Handling uncertainties of models inputs in inverse problem: the U-discrete PSO approach

The inverse problem of unconstrained physical systems consists in finding the set of continuous inputs for its model that minimize fitness function measuring the distance between a target and the result of the model. However either the characterization of the sensitivity of model to each input or a preliminary knowledge on the uncertainties on parameters can reveal different required precision on each input. Therefore a blind search of the best set of parameters could be unnecessarily costing in terms of computational effort on the determination of non significative digits. To overcome this problem a solution consists in searching the best parameter sets within a predefined discrete space of search that is designed from the significative number of digits. The strategy consists in transforming the continuous problem into a discrete one that falls within combinatorial optimization eventually before ending the search of the best continuous inputs if required. The performance of continuous and discrete PSO are compared by their application to the inverse problem of the purple color of the Lycurgus cup. The structure of this ancient glass is recovered from photograph by using a direct model of color generation. The discussion of results is also an opportunity to convince that the heuristic approaches provide solutions to the inverse problem unlike nested loops, from a pedagogical point of view.