Dr Scott McLachlan (Delft University of Technology) |
Computational Mathematics and Applications  |
Rutherford Appleton Laboratory, nr Didcot |
| The numerical study of lattice quantum chromodynamics (QCD) is an attempt to extract predictions about the world around us from the standard model of physics. Worldwide, there are several large collaborations on lattice QCD methods, with terascale computing power dedicated to these problems. Central to the computation in lattice QCD is the inversion of a series of fermion matrices, representing the interaction of quarks on a four-dimensional space-time lattice. In practical computation, this inversion may be approximated based on the solution of a set of linear systems. In this talk, I will present a basic description of the linear algebra problems in lattice QCD and why we believe that multigrid methods are well-suited to effectively solving them. While multigrid methods are known to be efficient solution techniques for many operators, those arising in lattice QCD offer new challenges, not easily handled by classical multigrid and algebraic multigrid approaches. The role of adaptive multigrid techniques in addressing the fermion matrices will be highlighted, along with preliminary results for several model problems. | |||
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Thu, 03/05/2007 14:00 |
Prof Gene Golub (Stanford University) |
Computational Mathematics and Applications |
Comlab |
| The "secular equation" is a special way of expressing eigenvalue problems in a variety of applications. We describe the secular equation for several problems, viz eigenvector problems with a linear constraint on the eigenvector and the solution of eigenvalue problems where the given matrix has been modified by a rank one matrix. Next we show how the secular equation can be approximated by use of the Lanczos algorithm. Finally, we discuss numerical methods for solving the approximate secular equation. | |||
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Thu, 10/05/2007 14:00 |
Prof Enrique Zuazua (Universidad Autonoma de Madrid) |
Computational Mathematics and Applications |
Comlab |
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In this talk we will mainly analyze the vibrations of a simplified 1-d model for a multi-body structure consisting of a finite number of flexible strings distributed along a planar graph. In particular we shall analyze how solutions propagate along the graph as time evolves. The problem of the observation of waves is a natural framework to analyze this issue. Roughly, the question can be formulated as follows: Can we obtain complete information on the vibrations by making measurements in one single extreme of the network? This formulation is relevant both in the context of control and inverse problems. Using the Fourier development of solutions and techniques of Nonharmonic Fourier Analysis, we give spectral conditions that guarantee the observability property to hold in any time larger than twice the total lengths of the network in a suitable Hilbert that can be characterized in terms of Fourier series by means of properly chosen weights. When the network graph is a tree these weights can be identified. Once this is done these results can be transferred to other models as the Schroedinger, heat or beam-type equations. This lecture is based on results obtained in collaboration with Rene Dager. |
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Thu, 17/05/2007 14:00 |
Prof Shiu-hong Lui (University of Manitoba) |
Computational Mathematics and Applications |
Comlab |
| Spectral methods are a class of methods for solving PDEs numerically. If the solution is analytic, it is known that these methods converge exponentially quickly as a function of the number of terms used. The basic spectral method only works in regular geometry (rectangles/disks). A huge amount of effort has gone into extending it to domains with a complicated geometry. Domain decomposition/spectral element methods partition the domain into subdomains on which the PDE can be solved (after transforming each subdomain into a regular one). We take the dual approach - embedding the domain into a larger regular domain - known as the fictitious domain method or domain embedding. This method is extremely simple to implement and the time complexity is almost the same as that for solving the PDE on the larger regular domain. We demonstrate exponential convergence for Dirichlet, Neumann and nonlinear problems. Time permitting, we shall discuss extension of this technique to PDEs with discontinuous coefficients. | |||
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Thu, 24/05/2007 14:00 |
Prof Gil Strang (MIT) |
Computational Mathematics and Applications |
Comlab |
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Thu, 31/05/2007 14:00 |
Dr Ekaterina Kostina (University of Heidelberg) |
Computational Mathematics and Applications |
Rutherford Appleton Laboratory, nr Didcot |
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The development and quantitative validation of complex nonlinear differential equation models is a difficult task that requires the support by numerical methods for sensitivity analysis, parameter estimation, and the optimal design of experiments. The talk first presents particularly efficient "simultaneous" boundary value problems methods for parameter estimation in nonlinear differential algebraic equations, which are based on constrained Gauss-Newton-type methods and a time domain decomposition by multiple shooting. They include a numerical analysis of the well-posedness of the problem and an assessment of the error of the resulting parameter estimates. Based on these approaches, efficient optimal control methods for the determination of one, or several complementary, optimal experiments are developed, which maximize the information gain subject to constraints such as experimental costs and feasibility, the range of model validity, or further technical constraints. Special emphasis is placed on issues of robustness, i.e. how to reduce the sensitivity of the problem solutions with respect to uncertainties - such as outliers in the measurements for parameter estimation, and in particular the dependence of optimum experimental designs on the largely unknown values of the model parameters. New numerical methods will be presented, and applications will be discussed that arise in satellite orbit determination, chemical reaction kinetics, enzyme kinetics and robotics. They indicate a wide scope of applicability of the methods, and an enormous potential for reducing the experimental effort and improving the statistical quality of the models. (Based on joint work with H. G. Bock, S. Koerkel, and J. P. Schloeder.) |
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Thu, 07/06/2007 14:00 |
Prof Uri Ascher (University of British Columbia) |
Computational Mathematics and Applications |
Comlab |
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Many recent algorithmic approaches involve the construction of a differential equation model for computational purposes, typically by introducing an artificial time variable. The actual computational model involves a discretization of the now time-dependent differential system, usually employing forward Euler. The resulting dynamics of such an algorithm is then a discrete dynamics, and it is expected to be ''close enough'' to the dynamics of the continuous system (which is typically easier to analyze) provided that small -- hence many -- time steps, or iterations, are taken. Indeed, recent papers in inverse problems and image processing routinely report results requiring thousands of iterations to converge. This makes one wonder if and how the computational modeling process can be improved to better reflect the actual properties sought. In this talk we elaborate on several problem instances that illustrate the above observations. Algorithms may often lend themselves to a dual interpretation, in terms of a simply discretized differential equation with artificial time and in terms of a simple optimization algorithm; such a dual interpretation can be advantageous. We show how a broader computational modeling approach may possibly lead to algorithms with improved efficiency. |
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Tue, 12/06/2007 14:00 |
Prof Ian Sloan (University of New South Wales) |
Computational Mathematics and Applications |
Comlab |
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Thu, 14/06/2007 14:00 |
Prof Tom Hou (Caltech) |
Computational Mathematics and Applications |
Comlab |
| Whether the 3D incompressible Euler or Navier-Stokes equations can develop a finite time singularity from smooth initial data has been an outstanding open problem. Here we review some existing computational and theoretical work on possible finite blow-up of the 3D Euler equations. We show that the geometric regularity of vortex filaments, even in an extremely localized region, can lead to dynamic depletion of vortex stretching, thus avoid finite time blowup of the 3D Euler equations. Further, we perform large scale computations of the 3D Euler equations to re-examine the two slightly perturbed anti-parallel vortex tubes which is considered as one of the most attractive candidates for a finite time blowup of the 3D Euler equations. We found that there is tremendous dynamic depletion of vortex stretching and the maximum vorticity does not grow faster than double exponential in time. Finally, we present a new class of solutions for the 3D Euler and Navier-Stokes equations, which exhibit very interesting dynamic growth property. By exploiting the special nonlinear structure of the equations, we prove nonlinear stability and the global regularity of this class of solutions. | |||
