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Numerical Analysis and Scientific Computing Seminars 2008/09

Semester One

Semester Two

  • 13 Feb
    2009
    A rigorously justified algebraic preconditioner for high-contrast diffusion
    Robert Scheichl (University of Bath)

    3.00 - Frank Adams Room 1, Alan Turing Building
    Abstract (click to view)

    In this talk I will analyse the robustness of a family of algebraic multigrid (AMG) preconditioners for elliptic PDEs with high-contrast coefficients. Problems with high-contrast coefficients are ubiquitous in porous media flow applications. Consequently, development of efficient solvers for high-contrast heterogeneous media has been a very active area of research. The family of preconditioners considered here exploits the binary character of high-contrast coefficients and is constructed in similar ways as AMG preconditioners by identifying strong and weak couplings in the stiffness matrix. However, for this new family we can rigorously prove robustness and we demonstrate this on a sequence of model problems. Moreover, our numerical experiments show that for sufficiently high contrast the performance of our new preconditioners is almost identical to classical AMG, both in terms of iteration count and CPU time. The theory is based on a singular perturbation analysis.

    This work was obtained partly in collaboration with B. Aksoylu (Louisiana State), I.G. Graham (Bath) and H. Klie (Conoco-Phillips).

  • 20 Feb
    2009
    GMRES and oscillatory differential equations
    Sheehan Olver (University of Oxford)

    3.00 - Frank Adams Room 2, Alan Turing Building
    Abstract (click to view)

    We investigate applying GMRES to differential operators.  This method works particularly well when the solutions to the differential equation are oscillatory, and by preconditioning the equation appropriately we can obtain asymptotic decay as the frequency increases.  For the case of oscillatory integrals, we will find general conditions for convergence; hence, the method has similar asymptotic properties to the asymptotic expansion, whilst converging for fixed frequencies.

  • 06 Mar
    2009
    From sparsity to block-sparsity: solving very large optimization problems with interior point methods
    Jacek Gondzio (University of Edinburgh)

    3.00 - Frank Adams Room 1, Alan Turing Building
    Abstract (click to view)

    Many real-life economic models involve system dynamics, spatial distribution or uncertainty and lead to large-scale optimization problems. Such problems usually have a hidden structure: they are constructed by replication of some small generic block. The linear algebra subproblems which need to be solved by optimization algorithms for such problems involve matrices which are not only sparse, but they additionally display a block-structure with many smaller blocks sparsely distributed in the large matrix.

    I will discuss several advantages of interior point methods and will focus on their implementation which involves the solution of symmetric indefinite system of equations. Next I will discuss how the special block-structure of very large problems can be exploited in the linear algebra operations. The optimization software OOPS ( Object-Oriented Parallel Solver: http://www.maths.ed.ac.uk/~gondzio/parallel/solver.html ) based on such a technique can efficiently handle very large problems and achieves scalability on a number of different computing platforms.

  • 13 Mar 2009 Parametric approximation of geometric evolution equations and their coupling to bulk equations
    Robert Nürnberg (Imperial College London)

    3.00 - Frank Adams Room 1, Alan Turing Building
    Abstract (click to view)

    Geometric flows, in which hypersurfaces move such that an energy, involving surface and bending terms, decreases appear in many situations in the natural sciences and in geometry. Classic examples are mean curvature, surface diffusion and Willmore flows. Computational methods to approximate such flows are based on one of three approaches (i) parametric methods, (ii) phase field methods or (iii) level set methods. The first tracks the hypersurface, whilst the other two implicitly capture the hypersurface. A key problem with the first approach, apart from the fact that it is does not naturally deal with changes of topology, is that in many cases the mesh has to be redistributed after every few time steps to avoid coalescence of mesh points.

    In this talk we present a variational formulation of the parametric approach, which leads to an unconditionally stable, fully discrete finite element approximation. In addition, the scheme has very good properties with respect to the distribution of mesh points, and if applicable volume conservation. We illustrate this for (anisotropic) mean curvature and (anisotropic) surface diffusion flows of closed curves in R^2 and closed hypersurfaces in R^3.

    Finally, we extend these approximations to the case when the hypersurface motion depends also on an underlying bulk equation, such as the Stefan problem with kinetic undercooling or the Mullins-Sekerka/Hele-Shaw flow with surface tension.

  • 27 Mar
    2009
    Data assimilation in numerical weather prediction: 4D-Var and links to other regularisation methods.
    Melina Freitag (University of Bath)

    3.00 - Frank Adams Room 1, Alan Turing Building
    Abstract (click to view)

    In this talk we will give an introduction to data assimilation techniques as they are used in modern numerical weather prediction. It is well-known that data assimilation using 4D-Var (4D Variation) can be interpreted as some form of Tikhonov regularisation, a very familiar method for solving ill-posed inverse problems. Such problems appear in a wide range of applications such as geosciences and image restoration, the process of estimating an original image from a given blurred image.

    From the latter work it is known that by replacing the $L_2$-norm penalty function by an $L_1$-norm penalty function the image restoration problems become edge-preserving as they do not penalise the edges of the image. The $L_1$-norm penalty regularisation then recovers sharp edges in the image better than the $L_2$-norm penalty regularisation. We apply this idea to 4D-Var for problems where shocks are present and give some examples where the $L_1$-norm penalty approach performs much better than the standard $L_2$-norm regularisation in 4D-Var.

    This work is supported by GWR (Great Western Research) and the UK MetOffice and joint with N. Nichols (Reading) and C. Budd (Bath).

  • 24 Apr 2009 Smith forms for even and odd matrix polynomials
    Christian Mehl (University of Birmingham)

    3.00 - Frank Adams Room 1, Alan Turing Building
    Abstract (click to view)

    Even and odd matrix polynomials, i.e., matrix polynomials with coefficient matrices alternating between symmetric and skew-symmetric matrices, frequently arise in applications.
    The corresponding eigenvalue problem is usually solved by first linearizing the problem in a structure-preserving way and then by solving the resulting linear even or odd eigenvalue problem.
    However, it has been observed that there exist even or odd matrix polynomials that do not allow a structure-preserving linearization.

    In this talk, we investigate the possible Smith forms of even and odd matrix polynomials. As a consequence, we will obtain necessary and sufficient condition when a given even or odd matrix
    polynomial allows a structure-preserving linearization.

    This is joint work with D. Steven Mackey, Niloufer Mackey, and Volker Mehrmann.

  • 03 Jul 2009 A sprint through the mathematical software landscape
    Mike Croucher (University of Manchester)

    3.00 - Frank Adams Room 1, Alan Turing Building
    Abstract (click to view)

    The world of mathematical software is becoming increasingly complicated with literally hundreds of different commercial and free products to choose from. In this talk I will discuss the relative merits of the main general-purpose commercial mathematical applications (and their open-source competitors) available to staff and students at Manchester University.

 

 

Further information

For further information please contact the seminar organiser Younes Chahlaoui.

Previous Seminars