Some remarks on the inverse Smoluchowski problem for cluster-cluster aggregation

TL;DR

This paper revisits the inverse Smoluchowski problem, proposing a regularised least squares method to reconstruct collision kernels from cluster size data, and analyzes its effectiveness and limitations through numerical experiments.

Contribution

It implements and tests a polynomial expansion-based regularisation method for kernel reconstruction, highlighting the use of L-curve plots for parameter selection and exploring improvements for complex kernels.

Findings

The method works well when the kernel is exactly polynomial-expressible.

L-curve plots help determine optimal regularisation parameters.

Performance can be improved with more complex regularisation functions for non-polynomial kernels.

Abstract

It is proposed to revisit the inverse problem associated with Smoluchowski's coagulation equation. The objective is to reconstruct the functional form of the collision kernel from observations of the time evolution of the cluster size distribution. A regularised least squares method originally proposed by Wright and Ramkrishna (1992) based on the assumption of self-similarity is implemented and tested on numerical data generated for a range of different collision kernels. This method expands the collision kernel as a sum of orthogonal polynomials and works best when the kernel can be expressed exactly in terms of these polynomials. It is shown that plotting an "L-curve" can provide an a-priori understanding of the optimal value of the regularisation parameter and the reliability of the inversion procedure. For kernels which are not exactly expressible in terms of the orthogonal polynomials it is found empirically that the performance of the method can be enhanced by choosing a more complex regularisation function.