Mathematical models and numerical methods for environmental applications of fast field cycling nuclear magnetic resonance

Fast Field-Cycling (FFC) Nuclear Magnetic Resonance (NMR) relaxometry is a non-destructive technique operating at low magnetic fields to investigate molecular dynamics and structures in environmental, biological, and food systems, exploring slow dynamics and revealing motion across diverse timescales. Despite its broad applicability, accurately identifying parameters from NMR Dispersion (NMRD) profiles remains a significant computational challenge. This thesis introduces and evaluates advanced inverse methods based on regularization and machine-learning approaches to enhance NMRD analysis. Within the Model-Free framework, NMRD profiles are represented as a linear combination of Lorentzian functions. To tackle the ill-conditioned inverse problem, three regularization methods are validated: (1) MF-UPen, a locally adaptive L₂ regularization; (2) MF-L1, an L₁-penalized approach; and (3) MF-MUPen, a hybrid of local L₂ and global L₁ penalties. Automated selection of regularization parameters via the Balancing and Uniform Penalty principles improves robustness and reproducibility. To model quadrupolar relaxation enhancement (QRE) arising from electric interactions of spin > ½ nuclei, a constrained L₁-regularized non-linear least squares framework is proposed. It decomposes relaxation profiles into dipole-dipole and quadrupolar contributions. The regularization parameter is iteratively computed with the Balancing Principle, and parameters are optimized using a non-linear Gauss–Seidel algorithm. Tests on synthetic and real datasets confirm convergence and effectiveness. A MATLAB tool implementing this method is freely available. Finally, a machine-learning framework based on Plug-and-Play (PnP) integrates a pre-trained feed-forward neural network into a coordinate-descent optimizer to extract QRE parameters and fit NMRD profiles. Its custom loss balances L₁ loss with quadrupolar prediction accuracy, yielding precise parameter extraction. Experimental validation against traditional inverse methods demonstrates accuracy and efficiency, particularly for large-scale industrial datasets. This work advances the computational toolkit for FFC-NMR relaxometry, offering robust algorithms and machine-learning solutions that deepen understanding of molecular dynamics across diverse systems.