Research Interests

My research is in Numerical Analysis and Scientific Computing, and concentrates mainly on the numerical solution of (systems of) partial differential equations with highly varying (discontinuous) coefficients including

Efficient and Robust Parallel Multilevel Solvers for Systems of PDEs
Mixed Finite Elements and Preconditioning
Analysis of Domain Decomposition and Multilevel Methods for Multiscale PDEs
Numerical Homogenisation and Upscaling
Multiscale Iterative Methods
Stochastic PDEs and Uncertainty Quantification
Modelling and Simulation of Flow in Porous Media
Modelling and Simulation of Neutronics in Nuclear Reactors
Bandgap Calculations in Photonic Crystal Fibres
Multigrid Methods in Data Assimilation for Weather and Climate Modelling

Research Group

Current Group Members:

Albert Ferreiro-Castilla, PDRA, since February 2012, "Multilevel Monte Carlo Methods for Levy Processes" (with Andreas Kyprianou, Probability Group; Newton Fellowship).
Sarah Cook, Undergraduate Project Student, since February 2012, "Multilevel Monte Carlo Methods for Atmospheric Dispersion" (with Eike Mueller).
Eike Mueller, PDRA, since September 2011, "Parallel Scalability of Elliptic Solvers in Weather and Climate Prediction" (with Met Office and STFC; NERC Grant NE/J005576/1).
Elisabeth Ullmann, PDRA, since July 2011, "Multilevel Monte Carlo Methods for Elliptic Problems with Applications to Radioactive Waste Disposal" (with Nottingham, Oxford, UK Nuclear Decomissioning Authority and Serco Assurances; EPSRC Grant EP/H051503/1).
Aretha Teckentrup, PhD Student, since October 2009, "Multilevel Monte Carlo Methods and Uncertainty Quantification" (with Andreas Kyprianou, Probability Group; EPSRC Studentship).
Chris Hart, PhD Student, since October 2008, "Numerical Computation of Spectra for PDEs on Infinite Domains" (with Marco Marletta, Cardiff; EPSRC Studentship)

Past Group Members:

Emily Dodgson, PhD Student, 2008 - 2011, "Thermoconvective Instabilities in Porous Media" (with Andrew Rees, Mech Eng, Bath; BICS studentship).
Sean Buckeridge, PhD Student, 2006 - 2010, "Numerical Methods for Weather and Climate Models" (with Jon Thuburn, Exeter, and Mike Cullen, MET Office; GWR studentship).
Jan Van lent, Postdoc, March 2006 - June 2008, "Domain Decomposition for Multiscale PDEs" (now lecturer at University of West of England).
Richard Norton, PhD Student, October 2004 - September 2008, Thesis: "Numerical Computation of Band Gaps in Photonic Crystal Fibres" (now at La Trobe University, Melbourne, Australia).
Graham Burton MSc Summer Project, June-September 2008, "Numerical Evaluation of Option Prices" (with Alex Cox, Maths, Bath).
Craig Dickinson MSc Summer Project, June-September 2008, "Numerical Solution of Richards Equation" (with Andrew Rees, Mech Eng, Bath, and Steve Benbow, Quintessa).
Tristan Brindle, Research Student, February-July 2007, "Accurate and Efficient Simulation of Emergent Behaviour in Microstructures" (with Darryl Almond,Mech Eng, Bath).
Peter Way MSc Summer Project, June-September 2006, "Numerical solution of coupled carrier diffusion and wave propagation in laser models" (with Federica Causa, Elec Eng, Bath).
Clare Judd, MSc Summer Project, June-September 2005, "Numerical Simulation of the Properties of Materials with Random Microstructures" (with Darryl Almond, Mech Eng, Bath).
David Marquez, MSc Summer Project, June-September 2004, "Identification of Gas Transport Parameters in Polymers" (with Z. Benjelloun-Dabaghi and M.-H. Klopffer, IFP, Paris).
Sally Harbinson, MSc Summer Project, June-September 2002, "Mathematical Modelling of the Permeation of Gases in Polymers" (with Zakia Benjelloun-Dabaghi, IFP, Paris).

Collaborators:

Ivan Graham (Bath), Ulrich Langer (Linz), Andrew Cliffe (Nottingham), Clemens Pechstein (Linz), Roland Masson (IFP, Paris), Eero Vainikko (Comput.Sci., Tartu, Estonia), Hector Klie (Conoco Philips), Mary Wheeler (Austin), Frederic Nataf (Paris VI), Peter Bastian (Heidelberg), Ian Sloan (UNSW), Francis Kuo (UNSW), Mike Cullen (MET Office, Exeter), Jon Thuburn (Exeter), Tom Hou (Caltech), Marco Marletta (Cardiff), Mike Giles (Oxford), David Kay (Oxford), Ludmil Zikatanov (Penn State), David Bird (Physics, Bath), Andrew Rees (Mech.Eng., Bath), Darryl Almond (Mater.Sci., Bath), Valery Smyshlyaev (Bath), Ilia Kamotski (Bath), Nick McCullen (Bath).

