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Department of Mathematics

Scientific computing

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Scientific computing is the study of the techniques that underpin discipline-specific fields of computational science.

Our research projects

Members of the group have wide-ranging expertise in designing numerical algorithms and computational models for solving problems in a variety of physics-based applications, such as computational fluid dynamics, solid mechanics and computational finance, as well as for uncertainty quantification (UQ) and deep learning.

They also carry out fundamental research in numerical linear algebra that underpins computer simulation, develop techniques for parallelising algorithms on specialised computer architectures and develop open-source software.

Finite Element Method (FEM)

The finite element method (FEM) is one of the most general computational techniques for approximating solutions of partial differential equations (PDEs).

Team members have world-leading expertise in applying FEMs to multi-physics models arising in continuum mechanics and the life sciences. They work on all aspects, from fundamental analysis of approximation properties to the design and implementation of efficient adaptive solution algorithms and software.

They are actively engaged in a wide range of research projects related to finite element approximation and PDEs, including:

  • computational methods for complex and deforming domains
  • fast solvers and robust preconditioning
  • reliable a posteriori error estimation techniques
  • efficient adaptive approximation strategies
  • stochastic FEMs for PDEs with uncertain inputs
  • efficient solution of nonlinear PDE systems

FEM software packages currently being developed include:


Group members also have expertise in developing algorithms based on spectral methods for the Navier-Stokes equations and other fluid flow problems.

In computational finance, some group members use finite difference methods to approximate solutions of stochastic optimal control problems. They have expertise in developing efficient numerical schemes to solve the associated PDEs and calibrate models against real data to enable simulations of projected future scenarios.

Stochastic models and models with uncertain inputs arise in a variety of real-world applications. Team members are developing a wide range of specialised algorithms for performing simulations using such models, as well as for predication and UQ, including:

  • computational methods for constructing accurate surrogates for expensive forward models (such as PDEs)
  • Markov chain Monte Carlo and other sampling methods for characterising complex probability distributions arising in Bayesian inverse problems
  • algorithms for simulating solutions of stochastic models in Biology.
  • computational methods for Bayesian optimisation.

In modern applications of data science, an important challenge is to develop efficient computational algorithms for training highly parameterised deep learning models such as neural networks. Our group members are working on fundamental as well as practical strategies for parallelising algorithms to improve efficiency.

Further information

More information about our research outputs and research-related activities can be found by browsing the webpages of the staff listed. Potential PhD students may email academic staff directly to discuss possible projects.