Techniques used in applying regularization to the ill‐posed problem of determining a molecular weight distribution from sedimentation equilibrium.

Abstract

Inferring the Molecular Weight Distribution (MWD) of a polymer in solution from ultracentrifuge sedimentation equilibrium data has been a problem of major concern during the past decades. Recently, it has been shown that this problem is of itself mathematically ill‐posed, i.e., leads to unstable and unreliable results. To combat this difficulty, a technique of regularization applicable to the sedimentation equilibrium equation has been formulated which casts this into a well‐posed problem. This paper discusses the criteria for calling a problem mathematically ill‐posed; a regularization technique for circumventing this ill‐posedness; a regularization incorporated into linear programming technique; and a recent method for incorporating regularization into a quadratic programming algorithm. All examples are presented using arbitrarily assumed MWD. Good results were obtained by applying this technique to a unimodal through a pentamodal distribution.