Numerical Analysis and Scientific Computing PhD projects

This page provides a (partial) list of specific (and not so specific) PhD projects currently offered around the Numerical Analysis and Scientific Computing topic.

Identifying an interesting, worthwhile and do-able PhD project is not a trivial task since it depends crucially on the interests and abilities of the student (and the supervisor!) The list below therefore contains a mixture of very specific projects and fairly general descriptions of research interests across the School. In either case you should feel free to contact the potential supervisors to find out more if you're interested.

You should also explore the School's research groups' pages, and have a look around the homepages of individual members of staff to find out more about their research interests. You may also contact us for general enquiries.

Title

Scheduling and Parallel Computing

Group Numerical Analysis and Scientific Computing
Supervisor
Description

Consider a large computational task, for instance, solving a large system of linear equations. This task can be split into many smaller jobs which are then scheduled and queued at a large number of different heterogeneous computing resources and are executed in parallel. The processing requirements of the different resources (for instance CPUs and GPUs) are different, and they may well have different communication costs. In this project we will be interested in understanding the stochastic effects before designing (and implementing) novel distributed scheduling algorithms.

We seek a student with skills in probability, optimization and mathematical modeling. A student with good programming skills (particularly in C/C++) would be preferred.

Title

Numerical Analysis and Computational Methods for Solving PDEs with Uncertainty

Group Numerical Analysis and Scientific Computing
Supervisor
Description

Uncertainty Quantification (UQ) is the science of accounting for uncertainty in mathematical models. Research in this area has undergone rapid growth in the last few years and is currently considered a 'hot topic'. This growth has been driven by the need for scientists in today's world to provide decision makers with ever more accurate and reliable predictions that are based on results obtained from mathematical models.

Many physical processes such as fluid flows are governed by partial differential equations (PDEs). In practical applications in the real world, it is unlikely that all the inputs (boundary conditions, geometry, coefficients) for the chosen PDE model will be known. One possibility is to model the quantities that we don't know as random variables. Solving these problems is not always hard in theory but solving them efficiently in practice is a massive challenge.

I am interested in working with students who want to develop numerical analysis and numerical methods for solving partial differential equations with uncertain inputs. I welcome any enquiries to work in this area. Specific projects could be theoretical or computational, according to the strengths of the student.

Projects on this topic would suit students who have taken undergraduate courses in numerical analysis and applied mathematics who have a keen interest in computational mathematics and developing practical algorithms. Some prior programming experience is essential.

Background reference:

An Introduction to Computational Stochastic PDEs (Cambridge Texts in Applied Mathematics), G. J. Lord, C.E. Powell and T. Shardlow, 2014.

Title

Integral Geometry of Cones

Group Numerical Analysis and Scientific Computing
Supervisor
Description

Science and technology depends increasingly on the efficient acquisition, storage and processing of vast amounts of data.
While the seemingly endless availability of data is a blessing from a statistical point of view, it poses enormous challenges from a computational perspective.

An exciting recent development has been the emergence of ideas for efficient information acquisition and identification that take advantage of simple underlying structure of the problems considered . This new genre of ideas, encompassing the burgeoning field of compressed sensing, has experienced tremendous growth in recent years.

Numerical optimization plays an important role in these developments, and its effectiveness depends crucially on deep geometric properties of the underlying problems. I am interested in studying such problems in convex geometry and geometric probability that help explain the reach and limitations of convex optimization. Besides background in numerical analysis, this project would also benefit from knowledge in computational complexity, differential geometry, and probability.

Literature:
{\bf Schneider, Weil}, Stochastic and Integral Geometry.
{\bf Foucart, Rauhut}, A Mathematical Introduction to Compressive Sensing.

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