Recovering aerosol particle size distribution function on the set of bounded piecewise-convex functions

In this article we discuss the determination of the aerosol particle size distribution function using the particle spectrum extinction equation. This is an ill-posed integral equation of the first kind, since we are often faced with limited or insufficient observations in remote sensing and the observations are contaminated. To overcome the ill-posed nature, using the a priori information, we study the inflection point regularization method and develop a conjugate gradient projection method for solving the corresponding optimization problem. In our work, regularization parameter can be considered as the number of inflection points and their position. The algorithms for constructing an approximate solution are described. Error estimates are also presented. Our numerical tests for both the synthetic problem and practical problem are given to show the efficiency and feasibility of the proposed algorithm.