The pragmatics of solving industrial (real-world) inverse problems with exemplification based on the molecular weight distribution problem

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, scien tifically and mathematically, the essential and generic nature of some unde rlying class of inverse problems, attention must focus on the specific inve rse problem which encapsulates the question. In an industrial inverse probl em context, the question comes first. The fact that it involves the solutio n of a specific inverse problem only comes at a later stage, as various fra meworks, within which the question can be examined, are formulated and asse ssed. 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 in verse problem can be solved depends crucially on the success with which the underlying inverse problem has already been investigated. The identificati on of the question is not the time to start the detailed investigation of t he underlying inverse problem, but to look for an alternative way of answer ing it, if the required information is not available to implement the curre nt approach being explored. Consequently, the success of any industrial inv erse problem endeavour depends crucially on exploiting the scientific and m athematical infrastructure which is already available about the particular problem (and question) under consideration. Thus, the formal study of indus trial inverse problems reduces to the comprehensive examination, for a part icular industrial activity such as the determination of the molecular weigh t distribution (MWD) of a polymer, of the various inverse problem considera tions which relate to this activity. It is therefore not correct to view the solution of an industrial inverse p roblem as an opportunity to initiate a general examination of the underlyin g inverse problems. However, the study of a particular type of industrial a ctivity can be undertaken to build an infrastructure on which the future so lution of related industrial inverse problems can be based. It is the indus trial question which has the priority. The specific inverse problem to be s olved 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 solutio n' and 'the study' of an industrial inverse problem. In addition, 'the stud y' is being defined to be an investigation of the various aspects of invers e problems as they specifically relate to the particular industrial activit y being examined. 'Consequently, the focus of a study is not a specific inv erse problem, but the nature of the various inverse problems which are spaw ned 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 industr ial applications and processes. For example, in the study of macromolecules, the single most important conc ept is, from various practical and theoretical points of view, their MWD. T he pivotal role played by the higher molecular weight components in determi ning the properties of synthetic polymers and biopolymers, such as wheat fl our dough, is well documented. In the everyday science and technology of po lymer processing, as well as related studies of macromolecules and the rheo logy of biopolymers, the determination and interpretation of the MWD plays a central and sometimes a crucial role in the associated scientific and ind ustrial decision-making. In such contexts, the MWD is often viewed and inte rpreted as the molecular characterization of the material being tested. A comprehensive discussion of the MWD problem is beyond the scope of this r eview, although it does include appropriate background material as well as a historical introduction. Here, the emphasis and focus will be on the vari ous ways in which the modelling, analysis, determination and interpretation of the MWD problem, for synthetic polymers and biopolymers, involves the s olution of inverse problems. They range from the very practical, such as th e direct determination of the MWD from a gel permeation chromatography (GPC ) experiment, to the very theoretical, such as the formulation of various r eptation mixing rules which relate the MWD to the observed relaxation modul us pf a polymer. Depending on the issues being discussed, the review involv es various levels of mathematical sophistication such as the analysis of fi eld-flow fractionation (FFF) experiments, the application of the linear fun ctional strategy to the reptation mixing rules, and the solution of the bas ic equations of viscoelasticity. In order to stress the industrial aspects of such activities, some of the motivation and exemplification for this rev iew is drawn from food rheology and technology, as well as industrial polym er processing.