Past Computational Mathematics and Applications Seminar

Prof Simon Chandler-Wilde
Abstract
We consider the problem of recovering the position of a scattering surface from measurements of the scattered field on a finite line above the surface. A point source algorithm is proposed, based on earlier work by Potthast, which reconstructs, in the first instance, the scattered field in the whole region above the scattering surface. This information is used in a second stage to locate the scatterer. We summarise the theoretical results that can be obtained (error bounds on the reconstructed field as a function of the noise level in the original measurements). For the case of a point source of the incident field we present numerical experiments for both a steady source (time harmonic excitation) and a pulse source typical of an antenna in ground penetrating radar applications. \\ This is joint work with Claire Lines (Brunel University).
• Computational Mathematics and Applications Seminar
4 December 2003
14:00
Dr Nik Petrinic
Abstract
The seminar will address issues related to numerical simulation of non-linear behaviour of solid materials to impact loading. The kinematic and constitutive aspects of the transition from continuum to discontinuum will be presented as utilised within an explicit finite element development framework. Material softening, mesh sensitivity and regularisation of solutions will be discussed.
• Computational Mathematics and Applications Seminar
27 November 2003
14:00
Prof Andreas Griewank
Abstract
To numerical analysts and other applied mathematicians Jacobians and Hessians are matrices, i.e. rectangular arrays of numbers or algebraic expressions. Possibly taking account of their sparsity such arrays are frequently passed into library routines for performing various computational tasks. \\ \\ A central goal of an activity called automatic differentiation has been the accumulation of all nonzero entries from elementary partial derivatives according to some variant of the chainrule. The elementary partials arise in the user-supplied procedure for evaluating the underlying vector- or scalar-valued function at a given argument. \\ \\ We observe here that in this process a certain kind of structure that we call "Jacobian scarcity" might be lost. This loss will make the subsequent calculation of Jacobian vector-products unnecessarily expensive. Instead we advocate the representation of the Jacobian as a linear computational graph of minimal complexity. Many theoretical and practical questions remain unresolved.
• Computational Mathematics and Applications Seminar
20 November 2003
14:00
Prof Javier Pena
Abstract
Condition numbers are a central concept in numerical analysis. They provide a natural parameter for studying the behavior of algorithms, as well as sensitivity and geometric properties of a problem. The condition number of a problem instance is usually a measure of the distance to the set of ill-posed instances. For instance, the classical Eckart and Young identity characterizes the condition number of a square matrix as the reciprocal of its relative distance to singularity. \\ \\ We present concepts of conditioning for optimization problems and for more general variational problems. We show that the Eckart and Young identity has natural extension to much wider contexts. We also discuss conditioning under the presence of block-structure, such as that determined by a sparsity pattern. The latter has interesting connections with the mu-number in robust control and with the sign-real spectral radius.
• Computational Mathematics and Applications Seminar
Dr Patrik Bosander
Abstract
The seminar will focus on mathematical modelling of physics phenomena, with applications in e.g. mass and heat transfer, fluid flow, and electromagnetic wave propagation. Simultaneous solutions of several physics phenomena described by PDEs - multiphysics - will also be presented and discussed. \\ \\ All models will be realised through the use of the MATLAB based finite element package FEMLAB. FEMLAB is a multiphysics modelling environment with built-in PDE solvers for linear, non-linear, time dependent and eigenvalue problems. For ease-of-use, it comprises ready-to-use applications for various physics phenomena, and tailored applications for Structural Mechanics, Electromagnetics, and Chemical Engineering. But in addition, FEMLAB facilitates straightforward implementation of arbitrary coupled non-linear PDEs, which brings about a great deal of flexibility in problem definition. Please see http://ww.uk.comsol.com for more info. \\ \\ FEMLAB is developed by the COMSOL Group, a Swedish headquartered spin-off from the Royal Institute of Technology (KTH) in Stockholm, with offices around the world. Its UK office is situated in The Oxford Science Park.
• Computational Mathematics and Applications Seminar
Dr Eric Fraga
Abstract
The process industries are one of the UK's major sectors and include petrochemicals, pharmaceuticals, water, energy and the food industry, amongst others. The design of a processing plant is a difficult task. This is due to both the need to cater for multiple criteria (such as economics, environmental and safety) and the use highly complex nonlinear models to describe the behaviour of individual unit operations in the process. Early in the design stages, an engineer may wish to use automated design tools to generate conceptual plant designs which have potentially positive attributes with respect to the main criteria. Such automated tools typically rely on optimization for solving large mixed integer nonlinear programming models. \\ \\ This talk presents an overview of some of the work done in the Computer Aided Process Engineering group at UCL. Primary emphasis will be given to recent developments in hybrid optimization methods, including the use of graphical interfaces based on problem specific visualization techniques to allow the engineer to interact with embedded optimization procedures. Case studies from petrochemical and water industries will be presented to demonstrate the complexities involved and illustrate the potential benefits of hybrid approaches.
