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Context.High-resolution magnetograms are crucial for studying solar flare dynamics because they enable the precise tracking of magnetic structures and rapid field changes. The Helioseismic and Magnetic Imager on board the Solar Dynamics Observatory (SDO/HMI) has been an essential provider of vector magnetograms. However, the spatial resolution of the HMI magnetograms is limited and hence is not able to capture the fine structures that are essential for understanding flare precursors. The Near InfraRed Imaging Spectropolarimeter on the 1.6 m Goode Solar Telescope (GST/NIRIS) at Big Bear Solar Observatory (BBSO) provides a better spatial resolution and is therefore more suitable to track the fine magnetic features and their connection to flare precursors. Aims.We propose DeepHMI, a machine-learning method for solar image super-resolution, to enhance the transverse and line-of-sight magnetograms of solar active regions (ARs) collected by SDO/HMI to better capture the fine-scale magnetic structures that are crucial for understanding solar flare dynamics. The enhanced HMI magnetograms can also be used to study spicules, sunspot light bridges and magnetic outbreaks, for which high-resolution data are essential. Methods.DeepHMI employs a conditional diffusion model that is trained using ground-truth images obtained by an inversion analysis of Stokes measurements collected by GST/NIRIS. Results.Our experiments show that DeepHMI performs better than the commonly used bicubic interpolation method in terms of four evaluation metrics. In addition, we demonstrate the ability of DeepHMI through a case study of the enhancement of SDO/HMI transverse and line-of-sight magnetograms of AR 12371 to GST/NIRIS data. 

Abstract Computational inverse problems utilize a finite number of measurements to infer a discrete approximation of the unknown parameter function. With motivation from the setting of PDE-based optimization, we study the unique reconstruction of discretized inverse problems by examining the positivity of the Hessian matrix. What is the reconstruction power of a fixed number of data observations? How many parameters can one reconstruct? Here we describe a probabilistic approach, and spell out the interplay of the observation size (r) and the number of parameters to be uniquely identified (m). The technical pillar here is the random sketching strategy, in which the matrix concentration inequality and sampling theory are largely employed. By analyzing a randomly subsampled Hessian matrix, we attain a well-conditioned reconstruction problem with high probability. Our main theory is validated in numerical experiments, using an elliptic inverse problem as an example.