Well Posedness and Convergence Analysis of the Ensemble Kalman Inversion

The ensemble Kalman inversion is widely used in practice to estimate unknown parameters from noisy measurement data. Its low computational costs, straightforward implementation, and non-intrusive nature makes the method appealing in various areas of application. We present a complete analysis of the ensemble Kalman inversion with perturbed observations for a fixed ensemble size when applied to linear inverse problems. The well-posedness and convergence results are based on the continuous time scaling limits of the method. The resulting coupled system of stochastic differential equations allows to derive estimates on the long-time behaviour and provides insights into the convergence properties of the ensemble Kalman inversion. We view the method as a derivative free optimization method for the least-squares misfit functional, which opens up the perspective to use the method in various areas of applications such as imaging, groundwater flow problems, biological problems as well as in the context of the training of neural networks

Superiorization and Perturbation Resilience of Algorithms: A Continuously Updated Bibliography

This document presents a, (mostly) chronologically ordered, bibliography of scientific publications on the superiorization methodology and perturbation resilience of algorithms which is compiled and continuously updated by us at: http://math.haifa.ac.il/yair/bib-superiorization-censor.html. Since the beginings of this topic we try to trace the work that has been published about it since its inception. To the best of our knowledge this bibliography represents all available publications on this topic to date, and while the URL is continuously updated we will revise this document and bring it up to date on arXiv approximately once a year. Abstracts of the cited works, and some links and downloadable files of preprints or reprints are available on the above mentioned Internet page. If you know of a related scientific work in any form that should be included here kindly write to me on: [email protected] with full bibliographic details, a DOI if available, and a PDF copy of the work if possible. The Internet page was initiated on March 7, 2015, and has been last updated on March 12, 2020.Comment: Original report: June 13, 2015 contained 41 items. First revision: March 9, 2017 contained 64 items. Second revision: March 8, 2018 contained 76 items. Third revision: March 11, 2019 contains 90 items. Fourth revision: March 16, 2020 contains 112 item

Computational Image Formation

At the pinnacle of computational imaging is the co-optimization of camera and algorithm. This, however, is not the only form of computational imaging. In problems such as imaging through adverse weather, the bigger challenge is how to accurately simulate the forward degradation process so that we can synthesize data to train reconstruction models and/or integrating the forward model as part of the reconstruction algorithm. This article introduces the concept of computational image formation (CIF). Compared to the standard inverse problems where the goal is to recover the latent image $\mathbf{x}$ from the observation $\mathbf{y} = \mathcal{G}(\mathbf{x})$, CIF shifts the focus to designing an approximate mapping $\mathcal{H}_{\theta}$ such that $\mathcal{H}_{\theta} \approx \mathcal{G}$ while giving a better image reconstruction result. The word ``computational'' highlights the fact that the image formation is now replaced by a numerical simulator. While matching nature remains an important goal, CIF pays even greater attention on strategically choosing an $\mathcal{H}_{\theta}$ so that the reconstruction performance is maximized. The goal of this article is to conceptualize the idea of CIF by elaborating on its meaning and implications. The first part of the article is a discussion on the four attributes of a CIF simulator: accurate enough to mimic $\mathcal{G}$, fast enough to be integrated as part of the reconstruction, providing a well-posed inverse problem when plugged into the reconstruction, and differentiable in the backpropagation sense. The second part of the article is a detailed case study based on imaging through atmospheric turbulence. The third part of the article is a collection of other examples that fall into the category of CIF. Finally, thoughts about the future direction and recommendations to the community are shared