A gradient method for regularizing retrieval of aerosol particle size distribution function

The determination of the aerosol particle size distribution function using the particle spectrum extinction equation is an ill-posed integral equation of the first kind ([15, 19]). Even for finite moment case, the problem is still discrete ill-posed, since as is known, in remote sensing the observations are often limited /insufficient or contaminated. To overcome the ill-posedness, various standard or non-standard regularization techniques were developed (see [18] and references therein). However, most of the literature focuses on the application of the Phillips-Twomey's regularization or its variants which is unstable in several cases. Recently in [17], the authors considered Tikhonov's smooth regularization method in $W^{1,2}$ space for ill-posed inversion. But the method still relies on the choice of the regularization parameter and the a priori estimation of the noise level. In addition, these methods do not consider the nonnegative constraints of the model problem. As is known, the particle size distribution is always nonnegative and we are often faced with incomplete data. Therefore, creation of data to establish well-posedness and development of suitable method are urgently needed. We first present a regularization model which incorporates smoothness constraint to the solution, and then propose an efficient gradient method for solving the regularizing problem. Numerical tests are performed to show the efficiency and feasibility of the proposed algorithms.