A Modified Landweber Algorithm for Inversion of Particle Size Distribution Combined With Tikhonov Regularization Theory

Abstract

Particle size distribution (PSD) measurement based on the static light scattering method has been widely used in the environmental field and combustion diagnostics, such as PM2.5 measurement and combustion process monitoring. The PSD inversion is mathematically related to the Fredholm integral equation of the first kind. Although the Tikhonov regularization algorithm is one of the effective inversion methods to solve the ill-posed linear equation, it still has the disadvantages of excessive smoothness and low accuracy. Thus, a preconditioned Landweber algorithm combined with the Tikhonov regularization theory (the improved algorithm) is proposed in this paper. The Tikhonov regularization theory is used to pretreat the preconditioner B contained in the preconditioned Landweber algorithm. Simulations are conducted to compare the inversion results of the conventional Landweber algorithm, the preconditioned Landweber algorithm, the Tikhonov Chahine algorithm, and the improved algorithm at different signal-to-noise ratios. A CCD-based small-angle forward scattering measurement system is built. A standardized polystyrene microsphere with a diameter of $35.05~\mu \text{m}$ is used to evaluate the above algorithms. Both numerical and experimental results show that the improved algorithm improves the accuracy of the inversion results and is insensitive to the ring parameter of the detector. The experimental results of the standardized polystyrene microsphere reveal that the relative errors for the median diameter $50~\mu \text{m}$ are better than 3%. The improved algorithm can provide a highly reliable and stable inversion result.