• Computational Mathematics and Applications Seminar
30 October 2003
14:00
Dr Robert Scheichl
Abstract
The simulation of sedimentary basins aims at reconstructing its historical evolution in order to provide quantitative predictions about phenomena leading to hydrocarbon accumulations. The kernel of this simulation is the numerical solution of a complex system of time dependent, three dimensional partial differential equations of mixed parabolic-hyperbolic type in highly heterogeneous media. A discretisation and linearisation of this system leads to large ill-conditioned non-symmetric linear systems with three unknowns per mesh element. \\ \\ In the seminar I will look at different preconditioning approaches for these systems and at their parallelisation. The most effective preconditioner which we developed so far consists in three stages: (i) a local decoupling of the equations which (in addition) aims at concentrating the elliptic part of the system in the "pressure block''; (ii) an efficient preconditioning of the pressure block using AMG; (iii) the "recoupling'' of the equations. Numerical results on real case studies, exhibit (i) a significant reduction of sequential CPU times, up to a factor 5 with respect to the current ILU(0) preconditioner, (ii) robustness with respect to physical and numerical parameters, and (iii) a speedup of up to 4 on 8 processors.
• Computational Mathematics and Applications Seminar
23 October 2003
14:00
Prof Arieh Iserles
Abstract
Rapidly oscillating problems, whether differential equations or integrals, ubiquitous in applications, are allegedly difficult to compute. In this talk we will endeavour to persuade the audience that this is false: high oscillation, properly understood, is good for computation! We describe methods for differential equations, based on Magnus and Neumann expansions of modified systems, whose efficacy improves in the presence of high oscillation. Likewise, we analyse generalised Filon quadrature methods, showing that also their error sharply decreases as the oscillation becomes more rapid.
• Computational Mathematics and Applications Seminar
16 October 2003
14:00
Prof Andrew Stuart
Abstract
In many applications of interest, such as the conformational dynamics of molecules, large deterministic systems can exhibit stochastic behaviour in a relative small number of coarse-grained variables. This kind of dimension reduction, from a large deterministic system to a smaller stochastic one, can be very useful in understanding the problem. Whilst the subject of statistical mechanics provides a wealth of explicit examples where stochastic models for coarse variables can be found analytically, it is frequently the case that applications of interest are not amenable to analytic dimension reduction. It is hence of interest to pursue computational algorithms for such dimension reduction. This talk will be devoted to describing recent work on parameter estimation aimed at problems arising in this context. \\ \\ Joint work with Raz Kupferman (Jerusalem) and Petter Wiberg (Warwick)
• Computational Mathematics and Applications Seminar
19 June 2003
14:00
Dr Austin Mack
Abstract
In recent times, research into scattering of electromagnetic waves by complex objects has assumed great importance due to its relevance to radar applications, where the main objective is to identify targeted objects. In designing stealth weapon systems such as military aircraft, control of their radar cross section is of paramount importance. Aircraft in combat situations are threatened by enemy missiles. One countermeasure which is used to reduce this threat is to minimise the radar cross section. On the other hand, there is a demand for the enhancement of the radar cross section of civilian spacecraft. Operators of communication satellites often request a complicated differential radar cross section in order to assist with the tracking of the satellite. To control the radar cross section, an essential requirement is a capability for accurate prediction of electromagnetic scattering from complex objects. \\ \\ One difficulty which is encountered in the development of suitable numerical solution schemes is the existence of constraints which are in excess of those needed for a unique solution. Rather than attempt to include the constraint in the equation set, the novel approach which is presented here involves the use of the finite element method and the construction of a specialised element in which the relevant solution variables are appropriately constrained by the nature of their interpolation functions. For many years, such an idea was claimed to be impossible. While the idea is not without its difficulties, its advantages far outweigh its disadvantages. The presenter has successfully developed such an element for primitive variable solutions to viscous incompressible flows and wishes to extend the concept to electromagnetic scattering problems. \\ \\ Dr Mack has first degrees in mathematics and aeronautical engineering, plus a Masters and a Doctorate, both in computational fluid dynamics. He has some thirty years experience in this latter field. He pioneered the development of the innovative solenoidal approach for the finite element solution of viscous incompressible flows. At the time, such a radical idea was claimed in the literature to be impossible. Much of this early research was undertaken during a six month sabbatical with the Numerical Analysis Group at the Oxford University Computing Laboratory. Dr Mack has since received funding from British Aerospace and the United States Department of Defense to continue this research.
• Computational Mathematics and Applications Seminar