Scottish Computational
Mathematics Symposium

18th Annual Meeting
Saturday 12th September 2009, 10:00-17:00

ICMS, 14 India Street, Edinburgh, EH3 6EZ

Finding ICMS

Speakers, titles and abstracts:

  • Lubomin Banas (Heriot-Watt University)
    A convergent finite element approximation of a phase field model for incompressible fluid flows.

    Abstract: We present a finite element approximation for a system of Cahn-Hilliard and Navier-Stokes equations which describes the flow of two immiscible and incompressible fluids. We show stability bounds for the approximation and prove convergence, and hence existence of a solution to the system of coupled equations. We propose an efficient implementation of the method and present some numerical experiments.

  • Coralia Cartis (University of Edinburgh)
    Adaptive regularization methods for nonlinear optimization

    Abstract: A new class of methods for nonlinear nonconvex optimization problems will be presented that approximately globally minimizes a quadratic model of the objective regularized by a cubic term, extending to practical large-scale problems earlier approaches by Nesterov (2007) and Griewank (1982). Preliminary numerical experiments show our method to perform better than a basic trust-region implementation, while our convergence and complexity results show it to be at least as reliable to the latter approaches. Extensions to problems with simple constraints will also be presented. Lastly, solving nonlinear least-squares problems and systems of equations will be discussed, and a lower-order regularization method will be analysed in this context, namely where a first-order model of the least-squares is regularized by a quadratic term. An overestimation property of functions with Lipschitz-continuous Hessians, and of systems with Lipschitz-continuous Jacobian underlies and justifies the entire work to be presented. This is joint work with Nick Gould (RAL), Philippe Toint (Namur, Belgium), and for the latter part of my talk, also with Stefania Bellavia and Benedetta Morini (Florence, Italy).

  • Jack Dongarra (University of Manchester, University of Tennessee at Knoxville and Oak Ridge National Laboratory)
    Five important concepts to consider when using computational high performance systems at scale

    Abstract: In this talk we examine how high performance computing has changed over the last 10-year and look toward the future in terms of trends. These changes have had and will continue to have a major impact on our software.? Some of the software and algorithm challenges have already been encountered, such as management of communication and memory hierarchies through a combination of compile--time and run--time techniques, but the increased scale of computation, depth of memory hierarchies, range of latencies, and increased run--time environment variability will make these problems much harder.?

    We will look at five areas of research that will have an importance impact in the development of software.

    We will focus on following themes:

    * Redesign of software to fit multicore architectures

    * Automatically tuned application software

    * Exploiting mixed precision for performance

    * The importance of fault tolerance

    * Communication avoiding algorithms

  • Ozgur S. Ergul (University of Strathclyde)
    Efficient and accurate solutions of large-scale electromagnetics problems using the multilevel fast multipole algorithm

    Abstract: Solutions of scattering and radiation problems are extremely important to analyze electromagnetic interactions of devices with each other and with their environments including living and nonliving objects. A vast variety of areas, such as antennas, radars, metamaterials, remote sensing, electronic packaging, optical imaging, nanotechnology, medical imaging, and wireless communications, involve scattering or radiation of electromagnetic waves. Mathematical formulations of physical applications lead to set of integral equations, which can be converted into matrix equations and solved numerically on computers. Unfortunately, accurate simulations of real-life problems usually require solutions of very large matrix equations, which cannot be achieved easily without using special acceleration algorithms. The multilevel fast multipole algorithm (MLFMA) is a powerful method for the fast and efficient solution of electromagnetics problems discretized with large numbers of unknowns. This method reduces the complexity of matrix-vector multiplications required by iterative solvers, and it enables the solution of large-scale problems that cannot be investigated by using traditional techniques. Nevertheless, efficiency and accuracy of solutions via MLFMA still depend on many parameters, such as the underlying integral-equation formulation, discretization, iterative solver, preconditioning, computing platform, parallelization, and many other details of the numerical implementation. This talk will provide an overview of our efforts to develop sophisticated implementations of MLFMA for rigorous solutions of real-life scattering and radiation problems involving three-dimensional complicated objects discretized with millions of unknowns.

  • Doron Levy (University of Maryland, College Park)
    Can mathematics cure leukemia? (Lessons on mathematics in medical sciences)

    Abstract: Leukemia is a cancer of the blood that is characterized by an abnormal production of white blood cells. The treatment of Chronic Myelogenous Leukemia (CML) was revolutionized over the past decade with the introduction of new molecular-targeted drugs such as Imatinib. While these drugs keep leukemia in most patients in remission, they do not cure the disease. In this talk we will show how mathematical modeling, analysis, and scientific computation, combined with new experimental data, suggest the feasibility of a low-risk, clinical approach to enhancing the effect of the drug therapy, possibly leading to a durable cure of the disease. We will use this scientific problem to demonstrate challenges that face the dialogue between applied mathematics and medical sciences.

  • Magnus Svard (University of Edinburgh)
    Shock capturing for high-order central difference schemes

    Abstract: During the last decade, there has been a considerable effort to derive linearly stable high-order finite difference schemes for flow problems. The key difficulty has been to design boundary closures and derive stability proofs by energy estimates for the initial-boundary value problem. The schemes have been shown effective for smooth flows in several benchmark cases. However, linear stability proofs does not guarantee stability nor convergence in the presence of discontinuous solutions, such as shocks.

    In this work, we derive suitable diffusion terms for high-order central schemes and prove entropy stability without compromising the linear stability proofs for the initial-boundary value problem. Furthermore, we propose a limiter function that localizes the diffusion terms near discontinuities. For systems of conservation laws, we prove entropy stability of the limiter scheme and for a scalar conservation law, we also prove convergence.

    Numerical computations demonstrate the properties of the proposed scheme and comparisons with the WENO schemes are also presented.

    Finally, the extension of entropy stability to the initial-boundary value problem will be discussed.