Characterizing and reducing the influence of model errors in inverse problems

Research output: Book/ReportPh.D. thesis

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Abstract

In natural sciences, one uses mathematics to model the interaction between things in the real world. However, no model is fully correct, as it is always an approximation of the actual physics. In most cases this model error is negligible by design and therefore does not negatively influence the application at hand. However, in other cases it may not be feasible to construct a model with negligible model error. One example of this is inverse problems, where the goal is to infer something about a system given an indirect measurement of said system. Inverse problems are well-known to utilize simplified mathematical models to model complex large-scale physical systems for this reason. In this thesis, we study methods for characterizing and reducing model errors when solving inverse problems. The driving application for this work is the inverse problem of Computed Tomography (CT), where the goal is to reconstruct the interior
structure of an object from measurements of X-ray attenuation from different view angles around the object. The model error we consider is caused by uncertainty in the actual view angles for which the data is acquired. We view this application as a general linear inverse problem with the addition of uncertain model parameters and review existing research for this type of problem. The main contribution of the thesis is the development of a new framework in the
form of models and algorithms for handling CT with uncertain view angles. The work can be condensed into two major components: 1) a new method for approximate marginalization of view angle uncertainty and 2) a new method for estimation of the view angles – including uncertainty quantification of the estimate. Ultimately both components are combined to achieve estimation of the view angles as well as marginalization of any remaining uncertainty – which in our simulated experiments completely alleviate the issue with uncertainty in the view angles. Both components are also general enough that they can be applied to other similar inverse problems. A huge part of making models and algorithms relevant to practical inverse problems is computational efficiency. In this thesis, computational efficiency has been considered throughout and we demonstrate our framework on both 2D and 3D CT problems with more than 106 unknowns and 106 data points.
Original languageEnglish
PublisherTechnical University of Denmark
Number of pages146
Publication statusPublished - 2021

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