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

Adaptive Numerical Algorithms for the Forward Propagation of Uncertainty in Time-Dependent CFD.

Group Numerical Analysis and Scientific Computing
Supervisor
Description

Fully-funded iCASE PhD project supported by IBM Research UK

Uncertainty quantification (UQ) is a rapidly-evolving field, incorporating several traditional mathematical disciplines. This project will develop new adaptive numerical algorithms for the forward propagation of uncertainty in large-scale time-dependent CFD (computational fluid dynamics) simulations and is a collaboration between the School of Mathematics at the University of Manchester and IBM Research UK. The project will be jointly supervised by Dr. Catherine Powell and Professor David Silvester from the School of Mathematics, and Dr. Malgorzata Zimon from IBM Research UK. The PhD student will be based in the School of Mathematics but will also have the opportunity to spend a minimum of 3 months working alongside researchers at IBM Research UK's premises in Daresbury.

Project Outline: In real-world applications, when using mathematical models to simulate real-world processes (such as fluid flows) we frequently encounter situations where we are uncertain about one or more of the inputs (viscosity, material parameters, initial conditions, geometry etc). In forward UQ, the main aim is to assess the impact of uncertainty in the model inputs on quantities of interest associated with the model's outputs. For this, we require computationally efficient numerical methods that can take in a probability distribution for the model's inputs and deliver accurate approximations of statistical quantities of interest related to the model's outputs. For time-dependent problems, and especially those with non-smooth solutions, the approximation space often needs to be adapted in time to maintain accuracy. How to design adaptive numerical algorithms with guaranteed error control is highly challenging.

Requirements: Candidates with a strong background in applied mathematics and numerical analysis with a passion for solving real-world problems efficiently on computers are encouraged to apply. Some prior experience in scientific computing (Python, MATLAB, C or Fortran etc) is desirable but not essential. Applicants should have (or be on track to to be awarded) either (i) a first class honours MMath degree or (ii) a first class honours BSc degree in Mathematics and a one-year MSc degree in a relevant mathematical discipline. Ideally, applicants should be available to start in September 2019 or shortly after.

How to apply: Informal email queries should be directed to Dr. Catherine Powell and/or Professor David Silvester in the first instance. Formal applications can then be submitted online. As well as transcripts and references, applicants should supply a cover letter describing their academic background and motivation for the project, as well as a complete CV (two pages maximum). These will be considered upon receipt and the PhD position will remain open until it is filled.

Funding: For eligible UK applicants, funding covers all tuition fees and annual maintenance payments at the standard EPSRC rate (£15,009 for the academic year 2019/20), plus a CASE top-up (amount TBC). For eligible EU applicants, funding is only available to cover tuition fees.

Title

Fast and Reliable Algorithms for High Peformance Numerical Linear Algebra

Group Numerical Analysis and Scientific Computing
Supervisor
Description

The project focuses on developing a new generation of numerical linear algebra algorithms that exploit current and future computers. The algorithms need to be fast and to be accompanied by rigorous error analysis to guarantee their reliability, even for the largest and most difficult problems. The target problems will be drawn from linear equations, linear least squares problems, eigenvalue problems, the singular value decomposition, and matrix function evaluation. These are the innermost kernels in many scientific and engineering applications---in particular, in data science and in machine learning---so it is essential that they are fast, accurate, and reliable.

A key aspect of this work is the exploitation of variable precision arithmetic. Low precision arithmetic is now available in hardware and is increasingly being used in machine learning and scientific computing more generally because of its speed, but its limited precision and narrow range require careful treatment. High precision (quadruple precision and above) is available in software and may be used in small amounts to speed up or stabilize an algorithm.

A strong background in numerical linear algebra and programming skills in MATLAB or a high level language are essential.

This project provides the opportunity to be part of a large and vibrant numerical linear algebra group that has several strongly committed industrial partners (see https://nla-group.org/).  

Funding Notes

Open to all. Funding is available and would provide fees and maintenance at RCUK level for home/EU students, or a fees-only bursary for overseas students.

Title

Fluid-structure interaction effects in the sedimentation of thin elastic sheets

Group Numerical Analysis and Scientific Computing
Supervisors
Description

There is much current interest in so-called two-dimensional materials 
because of their unusual and attractive mechanical and electrical properties. 
Much of their processing is performed in a fluid environment, e.g., during
the size selection of dispersed graphene flakes by centrifugation, or
their deposition by ink-jet printing. The flakes' large aspect ratio 
implies that despite their impressive in-plane stiffness they have a
very small bending stiffness and are therefore easily deformed by the 
traction that the surrounding fluid exerts on them. The resulting
strong fluid-structure interaction affects not only the dynamics of 
individual flakes but also their collective behaviour.

The aim of this project is to perform a systematic study of 
the behaviour of thin elastic sheets in a viscous fluid. 
Specifically, we wish to establish how the flow-induced 
deformation affects the sedimentation of such sheets, paying
particular attention to

-- the effect of the sheets' aspect ratio; long narrow sheets are
likely to behave in a manner similar to elastic rods: at which point
does their finite aspect ratio become significant?

-- the effect of wrinkling instabilities and the development of
symmetry-breaking frustrated patterns: how do they arise in sheets of 
canonical shapes (circular, rectangular, polygonal,...) and how do
they affect the sheets' sedimentation?

The focus of this specific project is on computational/semi-analytical
approaches and would suit a student with a good background in Applied
Mathematics (especially fluid and solid mechanics) and Scientific
Computing. There is an opportunity for hands-on involvement
in an associated experimental study in the School of Physics
and Astronomy.


MAIN SUPERVISOR: Professor Matthias Heil (School of Mathematics)

CO-SUPERVISORS: Professor Anne Juel (School of Physics and Astronomy)
Dr Draga Pihler-Puzovic (School of Physics and
Astronomy)

START DATE: September 2019 (or as soon as possible thereafter)

OTHER ASSOCIATED PROJECT AREAS: Physics

FUNDING: Funding is available and would provide fees and maintenance
at RCUK level for home/EU students, or a fees-only bursary
for overseas students.

DEADLINE: Applications are accepted at any time until the position
is filled.

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.

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