Computational Mathematics and Applications
|
Thu, 16/10/2008 14:00 |
Dr David Mayers (University of Oxford) |
Computational Mathematics and Applications  |
Comlab |
| This is not intended to be a systematic History, but a selection of highlights, with some digressions, including: The early days of the Computing Lab; How the coming of the Computer changed some of the ways we do Computation; A problem from the Study Groups; Influence of the computing environment (hardware and software); Convergence analysis for the heat equation, then and now. | |||
|
Thu, 23/10/2008 14:00 |
Dr Marc Baboulin (University of Coimbra) |
Computational Mathematics and Applications |
Rutherford Appleton Laboratory, nr Didcot |
| The advent of multicore processors and other technologies like Graphical Processing Units (GPU) will considerably influence future research in High Performance Computing. To take advantage of these architectures in dense linear algebra operations, new algorithms are proposed that use finer granularity and minimize synchronization points. After presenting some of these algorithms, we address the issue of pivoting and investigate randomization techniques to avoid pivoting in some cases. In the particular case of GPUs, we show how linear algebra operations can be enhanced using hybrid CPU-GPU calculations and mixed precision algorithms. | |||
|
Thu, 30/10/2008 14:00 |
Prof Simon Tavener (Colorado State University) |
Computational Mathematics and Applications |
Comlab |
| Operator decomposition methods are an attractive solution strategy for computing complex phenomena involving multiple physical processes, multiple scales or multiple domains. The general strategy is to decompose the problem into components involving simpler physics over a relatively limited range of scales, and then to seek the solution of the entire system through an iterative procedure involving solutions of the individual components. We analyze the accuracy of an operator decomposition finite element method for a conjugate heat transfer problem consisting of a fluid and a solid coupled through a common boundary. We derive accurate a posteriori error estimates that account for both local discretization errors and the transfer of error between fluid and solid domains. We use these estimates to guide adaptive mesh refinement. In addition, we show that the order of convergence of the operator decomposition method is limited by the accuracy of the transferred gradient information, and how a simple boundary flux recovery method can be used to regain the optimal order of accuracy in an efficient manner. This is joint work with Don Estep and Tim Wildey, Department of Mathematics, Colorado State University. | |||
|
Thu, 06/11/2008 14:00 |
Prof Divakar Viswanath (University of Michigan, USA) |
Computational Mathematics and Applications |
Comlab |
| The butterfly-shaped Lorenz attractor is a fractal set made up of infinitely many periodic orbits. Ever since Lorenz (1963) introduced a system of three simple ordinary differential equations, much of the discussion of his system and its strange attractor has adopted a dynamical point of view. In contrast, we allow time to be a complex variable and look upon such solutions of the Lorenz system as analytic functions. Formal analysis gives the form and coefficients of the complex singularities of the Lorenz system. Very precise (> 500 digits) numerical computations show that the periodic orbits of the Lorenz system have singularities which obey that form exactly or very nearly so. Both formal analysis and numerical computation suggest that the mathematical analysis of the Lorenz system is a problem in analytic function theory. (Joint work with S. Sahutoglu). | |||
|
Thu, 13/11/2008 14:00 |
Frederic Nataf (Universite Paris VI and CNRS UMR 7598) |
Computational Mathematics and Applications |
Comlab |
| We focus on domain decomposition methods for systems of PDEs (versus scalar PDEs). The Smith factorization (a "pure" algebra tool) is used systematically to derive new domain decompositions methods for symmetric and unsymmetric systems of PDEs: the compressible Euler equations, the Stokes and Oseen (linearized Navier-Stokes) problem. We will focus on the Stokes system. In two dimensions the key idea is the transformation of the Stokes problem into a scalar bi-harmonic problem. We show, how a proposed domain decomposition method for the bi-harmonic problem leads to a domain decomposition method for the Stokes equations which inherits the convergence behavior of the scalar problem. Thus, it is sufficient to study the convergence of the scalar algorithm. The same procedure can also be applied to the three-dimensional Stokes problem. | |||
|
Thu, 20/11/2008 14:00 |
Prof Soeren Bartels (University of Bonn) |
Computational Mathematics and Applications |
Comlab |
| Partial differential equations with a nonlinear pointwise constraint defined through a manifold occur in a variety of applications: The magnetization of a ferromagnet can be described by a unit length vector field and the orientation of the rod-like molecules that constitute a liquid crystal is often modeled by a vector field that attains its values in the real projective plane thus respecting the head-to-tail symmetry of the molecules. Other applications arise in geometric modeling, quantum mechanics, and general relativity. Simple examples reveal that it is impossible to satisfy pointwise constraints exactly by lowest order finite elements. For two model problems we discuss the practical realization of the constraint, the efficient solution of the resulting nonlinear systems of equations, and weak accumulation of approximations at exact solutions. | |||
|
Thu, 27/11/2008 14:00 |
Dr. Alicia Kim (University of Bath) |
Computational Mathematics and Applications |
Rutherford Appleton Laboratory, nr Didcot |
| As research in topology optimisation has reached a level of maturity, two main classes of methods have emerged and their applications to real engineering design in industry are increasing. It has therefore become important to identify the limitations and challenges in order to ensure that topology optimisation is appropriately employed during the design process whilst research may continue to offer a more reliable and fast design tool to engineers. The seminar will begin by introducing the topology optimisation problem and the two popular finite element based approaches. A range of numerical methods used in the typical implementations will be outlined. This will form the basis for the discussion on the short-comings and challenges as an easy-to-use design tool for engineers, particularly in the context of reliably providing the consistent optimum solutions to given problems with minimum a priori information. Another industrial requirement is a fast solution time to easy-to-set-up problems. The seminar will present the recent efforts in addressing some of these issues and the remaining challenges for the future. | |||
|
Thu, 04/12/2008 14:00 |
Jonathan Hogg (Rutherford Appleton Laboratory) |
Computational Mathematics and Applications |
Comlab |
| Multicore chips are nearly ubiquitous in modern machines, and to fully exploit this continuation of Moore's Law, numerical algorithms need to be able to exploit parallelism. We describe recent approaches to both dense and sparse parallel Cholesky factorization on shared memory multicore systems and present results from our new codes for problems arising from large real-world applications. In particular we describe our experiences using directed acyclic graph based scheduling in the dense case and retrofitting parallelism to a sparse serial solver. | |||
