Gibson Grd floor SR

Standard numerical schemes for acoustic scattering problems suffer from the restriction that the number of degrees of freedom required to achieve a prescribed level of accuracy must grow at least linearly with respect to frequency in order to maintain accuracy as frequency increases. In this talk, we review recent progress on the development and analysis of hybrid numerical-asymptotic boundary integral equation methods for these problems. The key idea of this approach is to form an ansatz for the solution based on knowledge of the high frequency asymptotics, allowing one to achieve any required accuracy via the approximation of only (in many cases provably) non-oscillatory functions. In particular, we discuss very recent work extending these ideas for the first time to non-convex scatterers.

We show that the steepest-descent and Newton's methods for unconstrained nonconvex optimization

under standard assumptions may both require a number of iterations and function evaluations

arbitrarily close to the steepest-descent's global worst-case complexity bound. This implies that

the latter upper bound is essentially tight for steepest descent and that Newton's method may be as

slow as the steepest-descent method in the worst case. Then the cubic regularization of Newton's

method (Griewank (1981), Nesterov & Polyak (2006)) is considered and extended to large-scale

problems, while preserving the same order of its improved worst-case complexity (by comparison to

that of steepest-descent); this improved worst-case bound is also shown to be tight. We further

show that the cubic regularization approach is, in fact, optimal from a worst-case complexity point

of view amongst a wide class of second-order methods. The worst-case problem-evaluation complexity

of constrained optimization will also be discussed. This is joint work with Nick Gould (Rutherford

Appleton Laboratory, UK) and Philippe Toint (University of Namur, Belgium).

We present progress on an Interior Point based multi-step solution approach for stochastic programming problems. Our approach works with a series of scenario trees that can be seen as successively more accurate discretizations of an underlying probability distribution and employs IPM warmstarts to "lift" approximate solutions from one tree to the next larger tree.

Nonnormality is a well studied subject in the context of partial differential operators. Yet, only little is known for boundary integral operators. The only well studied case is the unit ball, where the standard single layer, double layer and conjugate double layer potential operators in acoustic scattering diagonalise in a unitary basis. In this talk we present recent results for the analysis of spectral decompositions and nonnormality of boundary integral operators on more general domains. One particular application is the analysis of stability constants for boundary element discretisations. We demonstrate how these are effected by nonnormality and give several numerical examples, illustrating these issues on various domains.

The fundamental task in climate variability research is to eke

out structure from climate signals. Ideally we want a causal

connection between a physical process and the structure of the

signal. Sometimes we have to settle for a correlation between

these. The challenge is that the data is often poorly

constrained and/or sparse. Even though many data gathering

campaigns are taking place or are being planned, the very high

dimensional state space of the system makes the prospects of

climate variability analysis from data alone impractical.

Progress in the analysis is possible by the use of models and

data. Data assimilation is one such strategy. In this talk we

will describe the methodology, illustrate some of its

challenges, and highlight some of the ways our group has

proposed to improving the methodology.

The liquid crystal (LC) flow model is a coupling between

orientation (director field) of LC molecules and a flow field.

The model may probably be one of simplest complex fluids and

is very similar to a Allen-Cahn phase field model for

multiphase flows if the orientation variable is replaced by a

phase function. There are a few large or small parameters

involved in the model (e.g. the small penalty parameter for

the unit length LC molecule or the small phase-change

parameter, possibly large Reynolds number of the flow field,

etc.). We propose a C^0 finite element formulation in space

and a modified midpoint scheme in time which accurately

preserves the inherent energy law of the model. We use C^0

elements because they are simpler than existing C^1 element

and mixed element methods. We emphasise the energy law

preservation because from the PDE analysis point of view the

energy law is very important to correctly catch the evolution

of singularities in the LC molecule orientation. In addition

we will see numerical examples that the energy law preserving

scheme performs better under some choices of parameters. We

shall apply the same idea to a Cahn-Hilliard phase field model

where the biharmonic operator is decomposed into two Laplacian

operators. But we find that under our scheme non-physical

oscillation near the interface occurs. We figure out the

reason from the viewpoint of differential algebraic equations

and then remove the non-physical oscillation by doing only one

step of a modified backward Euler scheme at the initial time.

A number of numerical examples demonstrate the good

performance of the method. At the end of the talk we will show

how to apply the method to compute a superconductivity model,

especially at the regime of Hc2 or beyond. The talk is based

on a few joint papers with Chun Liu, Qi Wang, Xingbin Pan and

Roland Glowinski, etc.

Computational Fluid Dynamics (CFD) has become an

indispensable tool in designing turbomachinery components in all sectors of

Rolls-Royce business units namely, Aerospace, Industrial, Marine and Nuclear.

Increasingly sophisticated search and optimisation techniques are used based on

both traditional optimisation methods as well as, design of computer experiment

techniques, advanced surrogate methods, and evolutionary optimisation

techniques. Geometry and data representation as well as access, queuing and

loading control of large high performance computing clusters are areas of

research to establish the most efficient techniques for improving the

performance of an already highly efficient modern jet engine.

\\

\\

This presentation focuses on a high fidelity design

optimisation framework called SOPHY that is used in Rolls-Royce to provide

parametric geometry, automatic meshing, advanced design-space search

algorithms, accurate and robust CFD methodology and post-processing. The

significance of including the so-called real geometry features and interaction

of turbomachinery components in the optimisation cycle are discussed. Examples are drawn from real world

applications of the SOPHY design systems in an engine project.

We will study the question of whether the adjoint of a given matrix can be written as a rational function in the matrix. After showing necessary and sufficient conditions, rational interpolation theory will help to characterize the most important existing cases. Several topics related to our question will be explored. They range from short recurrence Krylov subspace methods to the roots of harmonic polynomials and harmonic rational functions. The latter have recently found interesting applications in astrophysics, which will briefly be discussed as well.