## Dr Oliver DunbarResearch Associate |

**Submitted Publications**

Binary recovery via phase field regularization for first traveltime tomography (with C M Elliott) [Preprint]

Phase field modelling of surfactants in multi-phase flow (with K F Lam, and B Stinner) [Preprint]

**Recent Contributed Talks:**

EQUIP meeting, 20th August 2018, University of Warwick.

SIAM-UKIE NSCC, 19th June 2018, University of Bath.

MASDOC Retreat, 28th April 2018.

British Applied Mathematics Colloquium, 29th March 2018, University of St Andrews.

EQUIP meeting, 24th November 2017, University of Warwick.

**Personal Webpage and Previous Research:**

Please be directed to my webpage for information regarding my previous research as a MASDOC PhD student.

My PhD thesis is can be found on WRAP.

**Career Details:**

Please view my LinkedIn profile.

**Current Research: (Collaboration with Prof. Charles Elliott and Prof. Andrew Stuart)**

Since October 2017, I have had the position of research associate under the EQUIP grant.

My current interests are in the deterministic and probabilistic solution to PDE based inverse problems. In particular, I work on finding solutions of the inverse Eikonal equation, and the application of this to first traveltime tomography problems in geophysics.

**Deterministic framework:** I am using phase field based techniques to develop and implement mathematically rigorous reconstructive techniques for these tomographic problems. The phase fields let the reconstructed image (with value corresponding to subsurface 'slowness') to favour a representation with large regions taking constant values seperated by thin interfacial layers where the value smoothly changes between the constants (a notion known as a "sparse gradient" solution). This is a desirable feature where one for example needs to know accurately the whereabouts of the boundaries of different (e.g rock/oil/gas) regions. The sparseness of the gradients lead to difficulties in computational schemes, but the phase field techniques help overcome these difficulties. I have produced a deterministic solver for the problem in mixed C++/ MATLAB code and carefully set out the problem in a paper with Charles Elliott. Below on the left: the true slowness; centre: sources (x) receivers (.) and on the right: recovered phase field slowness function.

**Bayesian phase field framework:** Recently I have developed some robust and efficient hierarchical MCMC methods for solving this problem within a Bayesian framework. The setting mimics the formulation in the deterministic case, and the statistical techniques are more suited to inverse problems with noisy measurements of the data. The MCMC is of infinity-MALA type and based on gradients of the forward problem. It is dimension independent, and sample choices are guided by the underlying wave equation. The hierachical nature allows for hyperparameters in the model to be explored, to exploit natural unknowns in the problem such as a lengthscale of objects being recovered. The methods provide not only a 'most likely' solution to for the inverse problem, but also they give levels of uncertainty in this solution, though this comes at a computational cost. Below on the left: True slowness with central source (x) and receivers (.), centre: mean of MCMC at 10^{5} iterations, right: the calculated of prior (blue) and posterior (orange) interfaces and truth (dotted line).

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**Bayesian level set framework:** I have additionally developed an implementation for the (continuous) Bayesian level set description of the problem which, although does not provide maximum a posteriori estimaters in limit as in the phase field setting, does not concentrate around its mean and so provides more valuable uncertainty data. Futhermore it is not based directly on perimeter regularisation and so one may have more flexibility in recovery.

Below: Simulation endstate at 10^{5} iterations. Including (i) true slowness (with source receiver configurations as above), (ii) the uncertainty of the mean field (iii) the mean field, (iv) the trace of the lengthscale hyperparameter over the MCMC run. Below this on the left: the prior (blue) and posterior (orange) calculated interfacial lengths with truth (dotted line) and on the right: the prior (blue) and posterior (orange) of the lengthscale hyperparameter.

I am also currently investigating whether one may use this uncertainty quantification provided by the Bayesian framework to optimize over the locations of wave sources and receivers.