Supervisory Team: Dr Silvia Gazzola and Dr Tony Shardlow (Department of Mathematical Sciences, University of Bath) Edmund Fordham (Schlumberger Gould Research)
Nuclear Magnetic Resonance is used to infer properties of porous media, such as rocks, through which oil can be extracted. This research project aims to surpass the current standard numerical methodology by providing efficient estimates together with uncertainty estimates. This will be achieved through applying new tools from numerical linear algebra and optimization, and through reformulating the problem in a Bayesian framework.
Current numerical methods for NMR rely on the solution of a Tikhonov-regularised problem, sometimes subject to additional constraints such as non-negativity. This overcomes the ill-posedness inherent in the model. In addition, when considering NMR in a two-dimensional setting, efficient solvers are needed to cope with large-scale computations. Krylov methods are highly effective tools in numerical linear algebra: the well-known conjugate gradient and GMRES algorithms are examples of Krylov methods. We will consider ways of enforcing upper and lower bounds into the solution computed by a Krylov method, as well as stabilising the behaviour of the solution. We believe that Krylov methods, used in such a way, will provide highly efficient solution methods suitable for on-site and on-line estimations. However, Krylov techniques do not as yet provide a way of quantifying uncertainty. To do this, we will look at ways of interpreting the Krylov regularisation in a Bayesian framework.
The Bayesian framework is a natural extension to the standard formulation of the problem as a constrained optimization problem. It encodes some of the structure of the problem, known independently of any data, in a prior probability. Application of Bayes’ theorem leads to posterior probabilities for the unknown function from which uncertainty estimates emerge naturally. Full calculation of the so-called “evidence” statistic offers rational comparison between competing theoretical models for the underlying physics.
The project is in collaboration with leading oilfield-service company Schlumberger, and Edmund Fordham, who is an expert on NMR in the oil industry at Schlumberger, will be a member of the supervisory team; regular visits to Cambridge offices of Schlumberger will be planned.
The project requires a student with a good undergraduate degree in a mathematics- or physics-based discipline, with experience of numerical analysis and scientific computing. The student will be expected to learn more about linear algebra, Bayesian statistics, and large-scale numerical optimization during their PhD studies. T he successful student will be given privileged access to the training activities of the SAMBa CDT. More information on the SAMBa CDT may be found here: go.bath.ac.uk/samba.
Please feel free to address any questions to Silvia Gazzola or Tony Shardlow.
Formal applications should be submitted via the University of Bath’s online application form https://www.bath.ac.uk/samis/urd/sits.urd/run/siw_ipp_lgn.login?process=siw_ipp_app&code1=RDUMA-FP03&code2=0012
More information about applying for a PhD at Bath may be found here: http://www.bath.ac.uk/guides/apply-for-a-doctorate/
Anticipated start date: 1 October 2018
UK and EU students applying for this project may be considered for a studentship funded by the University of Bath and Schlumberger Ltd. The studentship will cover Home/EU tuition fees, a training support fee of £1,000 per annum and a tax-free maintenance allowance at the RCUK Doctoral Stipend rate (£14,777 in 2018-19) for a period of 3.5 years.
Note: ONLY UK and EU applicants are eligible for this studentship; unfortunately, applicants who are classed as Overseas for fee paying purposes are NOT eligible for funding.
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