- Fractional Swift-Hohenberg Equation and Its Application to Modelling of Crystals
Mark Ainsworth, Brown University, USA
Recent years have seen a surge of interest in fractional partial differential equations driven, in part, by the
desire to develop models that are able to more accurately model systems which exhibit behaviour including
sub-and super-diffusion. In the current work, we shall discuss a fractional version of the Swift-Hohenberg
equation. The Swift-Hohenberg equation is a non-linear parabolic PDE which is known to have solutions
display pattern formation, and which has been used in the physics literature to develop macroscopic models
for crystals known as "phase field crystals." We consider a fractional version of the Swift-Hohenberg
equation (FSHE) which gives a non-linear fractional parabolic problem. In particular, we show that FSHE
is well-posed and admits a unique solution; develop a Fourier-Galerkin scheme and obtain error estimates;
analyse pattern formation properties and discuss the use of FSHE to develop fractional phase
field crystal models. This is joint work with Zhiping Mao (Brown University).
- Lattice rules with a twist
Ronald Cools KU Leuven, Belgium
Abstract coming soon
- From CFD to Monte Carlo methods -- the joys of a mid-career change of field
Mike Giles
University of Oxford, UK
One noteworthy aspect of Ian Sloan's career is his switch from theoretical physics to computational mathematics. In this talk I will discuss my own mid-career switch from CFD (computational fluid dynamics) to mathematical finance and Monte Carlo methods, with a visit to Ian in UNSW in early 2007 being a key part of my switch.
- Uncertainty quantification in partial differential and integral equation models
Ivan Graham University of Bath, UK
In late 1975 I received a very enthusiastic aerogramme from Ian Sloan telling me that
"he and his collaborators had just discovered some exciting new numerical
methods for solving integral equations", and offering me a PhD scholarship to work with
him on this at UNSW. Going to Sydney in 1976 to work with Ian was probably the most important
decision of my career and I'm pleased to say that we have kept in close touch and have continued
to work together down the years. In fact recently some of the things we were working on in 1976
became useful again in the analysis of uncertainty propagation in reactor modelling, an
industrial project I've been doing with Wood plc in the UK. I'll talk about this and, more
generally, I'll talk about uncertainty quantification for various PDE models and how recent work by
Ian and a number of other collaborators on high dimensional integration has allowed us to make
substantial progress in this area.
- Pattern-avoiding permutations
Tony Guttmann
Melbourne, Australia
There are n! permutations of the integers 1,2,...,n. We say that a permutation avoids a given
sub-permutation (called a pattern) if there is no sub-sequence of the permutation with digits
in the same relative order as the digits in the pattern. Thus 1,2,3,4,5 avoids the pattern 2,1
as there are no two integers in decreasing order in the permutation. However 1,3,4,2,5 does
not avoid the pattern, as the pairs 3,2, and 4,2 are both in decreasing order. We will
describe some open questions, and discuss some numerical methods for exploring them.
- The Advantages of Sampling with Integration Lattices
Fred Hickernell IIT Chicago, USA
Ian Sloan has been a key developer and proponent of integration lattices for numerical
computation. His 1994 monograph has over 700 citations, and his more recent 2013 Acta Numerica
article has over 200 citations. This talk highlights how well-chosen integration lattices can speed
up numerical multidimensional integration in contrast to tensor product rules and simple Monte
Carlo. It also describes recent work sampling with integration lattices when performing Bayesian
cubature, where the integrand is assumed to be an instance of a Gaussian random process.
- Minimal Riesz energy problems in Sobolev spaces
Wolfgang Wendland University of Stuttgart, Germany
The minimal energy problem for nonnegative charges on a closed surface $\Gamma$ in $\mathbb{R}^3$ goes back to C.F.~Gauss in 1839.
The corresponding Riesz kernel is then on $\Gamma$ weakly singular. More general, in $\mathbb{R}^n, n\geq 2$, the constructive and
numerical solution of minimizing the energy relative to the Riesz kernel then defines on $\Gamma$ a weakly singular single layer
boundary integral operator. The construction and numerical solution of minimizing the corresponding energy then provides the
distribution of charges on $\Gamma$ whose single layer potential is the solution of the Dirichlet problem inside $\Gamma$.
For more general Riesz kernels the corresponding minimizing problem leads to a charge distribution which provides the density
of Sloan’s integration points on $\Gamma$. If the surface consists of two separate parts $\Gamma_1$ and $\Gamma_2$ with
positive charges on $\Gamma_1$ and negative charges on $\Gamma_2$ then the minimizing charge distribution can be approximated
by piecewise constant charges via a Galerkin-Bubnov method. Wavelet matrix compression is applied for solving the discrete system.
This is joint work with Helmut Harbrecht (Basel), Günther Of (Graz) and Natalia Zorii (Kiev).