# Numerical Analysis and Scientific Computing Seminars 2005/06

Please follow the link on the navigation menu on the left hand side for information on the current Numerical Analysis and Scientific Computing Seminars.

## Semester One

- Fri

Oct 14

2005**The World's Most Fundamental Matrix Equation, and Early Numerical Linear Algebra in the UK**

Nick Higham (University of Manchester)

3.00pm - MSS M12 - Fri

Oct 28

2005**Nonsmooth Matrix Equations with Some Applications**

Houduo Qi (Southampton)

3.00pm - MSS M12 - Fri

Nov 4

2005**Fourth Generation Light Sources - Coherent, Ultra-bright X-rays**

Barry Dobson (Daresbury)

3.00pm - MSS M12 - Fri

Nov 11

2005**High order stochastic integrators for linear systems**

Simon Malham (Heriot-Watt)

3.00pm - MSS M12 - Fri

Nov 18

2005**Computation in the presence of high oscillation**

Arieh Iserles (Cambridge)

3.00pm - MSS M12 - Fri

Nov 25

2005**Convergence of the solution of a nonsymmetric matrix Riccati differential equation to its stable equilibrium solution**

Chun-Hua Guo (Manchester)

3.00pm - MSS M12 - Fri

Dec 2

2005**Stochastic and finite element modelling of heterogeneity in geomaterials**

Michael Hicks (Manchester)

3.00pm - MSS M12 - Fri

Dec 9

2005**An Adaptive Time Integrator: from ODEs to Advection-Diffusion**

David Griffiths (Dundee)

3.00pm - MSS M12

The talk begins with a brief introduction on how nonsmooth equations arise naturally from simple situations, via examples including the projection operator onto the positive semidefinite cone. The talk then turns to developing Newton's method for nonsmooth equations, extending the classical theory to the nonsmooth case. We demonstrate the powerfulness of the nonsmooth (matrix) equation approach against two classes of important problems, namely the nearest correlation matrix problem (made widely known by Higham's 2002 IMA paper) and the semidefinite complementarity problem. The issue of solving linear matrix equations in the Newton method is particularly addressed. We conclude the talk with a few of interesting future research topics including the H-weighted nearest correlation matrix problem and its application to kernel matrix completion problem in data mining.

The production of electromagnetic radiation has developed from laboratory based generators to synchrotrons over the last three decades. Each stage of development has resulted in a dramatic increase in brilliance enabling new science with each step. There are now initiatives underway across the world to take the next step by building free electron lasers with radically new levels of brilliance and for the first time with complete lateral coherence in the spectrum from the infra-red to x-rays.

However, these new sources have outstripped the codes used to design the optics which delivers the radiation from the source to the experiment.

This talk will review the development of the sources, describe in some detail the physics of the operation of free electron lasers and survey the available optics design codes. Finally, I will pose the problems facing the developers of new optics modeling codes by way of invitation to any interested parties who would be interested in participating in what will be a complex, challenging and hopefully intellectually stimulating international collaboration.

## Semester Two

- Fri

Feb 3

2006**Model Order Reduction for RF/Microwave Applications**

Slobodan Mijalkovic (Delft University of Technology )

3.00pm - MSS M12 - Fri

Feb 10

2006**Accurate polynomial evaluation in floating point arithmetic**

Stef Graillat (Universite de Perpignan Via Domitia)

3.00pm - MSS M12 - Fri

Mar 10

2006**ARK methods for Stiff and Non-stiff problems**

Nicolette Rattenbury (Auckland)

3.00pm - MSS M12 - Fri

Mar 17

2006**Parameter Estimation for Partially Observed Hypo-Elliptic Diffusions**

Yvo Pokern (Warwick University)

3.00pm - MSS M12 - Fri

Mar 24

2006**Max-algebra, max-eigenvector and pairwise comparison matrices**

Ludwig Elsner

3.00pm - MSS M12

Model-order reduction (MOR) explores the ways in which large-scale physical models and simulations can be reduced for the purpose of expedient, yet accurate, computer-aided design and optimization. Recently, MOR is aggressively pursued by the RF/Microwave community in order to develop a new class of efficient modeling tools capable of tackling the escalating complexity of the future circuit design and virtual prototyping. The principle objective of this seminar is to introduce the basic principles and practical techniques of MOR, with special emphasis on novel approaches and open problems.

Using error-free transformations, we improve the classic Horner Scheme(HS) to evaluate (univariate) polynomials in floating point arithmetic. We prove that this Compensated Horner Scheme (CHS) is as accurate as the classic Horner scheme performed with twice the working precision. Theoretical analysis and experiments exhibit a reasonable running time overhead being also more interesting than double-double implementations. We introduce a dynamic and validated error bound of the CHS computed value.

Runge-Kutta methods have been important and useful numerical methods for solving ordinary differential equations for more than 100 years. It is natural to ask how it is possible to generalise them without destroying their essential properties. Approaches to this have taken the form of pseudo Runge-Kutta methods and two-step Runge-Kutta methods. We believe that Almost Runge-Kutta (ARK) methods also have interesting prospects and potential advantages. Although they pass more than one piece of information from step to step they retain the simple stability properties of traditional Runge-Kutta methods. The information that is passed from step to step is in Nordseick form, ensuring that changing step-size is cheap and convenient.

A significant advantage of these new methods is that the number of stages required to obtain a certain order is not restrained by the Butcher barrier. A family of methods has been discovered that obtain fifth order with only five stages. Because this family of methods do not satisfy all of the annihilation conditions, an order reduction is seen for variable step-size. By carefully implementing the step-size change we can restore fifth order behaviour for variable step-size.

Most of the analysis for this class of methods has concentrated on explicit methods for solving non-stiff differential equations, however much of this analysis is easily adaptable to implicit methods for stiff problems. Some low order implicit methods will be introduced.

Hypo-elliptic diffusion processes can be used to model a variety of phenomena ranging from molecular dynamics to vibrations in train carriages. This talk examines parameter estimation for such processes in situations where some components of the solution are observed at discrete times. Since exact likelihoods for the transition densities are typically not known, approximations are used that are expected to work well in the limit of small inter-sample times dt and large total observation times N dt. Hypoellipticity together with partial observation leads to ill-conditioning requiring a judicious combination of approximate likelihoods for the various parameters to be estimated. These are combined in a deterministic scan Gibbs sampler alternating between missing data in the unobserved solution components and parameters. Numerical experiments display asymptotic consistency of the method when applied to simulated data. I may conclude with an application of the Gibbs sampler to molecular dynamics data.

One of the main results in the theory of the max-algebra, i.e. the set of nonnegative matrices, where addition is replaced by the max-operation, is the existence of a max-eigenvector and a max-eigenvalue. We discuss its numerical calculation. In the second part we consider pairwise comparison matrices which play a role in the AHP (analytic hierarchy problem). To attach in a senseful manner values to certain goods, where only paiwise comparisons are available, the Perron eigenvector of a pairwise comparison matrix is used. We show that replacing it by the max-eigenvector has several advantages. (Joint with P. van den Driessche)

## Further information

For further information please contact the seminar organiser Tony Shardlow.