Bridging and Improving Theoretical and Computational Electric Impedance Tomography via Data Completion
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In computational PDE-based inverse problems, a finite amount of data is collected to infer unknown parameters in the PDE. In order to obtain accurate inferences, the collected data must be informative about the unknown parameters. How to decide which data is most informative and how to efficiently sample it, is the notoriously challenging task of optimal experimental design (OED). In this context, the best, and often infeasible, scenario is when the full input-to-output (ItO) map, i.e., an infinite amount of data, is available: This is the typical setting in many theoretical inverse problems, which is used to guarantee the unique parameter reconstruction. These two different settings have created a gap between computational and theoretical inverse problems. In this manuscript we aim to bridge this gap while circumventing the OED task. This is achieved by exploiting the structures of the ItO data from the underlying inverse problem, using the electrical impedance tomography (EIT) problem as an example. We leverage the rank-structure of the EIT model, and formulate the discretized ItO map, as an H-matrix. This suggests that one can recover the full ItO matrix, with high probability, from a subset of its entries sampled following the rank structure: The data in the diagonal blocks is informative thus fully sampled, while data in the off-diagonal blocks can be sub-sampled. This recovered ItO matrix is then utilized to represent the full ItO map, allowing us to connect with the problem in the theoretical setting where the unique reconstruction is guaranteed. This strategy achieves two goals: I) it bridges the gap between the settings for the numerical and theoretical inverse problems and II) it improves the quality of computational inverse solutions. We detail the theory for the EIT model, and provide numerical verification to both EIT and optical tomography problems
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