2nd October 2012

Speaker: Coralia Cartis (Edinburgh).

Title: Optimal Newton-type methods for nonconvex smooth optimization.

Abstract: This talk addresses global rates of convergence and the worst-case evaluation complexity of methods for nonconvex optimization problems. We show that the classical 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 for nonconvex optimization. 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 ofNamur,Belgium).

30th October 2012

Speaker: Tony Shardlow (Bath).

Title: Approximation of Gaussian random fields.

Abstract: With the growing importance of stochastic PDEs, there is much interest in random fields generators. We look at random field generators through the eyes of a numerical analyst. The most well known generators give approximate realisations to a large class of Gaussian random fields. We will analyses the errors and give rates of convergence for several well known (e.g., circulant embedding, turning bands, quadrature, KL) methods.

19th November 2012 in Room S/0.38, School of Engineering

Speaker: Feng Xiao (Tokyo).

Title: High order conservative collocation method using multi-moment constraints.

Abstract: In this talk, I will present a general formulation to devise high order conservative collocation schemes by using multi-moment constraint conditions for flux function reconstruction. Different from the existing methods, such as the nodal type discontinuous Galerkin method and the spectral collocation method, where only the point values are used to construct the numerical flux, the new method makes use of different types of quantities (moments), such as cell integrated average, point value and derivatives. This formulation can be also interpreted as a blend of the Lagrange interpolation and the Hermite interpolation, which leads to a new class of high order schemes. Some representative schemes will be presented and evaluated through Fourier analysis and numerical tests.

12th March 2013

Speaker: Dr. Mikito Furuichi (Japan Agency for Marine-Earth Science and Technology (JAMSTEC).

Title: Development of fluid-particle coupled simulation method in the Stokes flow regime: toward 3-D geodynamic magma simulation including granular media.

Abstract: A fluid-particle two-phase flow has been widely studied in geodynamics, because particle-saturated fluid layer is important for understanding the dynamics of solidifying and melting process in the magma chamber or magma ocean.

In order to deal with such particle-fluid systems as the geodynamical modeling in 3-D geometry, we develop a new coupled simulation code of Finite Difference method (FDM) for fluid flow and Discrete Element method (DEM) for solid particles. In the geodynamic modeling with highly viscous fluid, the fluid motion can be treated as the Stokes flow. Although this type of coupled simulation method has been well developed in the engineering field especially for a fluidized bed with high Reynolds number, the method for viscous granular media over long time scales has not yet been fully addressed.

The normal DEM-fluid formulation requires a solution of dumped oscillation with a small time step dt ~1/? for high fluid viscosity ?. Thus the normal formulation is not suitable for our target problems. We therefore propose to drop off the inertial term from the equation of particle motion likewise the Stokes flow. With this approach, we can employ the large dt~? for the problems with highly viscos fluid. In the talk, we introduce the details of our coupled model treatment and its implementation on the vector/many core parallel architectures.

19th March 2013

Speaker: Prof Serafim Kalliadasis (Imperial).

Title: Recent progress on the moving contact line problem.

Abstract: The moving contact line problem is a long-standing and fundamental challenge in the field of fluid dynamics, occurring when one fluid replaces another as it moves along a solid surface. Moving contact lines occur in a vast range of applications, where an apparent paradox of motion of a fluid-fluid interface, yet static fluid velocity at the solid satisfying the no-slip boundary condition arises. In this talk we will review recent progress on the problem made by our group.

The motion of a contact line is examined, and comparisons drawn, for a variety of proposed models in the literature. We first scrutinise a number of models in the classic test-bed system of spreading of a thin two-dimensional droplet on a planar substrate, showing that slip, precursor film and interface formation models effectively reduce to the same spreading behaviour. This latter model, developed by Shikhmurzaev a few years ago, is a complex and somewhat controversial one, differentiating itself by accounting for a variation in surface layer quantities and having finite-time surface tension relaxation. Extensions to consider substrate heterogeneities in this prototype system for slip models are also considered, such as for surface roughness and fluctuations in wetting properties through chemical variability. Analysis of a solid-liquid-gas diffuse-interface model is then presented, with no-slip at the solid and where the fluid phase is specified by a continuous density field. We first obtain a wetting boundary condition on the solid that allows us to consider the motion without any additional physics, i.e. without density gradients at the wall away from the contact line associated with precursor films.

Careful examination of the asymptotic behaviour asthe contact line is approached is then shown to resolve the singularities associated with the moving contact line problem. Various features of the model are scrutinised alongside extensions to incorporate slip, finite-time relaxation of the chemical potential, or a precursor film at the wall. But these are not necessary to resolve the moving contact line problem.

8th April 2013 at 10:30 in M/0.40

Speaker: John Bush (MIT).

Title: Hydrodynamic quantum analogs: Droplets walking on the impossible pilot wave

Abstract: Yves Couder and coworkers have recently discovered that droplets walking on a vibrating fluid bath exhibit several features previously thought to be peculiar to the microscopic, quantum realm. Theoretical developments provide rationale for the complex behavior of the bouncing droplets, and yield a trajectory equation for the walking droplets. Experimental results reveal the emergence of wave-like statistics from pilot-wave dynamics for droplets walking in confined geometries, and for droplets walking on a rotating fluid bath. Theoretical results indicate the manner in which wave-like statistics emerge from pilot-wave dynamics for droplets walking on a rotating bath, or under the influence of a central force. The relation between this fluid system and de Broglie’s relativistic pilot-wave theory of quantum mechanics is discussed.

4th June 2013

Speaker: Christopher Davies (Cardiff University School of Mathematics)

Title: Evolution of disturbance wavepackets in an oscillatory Stokes layer

Abstract: Numerical simulation results are presented for the evolution of disturbances in a flat Stokes layer. The response to a spatially localised impulsive forcing is investigated and it is found that the spatial-temporal development of linearized disturbances displays an intriguing family tree-like structure. This involves the birth of successive generations of distinct wavepacket components. It is shown that some important features of the disturbance behaviour can be predicted using the results of a linear stability analysis based on Floquet theory. The ramifications of the simulation results are also discussed with regard to the interpretation of observations obtained from physical experiments.

11th June 2013

Speaker: Greg King (Population and Conservation Genetics Group, Instituto Gulbenkian de Ciência Oeiras, Portugal)

Title: The energy cascade in the atmospheric mesoscales at the bottom of the marine boundary layer: Is it upscale? Downscale? Or is nature not so simple?

Abstract: A long-standing question in atmospheric dynamics has been: Is horizontal kinetic energy transferred to small scales through a downscale cascade as in ideal three-dimensional (3D) turbulence? Or is it transferred to large scales via a two-dimensional (2D) inverse cascade?

The classic papers by Nastrom et al (1984, 1985) and more recent papers by Lindborg (1999) and Cho and Lindborg (2001) have addressed this question through an analysis of global datasets of winds near the tropopause measured by instruments carried on commercial aircraft. Here we use winds at the bottom of the marine boundary layer inferred from radar backscatter from the ocean surface measured by the Advanced Scatterometer (ASCAT) on the MetOp-A satellite and the SeaWinds scatterometer on the QuikSCAT satellite. Our results indicate that nature is not so simple.