Our research group is focused on the following areas:

### Data assimilation and inverse problems (Professor Sarah Dance, Dr Amos Lawless, Professor Nancy Nichols, Professor Roland Potthast)

We have a large programme within numerical analysis in data assimilation and inverse problems, so large that it has spawned a separate research group within the School and an interdisciplinary research centre (Data Assimilation Research Centre) of staff from the departments of Mathematics and Statistics and Meteorology, and staff from a 20-strong group of Met Office scientists based on the campus. One of our professors in Mathematics and Statistics – Roland Potthast – also leads data assimilation research at the German Met Office.

For further details of the work of this group, see the dedicated Data Assimilation and Inverse Problems research page.

### Analysis of boundary element methods (Professor Simon Chandler-Wilde)

Boundary integral equation methods involve reformulating a partial differential equation on a domain as an integral equation on the boundary of the domain. This reduces the dimension of the problem and can turn problems on unbounded domains into problems on bounded domains.

This approach leads to non-local integral operators, and the efficient solution of the resulting integral equations is a difficult task resulting in many mathematical and numerical challenges. These range from accurate computation of system matrix entries and efficient solution of the system of linear equations to error control and convergence theory.

Current work is focused on analysis and numerics for boundary element methods for domains with fractal boundaries, and on analysis questions for Galerkin boundary elements methods on general Lipschitz domains.

### Hybrid numerical-asymptotic boundary integral methods for high frequency scattering (Professor Simon Chandler-Wilde)

The computing time for standard numerical methods for wave scattering problems grows rapidly as the frequency of the wave increases (equivalently as the size of the wavelength, relative to the scattering obstacle, decreases). This renders the solution to many problems of practical interest impossible with current technology.

Our group is at the forefront of the development and analysis of boundary integral equation methods for scattering problems that have the property that the computational time does not grow significantly as the frequency of the incident wave increases. The key idea of our approach is to incorporate knowledge of the oscillatory behaviour of the solution on the boundary directly into our approximation space, and to do this requires new results on high frequency asymptotics.

Work at Reading in this area has been supported by EPSRC, the Leverhulme Trust, the European Union, the Swiss National Science Foundation, Arup Acoustics, BAE Systems, the BBC, the Institute for Cancer Research, the Met Office and Schlumberger.

### Adaptive moving mesh methods based on conservation (Professor Mike Baines)

For problems in which boundaries need to be tracked or resolution requirements vary with time, moving meshes are a natural choice.

However, there is normally no information in the equations themselves about optimal mesh movement strategies. Numerical approximation therefore consists of discretisation of the whole problem including the mesh. The consequent flexibility can be used to advantage, for example by exploiting combinations of empirical observation and conservation.

Current modelling collaborations are with CEH Wallingford on population dynamics and with the Met Office on moisture infiltration into soils.

### Numerical schemes for conservation laws (Professor Mike Baines, Dr Peter Sweby)

Conservation laws arise when some (physical) quantity is conserved, for example mass or momentum. Thus they are frequently used to model physical processes which involve movement of some medium, for example air, water or even road traffic.

A distinguishing feature of conservation laws is that they allow the formation of shocks, i.e. discontinuous solutions, such as sonic booms, bores in rivers or traffic jams.

Since, in general, conservation laws must be solved numerically, these discontinuous solutions present an additional challenge in numerical modelling as they can trigger instabilities in the more classical numerical schemes. This has generated the need for the design of adaptive (non-linear) schemes which avoid such failings, together with associated techniques for their implementation in complex situations.

### Computational neuroscience (Professor Roland Potthast)

We are working on the numerical modelling of problems arising in computational neurosciences as part of interdisciplinary work within the Centre for Integrative Neuroscience and Neurodynamics. We are interested in the modelling and the analysis of the spatiotemporal evolution of neural tissue activity, in particular in the direct and inverse problems arising in "neural field theory".

Several integro-differential models are available in this area and we address some of the open questions related to their validation, analysis and numerical treatment.

### Sparse linear systems (Amos Lawless, Nancy Nichols, Jennifer Scott)

The numerical solution of large sparse linear systems of equations is a key cornerstone of computational science and engineering, with applications arising in a wide and varied range of areas, both within the academic community and more widely within industry. It has been estimated that the solution of a linear systems of equations enters in at some stage in about 75 percent of all scientific problems.

Designing and developing numerical methods for solving such systems is a key part of our work, with both direct methods and iterative methods being of interest. We are particularly concerned with very large systems, such as those that arise in data assimilation. Our interest is in developing new algorithms that are accompanied by mathematical theory.

For iterative methods to achieve acceptable performance it is normally necessary to use a preconditioner. A preconditioner aims to transform the system into a nicer system that is easier to solve and from which the solution of the original problem can easily be recovered.

Deriving effective and robust preconditioners is highly problem dependent. We are concerned with developing preconditioners that are applicable either to a wide class of problems (this includes work on incomplete factorization preconditioners and on domain decomposition preconditioners) or to a special class of problems (including preconditioning systems that arise in data assimilation).

### Large-scale least squares problems (Amos Lawless, Nancy Nichols, Jennifer Scott)

Linear least squares problems occur in a wide variety of practical applications, both in their own right and as subproblems of nonlinear least squares problems. Solving large-scale least squares problems is typically much harder than solving systems of linear algebraic equations, in part because key issues such as ill-conditioning or dense structures within an otherwise sparse problem can vary significantly between different problem classes. In particular, only limited work has been done on developing effective preconditioners for large-scale least squares problems.

We have developed a class of limited memory incomplete factorization preconditioners and are exploring the use of domain decomposition-based preconditioners. We are particularly interested in sparse problems that contain a small number of dense rows and in least squares problems that are subject to linear constraints that must be tightly satisfied.

This work is in collaboration with the Computational Mathematics Group at the Science and Technology Facilities Rutherford Appleton Laboratory and with the Department of Numerical Mathematics in Charles University, Prague. In other work we have analysed the linear least squares problems that arise in data assimilation, identifying the components of the problem that contribute to their ill-conditioning.