Reconstruction of a distribution from a finite number of moments with an adaptive spline-based algorithm

Abstract

An adaptive algorithm suitable for reconstructing a distribution when knowing only a small number of its moments is presented. This method elaborates on a previous technique presented in John et al. (2007), but leads to many advantages compared with the original algorithm. The so-called “finite moment problem” arises in many fields of science, but is particularly important for particulate flows in chemical engineering. Up to now, there is no well-established algorithm available to solve this problem. The examples considered in this work come from crystallization processes. For such applications, it is of crucial interest to reconstruct the particle size distributions (PSD) knowing only a small number of its moments, obtained mostly from numerical simulations or from advanced experiments, but without any a priori knowledge concerning the shape of this PSD. This was already possible in many cases with the original algorithm of John et al. (2007), but complex shapes could not be identified appropriately. The key of the advanced algorithm is the adaptive criterion for positioning dynamically the nodes in an appropriate manner. It turns out that the adaptive algorithm shows considerable improvements in the reconstruction of distributions with a quickly changing curvature or for non-smooth distributions. Since such configurations are quite often found in practice, the new algorithm is more widely applicable compared with the original method.