Improved regularized solution of the inverse problem in turbidimetric measurements

We present results of simulation research on the constrained regularized least-squares (RLS) solution of the ill-conditioned inverse problem in turbidimetric measurements. The problem is formulated in terms of the discretized Fredholm integral equation of the first kind. The inverse problem in turbidimetric measurements consists in determining particle size distribution (PSD) function of particulate system on the basis of turbidimetric measurements. The desired PSD should satisfy two constraints: nonnegativity of PSD values and normalization of PSD to unity when integrated over the whole range of particle size. Incorporating the constraints into the RLS method leads to the constrained regularized least-squares (CRLS) method, which is realized by means of an active set algorithm of quadratic programming. Results of simulation research prove that the CRLS method performs considerably better with reconstruction of PSD than the RLS method in terms of better fidelity and smaller uncertainty.