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We describe and compare two formulations of inverse problems for a physics-based process model in the context of uncertainty and random variability: the Bayesian inverse problem and the stochastic inverse problem. We describe the foundations of the two problems in order to create a context for interpreting the applicability and solutions of inverse problems important for scientific and engineering inference. We conclude by comparing them to statistical approaches to related problems, including Bayesian calibration of computer models. 

In manufacturing processes, controlling system responses with uncertain system inputs, e.g., due to variations in material parameters of critical system sub-components, is a crucial task for performing reliable quality control and verification & validation (V&V) of system design. As a model for a manufacturing process, we consider the production of drums, that is, thin elastic membranes, whose properties are modeled via Dirichlet Laplacian eigenproblems with uncertain diffusion coefficients. Both a quality control and V&V problem are formulated within a data-consistent framework utilizing push-forward and pullback measures. In both problems, the uncertain diffusion coefficients are parameterized for every instance and the corresponding eigen-information defines correlated data streams. Subsequently, the quantities of interest required in the solution to the data-consistent inverse problems are determined by an a posteriori analysis of these data streams using feature extraction techniques. While the methodology proposed here is quite general, the specific efficacy of the proposed methodology is comprehensively explored in the numerical results for both the quality control and V&V problems associated with the manufacturing of drums.