Convergence analysis of a new robust iterative solution method

What is it about?

Robustness incorporates uncertainty in the model and data inputs, allowing the resulting system to provide valid, reasonable results even when errors in the data and model exist. This provides a level of resilience to perturbations in the solution. Classically such robust systems have been dense systems, and were thus limited in their size. In this work an sparse iterative solution method, that incorporates robustness is presented and convergence is proven. In the examples, we consider the problem of a large block of missing data in an image reconstruction problem. A continuous sequence of data representing a large number of angles in a CT problem is removed and the image is still reconstructed. Not only is this useful for missing data it also covers the case where measurements cannot be taken, for instance across the hips in proton imaging.

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Why is it important?

Iterative methods allow for solution of large sparse systems but have not formally included robustness. Robust methods are generally dense and thus don't handle large sparse systems. This paper presents a result that is the best of both worlds, it robustly handles large sparse systems. The examples show how to handle large blocks of missing data. Not only is this useful for missing data it also covers the case where measurements cannot be taken, for instance across the hips in proton imaging.

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