Computational Mathematics and Applications
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Thu, 15/10/2009 14:00 |
Prof. Gitta Kutyniok (University of Osnabruck) |
Computational Mathematics and Applications  |
3WS SR |
During the last two years, sparsity has become a key concept in various areas
of applied mathematics, computer science, and electrical engineering. Sparsity
methodologies explore the fundamental fact that many types of data/signals can
be represented by only a few non-vanishing coefficients when choosing a suitable
basis or, more generally, a frame. If signals possess such a sparse representation,
they can in general be recovered from few measurements using  minimization
techniques.
One application of this novel methodology is the geometric separation of data,
which is composed of two (or more) geometrically distinct constituents – for
instance, pointlike and curvelike structures in astronomical imaging of galaxies.
Although it seems impossible to extract those components – as there are two
unknowns for every datum – suggestive empirical results using sparsity
considerations have already been obtained.
In this talk we will first give an introduction into the concept of sparse
representations and sparse recovery. Then we will develop a very general
theoretical approach to the problem of geometric separation based on these
methodologies by introducing novel ideas such as geometric clustering of
coefficients. Finally, we will apply our results to the situation of separation
of pointlike and curvelike structures in astronomical imaging of galaxies,
where a deliberately overcomplete representation made of wavelets (suited
to pointlike structures) and curvelets/shearlets (suited to curvelike
structures) will be chosen. The decomposition principle is to minimize the
norm of the frame coefficients. Our theoretical results, which
are based on microlocal analysis considerations, show that at all sufficiently
fine scales, nearly-perfect separation is indeed achieved.
This is joint work with David Donoho (Stanford University). |
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Thu, 22/10/2009 14:00 |
Prof. Charalambos Makridakis (University of Crete) |
Computational Mathematics and Applications |
3WS SR |
| Self adjusted meshes have important benefits approximating PDEs with solutions that exhibit nontrivial characteristics. When appropriately chosen, they lead to efficient, accurate and robust algorithms. Error control is also important, since appropriate analysis can provide guarantees on how accurate the approximate solution is through a posteriori estimates. Error control may lead to appropriate adaptive algorithms by identifying areas of large errors and adjusting the mesh accordingly. Error control and associated adaptive algorithms for important equations in Mathematical Physics is an open problem. In this talk we consider the main structure of an algorithm which permits mesh redistribution with time and the nontrivial characteristics associated with it. We present improved algorithms and we discuss successful approaches towards error control for model problems (linear and nonlinear) of parabolic or hyperbolic type. | |||
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Thu, 29/10/2009 14:00 |
Dr. Wayne Hayes (UC Irvine and Imperial College London) |
Computational Mathematics and Applications |
3WS SR |
| The stability of our Solar System has been debated since Newton devised the laws of gravitation to explain planetary motion. Newton himself doubted the long-term stability of the Solar System, and the question has remained unanswered despite centuries of intense study by generations of illustrious names such as Laplace, Langrange, Gauss, and Poincare. Finally, in the 1990s, with the advent of computers fast enough to accurately integrate the equations of motion of the planets for billions of years, the question has finally been settled: for the next 5 billion years, and barring interlopers, the shapes of the planetary orbits will remain roughly as they are now. This is called "practical stability": none of the known planets will collide with each other, fall into the Sun, or be ejected from the Solar System, for the next 5 billion years. Although the Solar System is now known to be practically stable, it may still be "chaotic". This means that we may—or may not—be able precisely to predict the positions of the planets within their orbits, for the next 5 billion years. The precise positions of the planets effects the tilt of each planet's axis, and so can have a measurable effect on the Earth's climate. Although the inner Solar System is almost certainly chaotic, for the past 15 years, there has been some debate about whether the outer Solar System exhibits chaos or not. In particular, when performing numerical integrations of the orbits of the outer planets, some astronomers observe chaos, and some do not. This is particularly disturbing since it is known that inaccurate integration can inject chaos into a numerical solution whose exact solution is known to be stable. In this talk I will demonstrate how I closed that 15-year debate on chaos in the outer solar system by performing the most carefully justified high precision integrations of the orbits of the outer planets that has yet been done. The answer surprised even the astronomical community, and was published in _Nature Physics_. I will also show lots of pretty pictures demonstrating the fractal nature of the boundary between chaos and regularity in the outer Solar System. | |||
