Approximation errors in nonstationary inverse problems

Often, in nonstationary inverse problems, computing estimates with accurate high-dimensional models is not practically feasible. An approach called the Bayesian approximation error (BAE) has been shown to be able to handle highly approximate models. In the BAE approach, the approximation errors (AEs) of the approximate models are modelled through sampling, based on the state evolution model. The approach has previously been implemented for filtering problems. In this thesis, we extend the BAE approach to smoothing problems. We test the applicability of the BAE approach in smoothing problems with numerical simulations. We consider the imaging of convection-diffusion type phenomena. Applications of this, in the industrial setting, include the pipeline imaging problem. We consider electrical impedance tomography (EIT) as the specific imaging modality. In this thesis, we consider implementing the BAE approach for smoothing problems in which AEs arise due to sparse discretisations, partially unknown boundary conditions, and model linearisations. The computational complexity can also be reduced by truncating the domain in which the estimates are computed. In the context of stationary EIT imaging problems, the forward operator is modelled over the truncated domain with the domain truncation model. In the model, the boundary condition over the truncation boundary models the current flow across the boundary with the stochastic Dirichlet-to-Neumann (DtN) operator. The Karhunen–Loéve (KL) theorem has been adapted to compute a low-dimensional representation of the DtN operator. In this thesis, we extend domain truncation problems to the nonstationary setting. Furthermore, we consider the BAE approach to approximately marginalise over AEs arising from domain truncation and the other aforementioned AEs. The numerical results suggest that the computational complexity can be effectively reduced by domain truncation and the KL theorem in nonstationary inverse problems. They also suggest that AEs due to all the aforementioned model approximations can be effectively and simultaneously marginalised over with the BAE approach in nonstationary inverse problems. The filter and smoother estimates computed with the BAE approach are shown to be more accurate and the corresponding error estimates are shown to be more realistic than those computed without the approach.

Name thesis:degree_name

PhD

Level thesis:degree_level

Doctoral

Discipline thesis:degree_discipline

Mathematics

Grantor dc:publisher

ResearchSpace@Auckland

Year dc:date.issued

2019

Author dc:creator

• Cheon, Eun Sig

Advisors dc:contributor.advisor

• Kaipio, J
• Barry, A
• Evans, T

Statement dc:rights

• Items in ResearchSpace are protected by copyright, with all rights reserved, unless otherwise indicated. Previously published items are made available in accordance with the copyright policy of the publisher.

Licence dc:rights.uri

• https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm

Handle dc:identifier.uri

https://hdl.handle.net/2292/49365

OAI identifier oai:identifier

oai:researchspace.auckland.ac.nz:2292/49365