What distinguishes the solution of industrial inverse problems from that of inverse problems, more generally, is the requirement to address and answer a specific question about an industrial application, within which the need to solve a specific inverse problem has arisen. Rather than explore, scientifically and mathematically, the essential and generic nature of some underlying class of inverse problems, attention must focus on the specific inverse problem which encapsulates the question. In an industrial inverse problem context, the question comes first. The fact that it involves the solution of a specific inverse problem only comes at a later stage, as various frameworks, within which the question can be examined, are formulated and assessed. The goal always remains one of answering the question as quickly and as efficiently as possible as ongoing practical and economic considerations rest on its resolution. Consequently, the extent to which an industrial inverse problem can be solved depends crucially on the success with which the underlying inverse problem has already been investigated. The identification of the question is not the time to start the detailed investigation of the underlying inverse problem, but to look for an alternative way of answering it, if the required information is not available to implement the current approach being explored. Consequently, the success of any industrial inverse problem endeavour depends crucially on exploiting the scientific and mathematical infrastructure which is already available about the particular problem (and question) under consideration. Thus, the formal study of industrial inverse problems reduces to the comprehensive examination, for a particular industrial activity such as the determination of the molecular weight distribution (MWD) of a polymer, of the various inverse problem considerations which relate to this activity. It is therefore not correct to view the solution of an industrial inverse problem as an opportunity to initiate a general examination of the underlying inverse problems. However, the study of a particular type of industrial activity can be undertaken to build an infrastructure on which the future solution of related industrial inverse problems can be based. It is the industrial question which has the priority. The specific inverse problem to be solved is the means to this end, if a suitable infrastructure is already in place. In this way, a clear distinction is being drawn between 'the solution' and 'the study' of an industrial inverse problem. In addition, 'the study' is being defined to be an investigation of the various aspects of inverse problems as they specifically relate to the particular industrial activity being examined. 'Consequently, the focus of a study is not a specific inverse problem, but the nature of the various inverse problems which are spawned by the examined industrial activity. The determination, both explicitly and implicitly, of the molecular weight distribution (MWD) of a polymer represents not only a suitable activity for which the construction of the mentioned infrastructure is appropriate, but also involves a quite wide and novel range of practical inverse problems. In addition, it is an important activity within a broad spectrum of industrial applications and processes. For example, in the study of macromolecules, the single most important concept is, from various practical and theoretical points of view, their MWD. The pivotal role played by the higher molecular weight components in determining the properties of synthetic polymers and biopolymers, such as wheat flour dough, is well documented. In the everyday science and technology of polymer processing, as well as related studies of macromolecules and the rheology of biopolymers, the determination and interpretation of the MWD plays a central and sometimes a crucial role in the associated scientific and industrial decision-making. In such contexts, the MWD is often viewed and interpreted as the molecular characterization of the material being tested. A comprehensive discussion of the MWD problem is beyond the scope of this review, although it does include appropriate background material as well as a historical introduction. Here, the emphasis and focus will be on the various ways in which the modelling, analysis, determination and interpretation of the MWD problem, for synthetic polymers and biopolymers, involves the solution of inverse problems. They range from the very practical, such as the direct determination of the MWD from a gel permeation chromatography (GPC) experiment, to the very theoretical, such as the formulation of various reptation mixing rules which relate the MWD to the observed relaxation modulus pf a polymer. Depending on the issues being discussed, the review involves various levels of mathematical sophistication such as the analysis of field-flow fractionation (FFF) experiments, the application of the linear functional strategy to the reptation mixing rules, and the solution of the basic equations of viscoelasticity. In order to stress the industrial aspects of such activities, some of the motivation and exemplification for this review is drawn from food rheology and technology, as well as industrial polymer processing.
