Efficient Estimators for Sequential and Resolution-Limited Inverse Problems

TL;DR

This paper introduces an efficient estimator for reconstructing signals from sequences of related inverse problems, especially when observational resolution is limited, with applications to large astronomical data collection systems like LSST.

Contribution

The paper develops a novel estimator tailored for sequential inverse problems with resolution constraints, supported by theoretical analysis and practical demonstrations.

Findings

Effective reconstruction in simulated inverse problems

Theoretical guarantees for estimator performance

Applicable to large-scale astronomical data collection

Abstract

A common problem in the sciences is that a signal of interest is observed only indirectly, through smooth functionals of the signal whose values are then obscured by noise. In such inverse problems, the functionals dampen or entirely eliminate some of the signal's interesting features. This makes it difficult or even impossible to fully reconstruct the signal, even without noise. In this paper, we develop methods for handling sequences of related inverse problems, with the problems varying either systematically or randomly over time. Such sequences often arise with automated data collection systems, like the data pipelines of large astronomical instruments such as the Large Synoptic Survey Telescope (LSST). The LSST will observe each patch of the sky many times over its lifetime under varying conditions. A possible additional complication in these problems is that the observational resolution is limited by the instrument, so that even with many repeated observations, only an approximation of the underlying signal can be reconstructed. We propose an efficient estimator for reconstructing a signal of interest given a sequence of related, resolution-limited inverse problems. We demonstrate our method's effectiveness in some representative examples and provide theoretical support for its adoption.