A Software Package to Simplify Tikhonov Regularization with Examples for Matrix-Based Inversion of SMPS and HTDMA Data

Abstract. Tikhonov regularization is a tool for reducing noise amplification during data inversion. This work introduces RegularizationTools.jl, a general-purpose software package to apply Tikhonov regularization to data. The package implements well-established numerical algorithms and is suitable for systems of up to ~1000 equations. Included is an abstraction to systematically categorize specific inversion configurations and their associated hyperparameters. A generic interface translates arbitrary linear forward models defined by a computer function into the corresponding design matrix. This obviates the need to explicitly write out and discretize the Fredholm integral equation, thus facilitating fast prototyping of new regularization schemes associated with measurement techniques. Example applications include the inversion involving data from scanning mobility particle sizers (SMPS) and humidified tandem differential mobility analyzers (HTDMA). Inversion of SMPS size distributions reported in this work builds upon the freely-available software DifferentialMobilityAnalyzers.jl. The speed of inversion is improved by a factor of ~200, now requiring between 2 and 5 ms per SMPS scan when using 120 size bins. Previously reported occasional failure to converge to a valid solution is reduced by switching from the L-curve method to generalized cross-validation as the metric to search for the optimal regularization parameter. Higher-order inversions resulting in smooth, denoised reconstructions of size distributions are now included in DifferentialMobilityAnalyzers.jl. This work also demonstrates that an SMPS-style matrix-based inversion can be applied to find the growth factor frequency distribution from raw HTDMA data, while also accounting for multiply-charged particles. The outcome of the aerosol-related inversion methods is showcased by inverting multi-week SMPS and HTDMA datasets from ground-based observations, including SMPS data obtained at Bodega Bay Marine Laboratory during the Calwater 2/ACAPEX campaign, and co-located SMPS and HTDMA data collected at the U.S. Department of Energy observatory located at the Southern Great Plains site in Oklahoma, U.S.A. Results show that the proposed approaches are suitable for unsupervised, nonparametric inversion of large-scale datasets as well as inversion in real-time during data acquisition on low-cost reduced-instruction-set architectures used in single-board computers. The included software implementation of Tikhonov regularization is freely-available, general, and domain-independent, and thus can be applied to many other inverse problems arising in atmospheric measurement techniques and beyond.