A groundbreaking paper by Theme D Leader and Deputy Leader, I.G. Graham and R. Scheichl, on the application of multiscale methods in the construction of fast solvers for discrete PDEs via domain decomposition was published in Numerische Mathematik in 2007 and has already accumulated 20 citations on Google Scholar. This has been followed by a recent paper of Pechstein (Radon Institute) and Scheichl, accepted for Numerische Mathematik, which contains a similar breakthrough for the analysis of FETI domain decomposition methods. Additional recent papers include Graham and Scheichl (2007a), Graham and Scheichl (2007b), Scheichl and Vainikko (2007) Aksoylu (Baton Rouge), Graham, Klee (Conuco Phillips, Houston) with Scheichl (2008) , and (Theme D Postdoc) Jan Van lent , with Scheichl and Graham (2008). While the above links are to the preprints, publication history can be found here. This body of work has been presented at a number of major international meetings such as MAFELAP 2006 (London), SIAM Geosciences 2007 (Santa Fe, USA) and ECCOMAS 2008 (Venice). A Theme D PhD student Patrick Lechner also wrote a successful PhD thesis in this area. The ideas in this body of work have also been exploited in Bath MSc projects on the computation of emergent behaviour.
A related development is new work of Graham together with colleagues at Caltech: T.-Y. Hou (BICS International Advisory Board) and C.-C. Chu on the accuracy of numerical methods for high contrast elliptic interface problems. This work has recently been presented in invited talks by Graham at Mathematisches Forschungsinstitut Oberwolfach, Germany and by Tom Hou at University of Texas at Austin. V.P. Smyshlyaev, Ilia Kamotski (both Theme A) and Natalia Babych (Postdoc) have all been involved and future joint publications with members of Theme A are planned.
I.G. Graham, T.-Y. Hou and R. Scheichl will be the organisers of an LMS Durham Research Symposium on ``Numerical Analysis of Multiscale Problems'' in summer 2010.
Emergent behaviour is also one focus of a new project involving Graham and Scheichl together with Ian Sloan, Frances Kuo and Dirk Nuyens of MASCOS in Australia (a formal partner of BICS), which is in the general area of Quasi-Monte Carlo methods for very high dimensional problems, such as those arising in stochastic models. Graham and Scheich both visited University of New South Wales recently to develop this collaboration.
Recent BICS Theme D PhD theses include Stefano Giani (supervisor Graham) who will graduate in December 2008 and Richard Norton (supervisor Scheichl) who recently submitted his thesis. Both have worked on different aspects of numerical simulation of photonic crystals, Stefano on adaptive finite element methods and Richard on plane wave methods, in Richard's case in collaboration with colleagues in the Physics Department at Bath. Stefano is now a postdoc at Nottingham and Richard is a postdoc in Oxford.
The collaboration between Themes A and D has been substantially enhanced by the award of a major EPSRC responsive mode grant (worth about 700K pounds), to Graham and Smyshlyaev (Bath) together with Chandler-Wilde and Langdon (both of Reading University), to appoint two postdocs and provide three international visitors - Hiptmair (Zuerich), Ganesh (Colorado) and Dominguez (Navarra) in a multidisciplinary project involving four industrial/research bodies - BAE systems, Institute for Cancer Research, Schlumberger and the Met Office. Two high quality postdocs have recently been appointed to this project which will start in March 2009. Further details are given here.
Theme D is strongly involved in a substantial partnership
between BICS and the Met
Office and Exeter University
on numerical weather forecasting, with funds from
(i) EPSRC and (ii) HEFCE and SWRDA via
the Great Western Research
(GWR) Alliance. The partnership involves an EPSRC CASE student
(with C.J. Budd on adaptive methods),
a GWR student (with R. Scheichl, on fast solvers)
and a GWR Fellowship (to which M. Freitag has been appointed),
hosted by Bath in association with
Maths at Exeter and the Met Office.
M. Freitag has spent 4 weeks at the UK Met Office working with Mike Cullen
on data assimilation for numerical weather prediction. She has recently
presented her results at the annual Bath/RAL Numerical Analysis day with a
talk on the links between 4DVar, the assimilation scheme used at the Met
Office, and Tikhonov regularisation, a standard regularisation technique
used in the inverse problems community.
There is overlap with the Invert group , directed by Cathryn
Mitchell , which includes the former Theme E postdoc
Nathan Smith.
Further industrial involvement in Theme D has been provided by SERCO Assurance who have had a long collaboration with Graham on energy-related problems, including a successful PhD project of S. Mandica and a current CASE award.