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Fast Field-Cycling Nuclear Magnetic Resonance relaxometry is anon-destructive technique to investigate molecular dynamics and structure ofsystems having a wide range of applications such as environment, biology, andfood. Besides a considerable amount of literature about modeling andapplication of such technique in specific areas, an algorithmic approach to therelated parameter identification problem is still lacking. We believe that arobust algorithmic approach will allow a unified treatment of different samplesin several application areas. In this paper, we model the parametersidentification problem as a constrained $L_1$-regularized non-linear leastsquares problem. Following the approach proposed in [Analytical Chemistry 202193 (24)], the non-linear least squares term imposes data consistency bydecomposing the acquired relaxation profiles into relaxation contributionsassociated with 1H-1H and 1H-14N dipole-dipole interactions. The data fittingand the L1-based regularization terms are balanced by the so-calledregularization parameter.  For the parameters identification, we propose an algorithm that computes, ateach iteration, both the regularization parameter and the model parameters. Inparticular, the regularization parameter value is updated according to aBalancing Principle and the model parameters values are obtained by solving thecorresponding $L_1$-regularized non-linear least squares problem by means ofthe non-linear Gauss-Seidel method. We analyse the convergence properties ofthe proposed algorithm and run extensive testing on synthetic and real data. AMatlab software, implementing the presented algorithm, is available uponrequest to the authors.

An automatic L1-based regularization method for the analysis of FFC dispersion profiles with quadrupolar peaks