Analysis of the Regularization Parameters of Primal–Dual Interior Method for Convex Objectives Applied to 1H Low Field Nuclear Magnetic Resonance Data Processing

Abstract

Two-dimensional low-field hydrogen nuclear magnetic resonance (2D ¹H LF-NMR) analysis of chemical compounds measures T₁ and T₂ relaxation times observed as exponential decay curves. Once relaxation curves are measured and stored in the format of discrete digital signals, they must be transformed, mathematically, into spectra that can be read and interpreted. We used primal–dual interior method for convex objectives (PDCO) that provided more accurate reconstructions than the standard algorithms. It is the objective of this paper to estimate the most suitable PDCO parameterization that provide accurate and robust reconstructions of relaxation curves into 2D spectra for LF-NMR under different signal-to-noise ratios. Finding optimal regularization parameters is an active field of mathematical research. PDCO, however, by making use of two regularization parameters instead of a single one, presents a much harder task, where the consolidated search criteria of a single parameter cannot be extended to two parameters. We featured a method based on numerical experiments and simulations that identified optimal and unique model coefficients that maximize PDCO reconstruction accuracy under different signal-to-noise conditions. The coefficients, as determined for artificial signals, increase the confidence of accurate reconstruction for 2D LF-NMR analysis on real lab samples.