Approximations and Surrogates for Computational Inverse Boundary Value Problems

Lauri Mustonen

Research output: ThesisDoctoral ThesisCollection of Articles

Abstract

Inverse boundary value problems are closely related to imaging techniques where measurements on the surface are used to estimate, or reconstruct, inner properties of the imaged object. In this thesis, improved reconstruction methods and new computational approaches are presented for elliptic and parabolic inverse boundary value problems. Two imaging applications that are addressed are electrical impedance tomography (EIT) and thermal tomography. The inverse problems that are considered are nonlinear and the reconstructions are sought by least squares minimization. Algorithms for such minimization often rely on iterative evaluation of the target function and its partial derivatives. In this thesis, we approximate these target functions, which themselves are solutions to partial differential equations, with polynomial surrogates that are simple to evaluate and differentiate. In the context of EIT, this method is used to estimate the shape of the object in addition to its electrical properties. The method is also shown to be feasible for thermal tomography, including the case of uncertain object shape. We also present a novel logarithmic linearization method for EIT. Transforming the voltage measurements in a certain logarithmic way reduces the nonlinearity in the relationship between the electrical properties and the measurements, allowing a reconstruction with fewer or only one minimization step. Furthermore, we propose a modification to the complete electrode model for EIT. The new model is shown to be compatible with experimental measurement data, while the increased regularity of the predicted electromagnetic potential improves convergence properties of numerical methods.
Translated title of the contributionApproksimaatioita ja surrogaatteja laskennallisille käänteisreuna-arvo-ongelmille
Original languageEnglish
QualificationDoctor's degree
Awarding Institution
  • Aalto University
Supervisors/Advisors
  • Hyvönen, Nuutti, Supervising Professor
  • Hyvönen, Nuutti, Thesis Advisor
Publisher
Print ISBNs978-952-60-7611-9
Electronic ISBNs978-952-60-7610-2
Publication statusPublished - 2017
MoE publication typeG5 Doctoral dissertation (article)

Keywords

  • inverse problems
  • uncertainty quantification
  • partial differential equations
  • nonlinear least squares
  • polynomial surrogate
  • electrical impedance tomography
  • complete electrode model
  • thermal tomography

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