294 K. Worden et al. Given all of the above, the mathematical question of interest is: Given all of the above, is MC1 ⊕ MI MC2 an -mirror for Sd, and if so, what is the smallest value of for which this is true? This will usually be a probabilistic problem where the metrics are quantities like probability of misclassification or probability of detection, in which case it will probably be more appropriate to frame the problem in terms of α-mirrors. 34.3.4 Multi-Fidelity Models: Refinement and Relaxation This section considers the situation when one has multiple models of the same structure S, in a fixed context C. Suppose that a model MC is an -mirror for S. A modified model M C =Ref[MC] is a refinement of MC, if it is an -mirror with < . Similarly, A modified model M C = Rel[MC] is a relaxation of MC, if it is an -mirror with > . For finite element models, these operations can be carried out by refining or coarsening the mesh. In this simplest of situations, one might estimate the values of using analytical error estimates. This idea is one that can be used in order to answer Question (1) in the introduction. In principle one starts with a model MC which is provably fit-for-purpose and then relaxes the model until one arrives at M C with = T. Now, it is possible to consider what sort of propositions one might wish to prove in the theory i.e. consider the hypothesis: Assume a model MA = MC 1 1 ⊕MJ MC 2 2 is an A-twin for a joined structure S 1 ⊕ SJ S 2. Further suppose that MC1 1 is an 1-mirror. Now, if M A =M C 1 1 ⊕MJ MC 2 2 is obtained by refining the first submodel, thenM A is an A-mirror, with A < A. Another strategy for answering Question (1) would then be to relax submodels in an assembly until the result is marginally fit-for-purpose. 34.3.5 An Example Concerning Design This is one of the potential applications of digital twin technology that would produce large cost savings for industry. Suppose one has a existing structure S and a context C; further suppose that a virtualisation VC =(MhC 1 ,MEC 2 ) exists which has been validated and shown to be an -mirror for SC. Imagine now that one wished to design a new structure S and thus wanted to know how it would behave (either in the old context C, or in a new context C . In a situation where one wished to avoid building a prototype for S , there is no direct means of validating a new visualisation V C = (M hC ,M EC ), even though this would be ideal for conducting ‘what-if’ games for the new structure. The question of immediate interest is: Given a virtualisationVC =(MhC 1 ,MEC 2 ), which is an -mirror for SC; is V C =(M hC 1 ,M EC 2 ) a mirror for S C for any values of 1 and 2, and if so, what are the smallest possible values for which this true? 34.4 Discussion and Conclusions This paper proposes some ingredients for a mathematical theory which would allow a general framework for measuring the fidelity of computational models and for understanding the consequences of combining validated models or using them outside their original context. Such a theory would be invaluable in the design and construction of digital twins, because one of the main uses of digital twins will be to make predictions in circumstances where their core models have not been explicitly validated, and it will be critical to obtain estimates of how much models can be trusted when they are used to extrapolate or generalise; i.e. when they are used to make inferences about different structures or in different contexts. As discussed in the introduction, there are already attempts to define a unifying framework for model calibration and validation. In fact, these papers already go into greater detail on specific technical points than the current paper e.g. they go as far as to propose a Bayesian framework and define appropriate priors, likelihoods etc. [5, 6]. The techniques proposed can very much form part of the armoury of the more general methodology proposed here. The current paper deliberately draws back from some details because the authors believe that important discussions are still to be had. For example, it is not agreed within the broader V&V and uncertainty quantification communities that probability theory is the correct way to
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