Past Seminars › Seminar Series

Computational Mathematics and Applications (past)

Thu, 03/12/2009
14:00 		Prof. Andre Weideman (University of Stellenbosch) Computational Mathematics and Applications Add to calendar 3WS SR
Rational Approximations to the Complex Error Function
We consider rational approximations to the Faddeeva or plasma dispersion function, defined as $ w(z) = e^{-z^{2}} \mbox{erfc} (-iz) $. 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 $ w(z) $ 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 $ w(z) \sim i/(\sqrt{\pi} z), z \rightarrow \infty $ (in an appropriate sector). This is Joint work with: Stephan Gessner, Stéfan van der Walt
Thu, 26/11/2009
14:00 		Dr. Timo Betcke (University of Reading) Computational Mathematics and Applications Add to calendar Rutherford Appleton Laboratory, nr Didcot
Invariant pairs of matrix polynomials
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.
Thu, 19/11/2009
14:00 		Dr. Pedro Gonnet (ETH Zurich and Oxford University) Computational Mathematics and Applications Add to calendar 3WS SR
Molecular Dynamics Simulations and why they are interesting for Numerical Analysts
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.
Thu, 12/11/2009
14:00 		Dr. Leigh Lapworth (t.b.c.) (Rolls Royce) Computational Mathematics and Applications Add to calendar Rutherford Appleton Laboratory, nr Didcot
CFD in the Gas Turbine Industry
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.
Thu, 05/11/2009
14:00 		Dr. Joris van Deun (University of Antwerp and University of Oxford) Computational Mathematics and Applications Add to calendar 3WS SR
On rational interpolation
Thu, 29/10/2009
14:00 		Dr. Wayne Hayes (UC Irvine and Imperial College London) Computational Mathematics and Applications Add to calendar 3WS SR
Is the Outer Solar System Chaotic?
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.
Thu, 22/10/2009
14:00 		Prof. Charalambos Makridakis (University of Crete) Computational Mathematics and Applications Add to calendar 3WS SR
Mesh redistribution algorithms and error control for time-dependent PDEs
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.
Thu, 15/10/2009
14:00 		Prof. Gitta Kutyniok (University of Osnabruck) Computational Mathematics and Applications Add to calendar 3WS SR
Sparsity, $ \ell_1 $ Minimization, and the Geometric Separation Problem
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 $ \ell_1 $ 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 $ \ell_1 $ 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).
Thu, 18/06/2009
14:00 		Dr. Natasha Flyer (National Center for Atmospheric Research) Computational Mathematics and Applications Add to calendar Comlab
Radial Basis Functions Methods for Modeling Atmospheric and Solid Earth Flows
Current community models in the geosciences employ a variety of numerical methods from finite-difference, finite-volume, finite- or spectral elements, to pseudospectral methods. All have specialized strengths but also serious weaknesses. The first three methods are generally considered low-order and can involve high algorithmic complexity (as in triangular elements or unstructured meshes). Global spectral methods do not practically allow for local mesh refinement and often involve cumbersome algebra. Radial basis functions have the advantage of being spectrally accurate for irregular node layouts in multi-dimensions with extreme algorithmic simplicity, and naturally permit local node refinement on arbitrary domains. We will show test examples ranging from vortex roll-ups, modeling idealized cyclogenesis, to the unsteady nonlinear flows posed by the shallow water equations to 3-D mantle convection in the earth’s interior. The results will be evaluated based on numerical accuracy, stability and computational performance.
Wed, 17/06/2009
14:00 		Prof Gil Strang (MIT) Computational Mathematics and Applications Add to calendar Comlab
Random triangles: are they acute or obtuse?
This is a special talk outside the normal Computational Mathematics and Application seminar series. Please note it takes place on a Wednesday.
Thu, 11/06/2009
14:00 		Dr. Atsushi Suzuki (Czech Technical University in Prague / Kyushu University) Computational Mathematics and Applications Add to calendar Comlab
A fast domain decomposition solver for the discretized Stokes equations by a stabilized finite element method
An iterative substructuring method with balancing Neumann-Neumann preconditioner is known as an efficient parallel algorithm for the elasticity equations. This method was extended to the Stokes equations by Pavarino and Widlund [2002]. In their extension, Q2/P0-discontinuous elements are used for velocity/pressure and a Schur complement system within "benign space", where incompressibility satisfied, is solved by CG method. For the construction of the coarse space for the balancing preconditioner, some supplementary solvability conditions are considered. In our algorithm for 3-D computation, P1/P1 elements for velocity/pressure with pressure stabilization are used to save computational cost in the stiffness matrix. We introduce a simple coarse space similar to the one of elasticity equations. Owing to the stability term, solvabilities of local Dirichlet problem for a Schur complement system, of Neumann problem for the preconditioner, and of the coarse space problem are ensured. In our implementation, local Dirichlet and Neumann problems are solved by a 4x4-block modified Cholesky factorization procedure with an envelope method, which leads to fast computation with small memory requirement. Numerical result on parallel efficiency with a shared memory computer will be shown. Direct use of the Stokes solver in an application of Earth's mantle convection problem will be also shown.
Thu, 04/06/2009
14:00 		Dr. Amos Lawless (University of Reading) Computational Mathematics and Applications Add to calendar Rutherford Appleton Laboratory, nr Didcot
Approximate Gauss-Newton methods using reduced order models
Work with N.K. Nichols (Reading), C. Boess & A. Bunse-Gerstner (Bremen) The Gauss-Newton (GN) method is a well known iterative technique for solving nonlinear least squares problems subject to dynamical system constraints. Such problems arise commonly from applications in optimal control and state estimation. Variational data assimilation systems for weather, ocean and climate prediction currently use approximate GN methods. The GN method solves a sequence of linear least squares problems subject to linearized system constraints. For very large systems, low resolution linear approximations to the model dynamics are used to improve the efficiency of the algorithm. We propose a new method for deriving low order system approximations based on model reduction techniques from control theory. We show how this technique can be combined with the GN method to give a state estimation technique that retains more of the dynamical information of the full system. Numerical experiments using a shallow-water model illustrate the superior performance of model reduction to standard truncation techniques.
Thu, 28/05/2009
14:00 		Prof. Bengt Fornberg (University of Colorado) Computational Mathematics and Applications Add to calendar Comlab
Radial Basis Functions for Solving Partial Differential Equations
For the task of solving PDEs, finite difference (FD) methods are particularly easy to implement. Finite element (FE) methods are more flexible geometrically, but tend to be difficult to make very accurate. Pseudospectral (PS) methods can be seen as a limit of FD methods if one keeps on increasing their order of accuracy. They are extremely effective in many situations, but this strength comes at the price of very severe geometric restrictions. A more standard introduction to PS methods (rather than via FD methods of increasing orders of accuracy) is in terms of expansions in orthogonal functions (such as Fourier, Chebyshev, etc.). Radial basis functions (RBFs) were first proposed around 1970 as a tool for interpolating scattered data. Since then, both our knowledge about them and their range of applications have grown tremendously. In the context of solving PDEs, we can see the RBF approach as a major generalization of PS methods, abandoning the orthogonality of the basis functions and in return obtaining much improved simplicity and flexibility. Spectral accuracy becomes now easily available also when using completely unstructured meshes, permitting local node refinements in critical areas. A very counterintuitive parameter range (making all the RBFs very flat) turns out to be of special interest. Computational cost and numerical stability were initially seen as serious difficulties, but major progress have recently been made also in these areas.
Thu, 21/05/2009
14:00 		Dr. Christoph Ortner (Computing Laboratory, Oxford) Computational Mathematics and Applications Add to calendar Comlab
Introduction to Quasicontinuum Methods: Formulation, Classification, Analysis
Quasicontinuum methods are a prototypical class of atomistic-to-continuum coupling methods. For example, we may wish to model a lattice defect (a vacancy or a dislocation) by an atomistic model, but the elastic far field by a continuum model. If the continuum model is consistent with the atomistic model (e.g., the Cauchy–Born model) then the main question is how the interface treatment affects the method. In this talk I will introduce three of the main ideas how to treat the interface. I will explain their strengths and weaknesses by formulating the simplest possible non-trivial model problem and then simply analyzing the two classical concerns of numerical analysis: consistency and stability.
Thu, 30/04/2009
14:00 		Prof. Andrew Stuart (University of Warwick) Computational Mathematics and Applications Add to calendar Comlab
Approximation of Inverse Problems
Inverse problems are often ill-posed, with solutions that depend sensitively on data. Regularization of some form is often used to counteract this. I will describe an approach to regularization, based on a Bayesian formulation of the problem, which leads to a notion of well-posedness for inverse problems, at the level of probability measures. The stability which results from this well-posedness may be used as the basis for understanding approximation of inverse problems in finite dimensional spaces. I will describe a theory which carries out this program. The ideas will be illustrated with the classical inverse problem for the heat equation, and then applied to so more complicated inverse problems arising in data assimilation, such as determining the initial condition for the Navier-Stokes equation from observations.
Thu, 23/04/2009
14:00 		Dr. Coralia Cartis (University of Edinburgh) Computational Mathematics and Applications Add to calendar Rutherford Appleton Laboratory, nr Didcot
How sharp is the restricted isometry property? An investigation into sparse approximation techniques
A simple, yet efficient, model for data acquisition is to measure signals $ x $ linearly through the action of a measurement matrix $ A $ of size $ n\times N $, namely $ b = Ax $. Surprisingly, only $n
Thu, 12/03/2009
14:00 		Prof Ke Chen (The University of Liverpool) Computational Mathematics and Applications Add to calendar Rutherford Appleton Laboratory, nr Didcot
On fast multilevel algorithms for nonlinear variational imaging models

