Ensemble based methods for geometric inverse problems
Since the development of the ensemble Kalman filter, it has seen a wide application to many scientific fields ranging from signal processing to weather forecasting and reservoir simulation. One field which has recently seen a keen interest towards filtering techniques is that of inverse problems. En...
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University of Warwick
2018
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| Online Access: | https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.767113 |
| Summary: | Since the development of the ensemble Kalman filter, it has seen a wide application to many scientific fields ranging from signal processing to weather forecasting and reservoir simulation. One field which has recently seen a keen interest towards filtering techniques is that of inverse problems. Ensemble-based methods are a popular choice of filtering techniques as they provide a computational advantage over traditional methods whilst retaining a good level of accuracy. This thesis is concerned with developing analysis and numerics of ensemble Kalman inversion (EKI) in the context of Bayesian inverse problems. In particular we are interested in quantifying the uncertainty that can arise for problems where our unknown is defined through geometric features. In the first part of this work we are interested in developing hierarchical approaches for EKI. This motivation is taken from hierarchical computational statistics for Gaussian processes where we are interested in a number of further unknowns such as hyperparameters that define the underlying unknown for the model problem. We present numerics of these hierarchical approaches whilst understanding its long term effect through continuous-time limits. The second part of this work is aimed at improving the computational burden of the forward solver within inverse problems. This improved forward solver is based on the reduced basis method which was designed for parameterized partial differential equations. The final part of the thesis concludes with an application of EKI where we adopt a Bayesian approach of the inverse eikonal equation. Our motivation is to extend the current work to Hamilton-Jacobi equations, where there exists a rich mathematical theory. A key understanding of how to tackle the uncertainty for this equation is addressed. |
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