James Rynn

Office 2.121 · Alan Turing Building · University of Manchester, M13 9PL · james.rynn@manchester.ac.uk

I have recently completed my PhD, working in Uncertainty Quantification in the Numerical Analysis group of the School of Mathematics at the University of Manchester (UK). My area of research is 'Efficient Inverse Uncertainty Quantification for PDEs with Uncertain Data'. I am the recipient of a Smith Institute CASE studentship award part funded by the National Physical Laboratory (Teddington, UK). My academic supervisors are Professor Catherine Powell and Dr Simon Cotter, my industrial supervisor is Louise Wright. My research interests include Bayesian Inverse Problems, Markov chain Monte Carlo methods, Finite Element Methods and Stochastic Galerkin Finite Element Methods.


Image credits: Nick Higham (Alan Turing Building - left) & Nuclear Engineering International Magazine (NPL Reception - right)

Research

Partial Differential Equations with Uncertain Data

Many real world phenomena may be modelled through the use of partial differential equations (PDEs). It is often the case however that inputs to the PDE such as coefficients, boundary conditions and even domains are unknown or, 'uncertain'. In such scenarios, one may choose to model such unknowns as random variables, allowing for forward propogation of uncertainty through the PDE model. An example of such a PDE could be the transient heat flow equation $$ \rho c_{\scriptsize \mbox{p}} \frac{\partial u}{\partial t}(\mathbf{x},t) = \nabla \cdot \left( \lambda(\mathbf{x},t) \nabla u(\mathbf{x},t)\right) + Q(\mathbf{x},t), \quad \quad \quad \mathbf{x} \in D, \quad t \in [0,T], $$ which models the flow of heat through a homogeneous isotropic material subject to a heat source \(Q\). We may wish to propogate uncertain inputs through such a model to quantify uncertainty in the solution, this is known as forward uncertainty quantification. Alternatively, given (often indirect and/or noisy) observations of the solution, we may wish to quantify the uncertainty on the inputs, this is known as inverse uncertainty quantification). I am particularly interested in the latter approach, tackling these problems using the Bayesian approach outlined below.

Bayesian Inverse Problems

Consider the inverse uncertainty quantification problem of recovering an unknown input given noisy, indirect measurements of the output. For example, we may wish to know the value of a PDE input such as the thermal conductivity of a material (\(\lambda\), in the equation above) but only have access to noisy observations of the temperature of the material at points in time at various spatial locations. Assuming the observations are subject to some form of noise, modelling this probabilistically allows us to define a posterior probability distribution for the unknown quantity given the data. This may be formulated in terms of our model and prior knowledge using Bayes' rule as $$ \mathbb{P}(\lambda|D) \propto \mathbb{P}(D|\lambda)\mathbb{P}(\lambda) \quad \quad \mbox{or} \quad \quad \pi(\lambda|D) \propto L(D|\lambda) \pi_{0}(\lambda). $$

Here our likelihood function, \(L(D|\lambda)\), is formulated using our forward model (for this example the PDE above) and our assumptions on the distribution of the noise. Our prior \(\pi_{0}(\lambda)\) encodes all our existing information about the unknown, usually from previous experiments or expert information. Using Markov chain Monte Carlo (MCMC) methods one is then able to sample from the posterior distribution allowing estimation of the posterior density itself, mean values or any other quantity of interest.


My research revolves around the use of MCMC to efficiently perform inverse uncertainy quantification for PDEs with uncertain data. I aim to find efficient ways of solving the forward problem and use these along with intelligent MCMC algorithms to reduce the cost of sampling from the posterior.

Research Groups

My research involves concepts studied in a variety of the University of Manchester's research groups. I am a member of the following research groups:

I am a student member of SIAM, a student member of the IMA, an e-student member of the RSS and a member of the Manchester SIAM-IMA Student Chapter.

Publications

PhD Thesis

Metrologia Paper

Certificates

Coursera

Conferences and Presentations

Conferences

2018

2017

2016

2015

Presentations

Posters

Teaching

Year 2018–19, Semester 2

Year 2017–18, Semester 1

Year 2016–17, Semester 2

Year 2016–17, Semester 1

Contact

@manchester.ac.uk

Telephone:

+44 (0)161 275 5882

Address:

Alan Turing Building School of Mathematics
Office 2.121
University of Manchester
Oxford Road
Manchester
M13 9PL

Online:

I have a University of Manchester profile available here and a Linkedin page here.

Curriculum Vitae

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Department Football

To pay please go to Michael's website from now on, thanks.