In recent years, the interdisciplinary field of imaging science has been experiencing an explosive growth in research activities including more models being developed, more publications generated, and above all wider applications attempted.
In this talk I shall first give an overview of the various imaging work carried out in our Liverpool group, some with collaborations with UCLA (T F Chan), CUHK (R H Chan) and Bergen (X C Tai) and several colleagues from other departments in Liverpool. Then I shall focus on two pieces of recent work, denoising and segmentation respectively:
(i) Image denoising has been a research topic deeply investigated within the last two decades. Even algorithmically the well-known ROF model (1992) can be solved efficiently. However less work has been done on models using high order regularization. I shall describe our first and successful attempt to develop a working multilevel algorithm for a 4th order nonlinear denoising model, and our work on solving the combined denoising and deblurring problem, different from the reformulation approach by M N Ng and W T Yin (2008) et al.
(ii) the image active contour model by Chan-Vese (2001) can be solved efficiently both by a geometric multigrid method and by an optimization based multilevel method. Surprisingly the new multilevel methods can find a solution closer to the global minimize than the existing unilevel methods. Also discussed are some recent work (jointly with N Badshah) on selective segmentation that has useful medical applications.

For any corrections/updates to these listings, please contact seminars [-at-] maths [dot] ox [dot] ac [dot] uk.

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