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Thu, 05/11/2009 14:00 |
Dr. Joris van Deun (University of Antwerp and University of Oxford) |
Computational Mathematics and Applications |
3WS SR |
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Thu, 12/11/2009 14:00 |
Dr. Leigh Lapworth (t.b.c.) (Rolls Royce) |
Computational Mathematics and Applications |
Rutherford Appleton Laboratory, nr Didcot |
| CFD is an indispensible part of the design process for all major gas turbine components. The growth in the use of CFD from single-block structured mesh steady state solvers to highly resolved unstructured mesh unsteady solvers will be described, with examples of the design improvements that have been achieved. The European Commission has set stringent targets for the reduction of noise, emissions and fuel consumption to be achieved by 2020. The application of CFD to produce innovative designs to meet these targets will be described. The future direction of CFD towards whole engine simulations will also be discussed. | |||
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Thu, 19/11/2009 14:00 |
Dr. Pedro Gonnet (ETH Zurich and Oxford University) |
Computational Mathematics and Applications |
3WS SR |
| Molecular Dynamics Simulations are a tool to study the behaviour of atomic-scale systems. The simulations themselves solve the equations of motion for hundreds to millions of particles over thousands to billions of time steps. Due to the size of the problems studied, such simulations are usually carried out on large clusters or special-purpose hardware. At a first glance, there is nothing much of interest for a Numerical Analyst: the equations of motion are simple, the integrators are of low order and the computational aspects seem to focus on hardware or ever larger and faster computer clusters. The field, however, having been ploughed mainly by domain scientists (e.g. Chemists, Biologists, Material Scientists) and a few Computer Scientists, is a goldmine for interesting computational problems which have been solved either badly or not at all. These problems, although domain specific, require sufficient mathematical and computational skill to make finding a good solution potentially interesting for Numerical Analysts. The proper solution of such problems can result in speed-ups beyond what can be achieved by pushing the envelope on Moore's Law. In this talk I will present three examples where problems interesting to Numerical Analysts arise. For the first two problems, Constraint Resolution Algorithms and Interpolated Potential Functions, I will present some of my own results. For the third problem, using interpolations to efficiently compute long-range potentials, I will only present some observations and ideas, as this will be the main focus of my research in Oxford and therefore no results are available yet. | |||
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Thu, 26/11/2009 14:00 |
Dr. Timo Betcke (University of Reading) |
Computational Mathematics and Applications |
Rutherford Appleton Laboratory, nr Didcot |
| Invariant subspaces are a well-established tool in the theory of linear eigenvalue problems. They are also computationally more stable objects than single eigenvectors if one is interested in a group of closely clustered eigenvalues. A generalization of invariant subspaces to matrix polynomials can be given by using invariant pairs. We investigate some basic properties of invariant pairs and give perturbation results, which show that invariant pairs have similarly favorable properties for matrix polynomials than do invariant subspaces have for linear eigenvalue problems. In the second part of the talk we discuss computational aspects, namely how to extract invariant pairs from linearizations of matrix polynomials and how to do efficient iterative refinement on them. Numerical examples are shown using the NLEVP collection of nonlinear eigenvalue test problems. This talk is joint work with Daniel Kressner from ETH Zuerich. | |||
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Thu, 03/12/2009 14:00 |
Prof. Andre Weideman (University of Stellenbosch) |
Computational Mathematics and Applications |
3WS SR |
We consider rational approximations to the Faddeeva or plasma dispersion function, defined
as
 .
With many important applications in physics, good software for
computing the function reliably everywhere in the complex plane is required. In this talk
we shall derive rational approximations to  via quadrature, Möbius transformations, and best approximation. The various approximations are compared with regard to speed of convergence, numerical stability, and ease of generation of the coefficients of the formula.
In addition, we give preference to methods for which a single expression yields uniformly
high accuracy in the entire complex plane, as well as being able to reproduce exactly the
asymptotic behaviour
(in an appropriate sector).
This is Joint work with: Stephan Gessner, Stéfan van der Walt |
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