Many areas of science and engineering rely on the fast solution of very large systems of equations with millions or even billions of unknowns. For example, in numerical weather- and climate-prediction, the equations of fluid dynamics are discretised and solved on a computational grid with hundreds of millions of grid points. They have to be solved in less than an hour—tomorrow's weather forecast would be useless if it took a week to produce it!
Computational problems of this size can only be solved with the fastest available numerical algorithms and on massively parallel supercomputers with a hundred thousand or more compute cores.
In collaboration with the UK Met Office, we develop fast and massively parallel geometric multigrid solvers for partial differential equations (PDEs) in numerical weather prediction. We design new multilevel methods for solving stochastic partial differential equations (SDEs) in atmospheric-dispersion modelling. We have a strong interest in the implementation of these algorithms on novel chip architectures such as GPUs, which require the application of state-of-the-art software development techniques to achieve performance-portability.
- Mueller, Eike H. and Scheichl, Robert Massively parallel solvers for elliptic PDEs in Numerical Weather- and Climate Prediction. Quart. J Royal Met. Soc. Vol.140 (2014). link
Moving mesh methods
Numerical methods for PDEs usually require the construction of a mesh on which the PDE is then discretised. However, if a fixed mesh is used during the computation then this can lead to poor resolution of the evolving features of the PDE and also the loss of important qualitative formation such as conservation and balance laws. This is a major problem in many areas of computation of the inherently multi-scale problems that arise in nature, a particular example being meteorology. Budd is actively working in the construction of moving meshes in 3D which progressively move mesh points to resolve emergent features. These have already proved very effective for a wide range of problems and are now used as part of a data assimilation calculation in the UK Met Office Operational Code. Much new work needs to be done to extend these methods to work on the whole sphere and to study the complex PDEs encountered in nature. Moving mesh methods are a lovely mix of differential geometry, dynamical systems theory and numerical analysis with major practical applications.
- Budd, C. and Huang, W. and Russel, R. Adaptivity with moving grids. Acta Numerica Vol.18 (2009). link
Data assimilation problems arise for example in weather prediction and hydrology where observations of variables are incorporated into a mathematical model. The goal is to improve the accuracy of the model and make optimal predications of the future state of the system. Current data assimilation algorithms such as variational data assimilation and Kalman filters lead to large-scale problems in numerical linear algebra that need to be solved efficiently. Data assimilation belongs to the class of inverse problems, which are typically ill-conditioned and require regularisation methods such as Tikhonov regularisation to stabilise the problem. We collaborate with the UK Met Office on data assimilations in weather prediction.
Numerical linear algebra
Eigenvalue problems are central to a number of applications in science and industry, such as structural dynamics, quantum mechanics, material science and stability analysis. Often the problems are very large and only a few eigenvalues close to a target value are required. Many methods for solving eigenvalue problems iteratively are available, such as the inverse iteration, the Jacobi-Davidson method, and Krylov-methods such as the Lanczos and Arnoldi method. Linear and nonlinear eigenvalue problems that depend on a scalar parameter arise, for example in aerodynamics, electrical power systems and matrix perturbation theory, and are important in determining bifurcations numerically. It is an important research problem to find critical parameter values for parameter dependent eigenproblems such that, for example, eigenvalues coalesce (and become defective, that is, form a Jordan block), eigenvalues cross the imaginary axis (that is, stable matrices become unstable) or move to another critical region in the complex plane.
Stochastic differential equations
Uncertainty quantification is a fundamental problem in all types of mathematical modelling. Due to the rapid increase in computer power, it is now possible to incorporate uncertainty into many types of differential equations and compute accurate numerical approximations to sample paths or distributions. The group in Bath has interests in stochastic differential equations as an ordinary, delay, or reaction diffusion equation with Brownian forcing, where stochastic calculus is necessary to understand the model. A different type of stochastic differential equation are problems with random data, such as an elliptic PDEs with random porosity coefficient modelling flow through a porous media. The PDE can be understood in the conventional sense for each sample of the coefficients, while advanced techniques from probability yield accurate approximation techniques.
