Abstract: Objective　The research and development of miniaturized nanoscale particle size spectrometer is of great significance in the field of particle monitoring of moving source at roadside. While in the indirect measurement of particle size distribution in nanometer scale, the inversion algorithm of ill-posed problem is crucial. Therefore, an alternate direction multiplier method (ADMM) with non-negative constraints and L2 regularization is proposed in this work to solve the ill-posed inversion problem of particle size distribution at the nanoscale. Methods　In order to improve the numerical stability and solving accuracy of the algorithm, a strategy of adaptive adjustment of the augmented parameter ρ is introduced on the ADMM frame firstly. And then the modeling of fine particulate matter particle size spectrometer based on electromigration principle is carried out, and an inversion model of the improved ADMM algorithm is constructed. The inversion experiment results show that, compared with the traditional nonnegative least square (NNLS) method, the improved ADMM has significant advantages in solving accuracy in particle size distribution inversion, and effectively avoids oscillations caused by non-negative constraints. Especially when dealing with ill-condition and high-noise data, the improved ADMM inversion method for particle size distributions can approach the true solution more stably and significantly reduce the relative error, and has excellent anti-noise and ill-condition suppression ability while ensuring the physical feasibility. This indicates that the improved ADMM has higher robustness and accuracy in solving non-negative constrained particle size spectrum inversion problems. In the comparative experiment with commercial instruments, the data measured by scanning mobility particle sizer (SMPS) is used as a reference. Results and Discussion　This method not only retains the basic requirement that particle size concentration must be nonnegative in physical sense, but also effectively suppresses the known oscillation phenomenon by introducing L2 regularization term, thereby improving the smoothness and rationality of inversion results. More importantly, in order to further improve the algorithm's adaptability under different data conditions, this study innovatively introduces a strategy of adaptively adjusting the augmented Lagrange parameter ρ on the basis of the standard ADMM framework. This strategy can dynamically adjust the value of parameter ρ according to the residual changes during the current iteration process. A larger ρ is adopted in the early stage of convergence to accelerate the convergence speed, and a smaller ρ is adopted when approaching the optimal solution to improve the accuracy, thereby significantly improving the numerical stability while ensuring the computational efficiency. To verify the effectiveness of the proposed method, we first establish a complete mathematical model of the miniaturized particle size spectrometer based on the principle of electromigration, simulating the common multi-peak distribution, wide particle size range and different degrees of noise interference in actual measurements. Subsequently, an inversion model of the improved ADMM algorithm is constructed and systematically compared with the widely used NNLS. Conclusions　The experimental results show that under the same test conditions, the improved ADMM exhibits higher resolution and fidelity in restoring the true particle size distribution. Especially when dealing with bimodal or multi-modal particle distributions, the improved ADMM can accurately identify the position and relative intensity of each peak, while NNLS is prone to peak position offset or false peak generation problems. In addition, when Gaussian white noise is added to simulate actual sensor noise, the improved ADMM can still maintain relatively stable inversion performance, with an average relative error reduction of over 30%, demonstrating excellent noise resistance and robust control ability for pathological systems.

Key words: particulate emissions from mobile sources, particle size spectrum inversion algorithm, alternating direction multiplier method, non-negative constraint, L2 regularization