Here are the main projects I am currently working on (NOT UP-TO-DATE!):

Domain Decomposition for Multiscale PDEs

The work on overlapping additive Schwarz is joint work with Ivan Graham (Bath), Eero Vainikko (University of Tartu, Estonia) and Jan Van lent. Recently I have extended the work, together with Clemens Pechstein (Linz), also to FETI methods, and started to work with Frederic Nataf (Paris) on two-level DD methods with Robin-type interface conditions. The novel coarsenings and the theoretical results are also of interest in the context of multiscale numerical approximation and have lead to recent collaborations with Tom Hou (Caltech) and Ludmil Zikatanov (Penn State). [Further Details]

Preconditioning for Multiphase Flow in Porous Media

This is a project in collaboration with the Institut Francais du Petrole (IFP), Paris, France, notably Roland Masson, Philippe Quandalle, Stephane Requena (all three Division Informatique Scientifique et Mathematiques Appliquees), and Johannes Wendebourg (Division Geologie-Geochimie). [Further Details]

Algebraic Preconditioners for High-Contrast Media

This is joint work with Ivan Graham (Bath), Burak Aksoylu (Louisiana State) and Hector Klie (Austin) on rigorously justified algebraic preconditioners for high-contrast media using singular perturbation arguments, and initiated from a visit in Mary Wheeler's group at Austin. For first results see

B. Aksoylu, I.G. Graham, H. Klie and R. Scheichl, Towards a Rigorously Justified Algebraic Preconditioner for High-Contrast Diffusion Problems, Computing and Visualization in Science 11 (Special Issue Dedicated to Prof. W. Hackbusch on the Occasion of his 60th Birthday), pp. 319-331, 2008. [Preprint]

Further work and some mathematical details will be available soon in

R. Scheichl, A Rigorously Justified Algebraic Preconditioner for High-Contrast Diffusion Problems, in preparation.

Numerical Computation of Spectra for PDEs on Infinite Domains - Application in Bandgap Calculations for Photonic Crystal Fibres

This is work that initiated through discussion with David Bird (Physics, Bath). Together with my PhD student Richard Norton (ORS Award) we carried out a rigorous numerical analysis of planewave expansion (or spectral Galerkin) methods for elliptic eigenproblems in 1D and 2D as they are used in the physics community in bandgap computations for photonic crystal fibres. Three papers on this work are in preparation. For first results see [Extended Abstract for Waves '07].
I have recently started another collaboration on the subject with Marco Marletta (Cardiff) to exploit novel theoretical ideas to numerical compute localised modes of Schroedinger-type operators avoiding spectral pollution. A new PhD student, Chris Hart, has started to work on this project in October 2008, under our joint supervision (funded from/based in Bath).

Numerical Methods for Weather and Climate Models

This is joint work with Mike Cullen (The Met Office), Jon Thuburn (Exeter), and my PhD student Sean Buckeridge (GWR Grant). It concentrates on the development and analysis of fast parallel multigrid preconditioners for elliptic PDEs in spherical geometries arising in weather, climate, and data assimilation models.

Quasi-Monte Carlo and Multilevel Monte Carlo Techniques for Stochastic PDEs

These are two very recently started projects with Ivan Graham (Bath), Ian Sloan, Dirk Nuyens, Frances Kuo (all UNSW, Sydney) and Mike Giles (Oxford).

Other past and current projects include:

Preconditioning Techniques for Discontinuous Galerkin Methods

This is a recent project in collaboration with Peter Bastian, Markus Blatt, Christian Engwer (all Stuttgart) and with David Kay (Oxford).

Emergence in Random Microstructures

This is project in collaboration with Darryl Almond (Mech Eng, Bath) and Nick McCullen (Maths, Bath) from BICS Theme C, which I have worked on in three MSc summer projects Clare Judd (2005), Sean Buckeridge (2006), Tristan Brindle (2007).

Iterative Solution of Saddle Point Problems with Applications to Uncertainty Quantification in Groundwater Flow

This is a project I started during my PhD at the University of Bath under the joint supervision of Ivan Graham (University of Bath) and Andrew Cliffe (University of Nottingham). Part of the work has also been done in collaboration with Linda Stals (Old Dominion University, Norfolk, VA). [Further Details]

Permeation of Gases in Polymers: Parameter Identification and Nonlinear Regression Analysis

This is a project in collaboration with the Institut Francais du Petrole (IFP), Paris, France, in particular Zakia Benjelloun-Dabaghi (Applied Mathematics Department), Marie-Helene Klopffer and Bruno Flaconneche (both Applied Chemistry and Physical Chemistry Department). [Further Details]

Parallel Solution of the Multigroup Neutron Diffusion Equations

This is work I did for my master's thesis during a one-year undergraduate placement at Siemens, Munich, under the joint supervision of Ulrich Langer (University of Linz, Austria) and Meinhard Paffrath (Siemens ZT, Munich, Germany), and in collaboration with Herbert Finnemann (Siemens KWU, Erlangen, Germany). [Further Details]

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Last updated 03/12/2008