- Lord, G. and Powell, C. and Shardlow, T. An introduction to Computational Stochastic PDEs. CUP (2014). link
Inverse problems are among the most important computational problems in physics, biology, medicine, engineering, and finance. An example of an inverse problem is the computation of an image from indirect measurements in positron emission tomography; see the image above. While the measured data is not very informative to a medical doctor, the reconstructed image may reveal the location and size of a tumor. Mathematically, inverse problems are not well-posed and they require careful formulation. Here in Bath our research spans various aspects of inverse problems, ranging from mathematical modelling for regularization, over optimization to solve inverse problems, to applications in imaging and machine learning.
Our understanding of acoustic and electromagnetic wave scattering underpins technologies such as radar, sonar, telecommunications, ultrasound, and seismic imaging. Many of these applications require the accurate computation of high-frequency waves. However, standard numerical methods become increasingly expensive as the frequency increases, putting very high-frequency problems out of range of standard methods. Recent research in Bath has looked at ways to combat this, including the design of 'hybrid numerical-asymptotic methods', and novel preconditioners for use in conjunction with domain decomposition methods. This research currently involves collaboration with Schlumberger, the UK Met Office, BAE Systems, and the Institute for Cancer Research.
- Chandler-Wilde, Simon N. and Graham, Ivan G. and Langdon, Stephen and Spence, Euan A. Numerical-asymptotic boundary integral methods in high-frequency acoustic scattering. Acta Numer. Vol.21 (2012). link
The image shows snapshots of the evolution of a cat shape under a Hamiltonian flow map. The map is symplectic and while the shape of the cat changes its area is preserved. To explore the delicate dynamics of such systems, it is necessary that such geometric features are preserved by the numerical approximation in order to respect the underlying physics. The group in Bath looks at developing general linear methods for Hamiltonian differential equations and adaptive numerical methods for capturing shocks and singularities in PDEs with conservation laws.
- Budd, C.J. and Piggott, M.D. Geometric Integration and its Applications. Handbook of Numerical Analysis Vol.11 (2003). link
- Parkinson, Matt Uncertainty Quantification for the Neutron Transport Equation. (2018)
- Katsiolides, Grigoris Multilevel Monte Carlo Methods in Atmospheric Dispersion Modelling. (2018)
- Blake, J. Domain Decomposition Methods for Nuclear Reactor Modelling with Diffusion Acceleration. (2016)
- Cook, S. Adaptive Mesh Methods for Numerical Weather Prediction. (2016)
- Shanks, D. Robust solvers for large indefinite systems in seismic inversion. (2015)
- Jenkins, S. Numerical Model Error in Data Assimilation. (2015)
- Mora, K. Non-smooth dynamical systems and applications. (2014)
- Teckentrup, A. Multilevel Monte Carlo methods and uncertainty quantification. (2013)
- Browne, P. Topology optimization of linear elastic structures. (2013)
- Kim, T. Asymptotic and Numerical Methods for High-Frequency Scattering Problems. (2012)
- Scheben, F. Iterative Methods for Criticality Computations in Neutron Transport Theory. (2011)
- Millward, R. R. A New Adaptive Multiscale Finite Element Method with Applications to High Contrast Interface Problems. (2011)
- Buckeridge, S. D. Numerical Solution of Weather and Climate Systems. (2010)
- Walsh, E. J. Moving Mesh Methods for Problems in Meteorology. (2010)
- Akinola, R. O. Numerical Solution of Linear and Nonlinear Eigenvalue Problems. (2010)
- Hewitt, L. L. General Linear Methods for the Solution of ODEs. (2009)
- Pring, S. Discontinuous maps with applications to impacting systems. (2009)
- Green, S. Statics and dynamics of mechanical lattices. (2009)
- Giani, S. Convergence of Adaptive Finite Element Methods for Elliptic Eigenvalue Problmes with Applications to Photonic Crystals. (2008)
- Akinola, R. Numerical solution of linear and nonlinear eigenvalue problems. (2008)
- Norton, R. Numerical Computation of Band Gaps in Photonic Crystal Fibres. (2008)
- Stoyanov, Z. The symmetric eigenvalue problem: stochastic perturbation theory and some network applications. (2008)