This work tackles the problem of image restoration, a crucial task in many fields of applied sciences, focusing on removing degradation caused by blur and noise during the acquisition process. Drawing inspiration from the multi-penalty approach based on the Uniform Penalty principle introduced in [Bortolotti et al. arXiv.math.NA/2309.14163], we develop here a new image restoration model and an iterative algorithm for its effective solution. The model incorporates pixel-wise regularization terms and establishes a rule for parameters selection, aiming to restore images through the solution of a sequence of constrained optimization problems. To achieve this, we present a modified version of the Newton Projection method, adapted to multi-penalty scenarios, and prove its convergence. Numerical experiments demonstrate the efficacy of the method in eliminating noise and blur while preserving the image edges.

Due to the extreme complexity of Alzheimer's disease (AD), the aetiology of which is not yet known, nor are there any known effective treatments, mathematical modelling can be very useful. Indeed, mathematical models, if deemed reliable, can be used to test medical hypotheses that could be difficult to verify directly. In this context, it is important to understand how $A\beta$ and $\tau$ proteins, which in abnormal aggregate conformations are hallmarks of the disease, interact and spread. We are particularly interested in this paper in studying the spreading of misfolded $\tau$. To this end, we present four different mathematical models, all on networks on which the protein evolves. The models differ in both the choice of network and diffusion operator. Through comparison with clinical data on $\tau$ concentration, that we carefully obtained with multimodal analysis techniques, we show that some models are more adequate than others to simulate the dynamics of the protein. This type of study may suggest that, when it comes to modelling certain pathologies, the choice of the mathematical setting must be made with great care if comparison with clinical data is considered decisive.

Fast Field-Cycling (FFC) Nuclear Magnetic Resonance (NMR) relaxometry is a powerful non-destructive magnetic resonance technique that enables, among other things, the investigation of slow molecular dynamics at low magnetic field intensities. FFC-NMR relaxometry measurements provide insight into molecular motion across various timescales within a single experiment. This study focuses on a \textit{model-free} approach, representing the NMRD profile $R_1$ as a linear combination of Lorentzian functions, thereby addressing the challenges of fitting data within an ill-conditioned linear least-squares framework. Tackling this problem, we present a comprehensive review and experimental validation of three regularization approaches to implement the model-free approach to analysing NMRD profiles. These include: (1) \nameUpe, utilizing locally adapted $L_2$ regularization; (2) \nameL, based on $L_1$ penalties; and (3) a hybrid approach combining locally adapted $L_2$ and global $L_1$ penalties. Each method's regularization parameters are determined automatically, according to the Balancing and Uniform Penalty principles. % Our contributions include the implementation and experimental validation of the \nameUpe{} and \nameUp{} algorithms, and the development of a "dispersion analysis" technique to assess the existence range of the estimated parameters. The objective of this work is to delineate the variance in fit quality and correlation time distribution yielded by each algorithm, thus broadening the set of software tools for the analysis of sample structures in FFC-NMR studies. The findings underline the efficacy and applicability of these algorithms in the analysis of NMRD profiles from samples representing different potential scenarios. 1