Inverse problems and numerical approximation for high-energy solar imaging

The Spectrometer/Telescope for Imaging X-rays (STIX) is the instrument onboard Solar Orbiter mission designed to observe solar flares and, in particular, to collect hard X-ray radiation emitted during these phenomena, with the ultimate goal of understanding the physics behind them. STIX adopts an indirect imaging technique: it is based on a double layer of grids to modulate the incoming radiation. The photon counts measured by its detectors are related to the so-called visibilities, i.e., a finite set of Fourier frequencies of the emitted X-ray radiation flux, that is the quantity of interest for image reconstruction. Due to the limited number of visibilities available (only 30 in the case of STIX), the reconstruction of X-ray images is an ill-posed inverse problem which must be treated properly in order to obtain reasonable reconstructions. In this thesis, after a detailed explanation of the design and hardware of the STIX instrument, the mathematical formulation of the data formation process is derived and studied. This allows explaining how the recorded data are mathematically related to a particular sampling of the Fourier transform of the radiation flux emitted during solar flares. Two different image reconstruction techniques are developed and investigated: a parametric method based on global optimization strategies, and a multi-scale formulation of CLEAN, a well-known algorithm, widely used in both solar physics and radio astronomy. Both methods are validated on STIX data as well as on data collected by RHESSI, a NASA satellite and the predecessor of STIX. In the last part of the thesis, we propose two methods to obtain regularization in direction of contiguous energy levels, to correlate the related visibilities, and thus the maps from them reconstructed. The first approach is based on the construction of visibility spectra as a function of energy and their inversion through Tikhonov regularization. Two consecutive ill-posed problem are solved: the first inversion allows obtaining information on visibilities in the electron space, where the physics of interest resides. By solving a second inverse ill-posed problem, it is possible now to obtain, from electron visibilities, electron maps whose pixel values are proportional to the number of accelerated electrons. Thus, from the electron flux images, important physical information on the evolution of solar flares can be obtained. Finally, the second approach is based on advanced interpolation techniques, focusing on the use of Variably Scaled Kernel, and on the projected Landweber method to perform extrapolation. We present theoretical analysis detailing the error bounds that highlight the dependence on the scale function for VSKs, and the convergence results of the Landweber scheme for specific definitions of the scale function. In this framework, the images are reconstructed by an interpolation approach in which the reconstructions of contiguous energy channels are used as regularization priors. Finally, this thesis is supplemented with an appendix describing the work done to take into account all the aspects that make STIX a real and not an ideal instrument.