Physics Open (May 2021)
Assessing the credibility of the solutions of incomplete-data inverse problems
Abstract
This paper proposes an approach to measure the credibility of solutions of inverse problems with incomplete or missing data, encountered in some physical problems. In such problems, the same set of data can produce multiple solutions, depending on the constraints or assumptions used to compensate for the missing information. The actual “true” solution, against which the quality of a constrained solution can be measured, is not usually available. In this work, we propose to obtain a complete but coarse reference solution that reliably possesses attributes of the actual solution. This is done by solving an over-complete problem with the same set of data but with a coarser structure. A number of residual, composition and fidelity metrics are then used to quantitatively measure the quality of a solution obtained from incomplete data, against the coarse complete reference. The approach was tested with various degrees of incompleteness and different levels of uncertainty (noise) in data in the linear problem of image reconstruction in computed-tomography. The results show that the approach was successful in identifying credible solutions, as well as those that were not trustworthy.
