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For this situation, the temporal discretization becomes an additional decision variable, and the problem can be posed in two different contexts: (1) optimize cost and discrepancy jointly (Section 3.4.1) or (2) specify an allowable computational budget and minimize the uncertainty in the prediction within that budget (Section 3.4.2).
3.4.1 Minimizing the product of cost and error
Suppose ( ) is a possible model combination that predicts the output for a single cycle of a given input. The FOSM procedure (as in Section 3.2) can be applied to account for the uncertainty in by taking a Latin hypercube sample of and values and propagating the distribution of through all possible model combinations at each pair . A GP surrogate model is trained for the mean prediction and the variance of the prediction over the input space for each model combination. One advantage of the GP is that its efficiency allows the model to be evaluated on a cycle-by-cycle basis without discretizing into blocks as is necessary for the higher fidelity models. Thus, starting from cycle , the mean output after cycles can be approximated for each of the model combinations as
∑ for (3.14) The variance for each model combination can also be accumulated under the normality assumption. Therefore, the standard deviation of after cycles have passed starting from cycle can be approximated for each of the model combinations as
√∑ for (3.15)
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Frequently, there may be reason to assume that a particular model combination is more accurate than the others if, for example, it uses a finer spatial resolution or a more sophisticated physics model. In Sections 3.2 and 3.3, no such assumption was made (though it could be included by introducing weights as was previously mentioned), and hence all models were given equal weights in constructing the average distribution. A similar approach could be utilized in the time-dependent problem if no information were available about the ranking of fidelities among the candidate models. However, in some cases, it might be obvious that one model should be trusted more than the others because this maximum fidelity model includes all of the physics described by its competitors in addition to incorporating additional physical complexity or providing higher resolution. Even so, it might not be necessary to use the maximum fidelity model for every realization or for every instant and location in order to meet a given accuracy target. Since this scenario poses a tradeoff decision between accuracy and complexity, the methodology that follows here is a technique for selecting the model (among several cheaper models and the highest fidelity model) to evaluate over each block discretization of the input by considering the expense of a model and its discrepancy from the highest fidelity choice. A normal distribution can be constructed for the output of each model combination with the mean and standard deviation estimated by Eq. (3.14) and (3.15). The highest fidelity model combination (among the possible candidates ) is assumed to be the maximum fidelity model for each subsystem.
Given that the most accurate (and expensive) model is known, the analyst must decide how much deviation from this model is acceptable. From a decision maker’s perspective, it is often possible to establish some acceptable error bars on the prediction (e.g. based on the precision of experimental instrumentation or the width of the maximum fidelity model’s uncertainty).
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Therefore, a tolerance for the discrepancy between the most accurate (and expensive) combination and the other combinations is introduced. Given that the computation time associated with each model combination is known, the optimal model combination can then be selected by taking the model combination with the lowest product of computational time and probability of discrepancy greater than the specified tolerance as shown in Eq. (3.16) and (3.17).
( ) (3.16)
(3.17)
The implication of this treatment is that a less expensive model combination will be selected when its mean prediction agrees strongly with the mean prediction of the highest fidelity model and the variance of the prediction is small. Once a model combination is selected, it cannot be evaluated cycle-by-cycle as the GP was, so only one input value can be chosen for the entire duration of the block . Since the GP corresponding to the selected combination has already been evaluated at all between and , Eq. (3.13) can be applied to guide the selection decision. In particular, the discrete input point from that range with mean GP prediction, i.e.
, closest to the accumulated mean GP prediction for the maximum fidelity model combination, should be selected.
This selection procedure continues until the number of cycles that have been discretized and analyzed is equal to the desired total simulation length . This procedure can then be repeated for many realizations of the input and initial output values . From these samples, the distribution of interest will describe , the final value of the output for each realization.
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3.4.2 Variance minimization for a fixed simulation time
If instead of simultaneously considering time and discrepancy there is a fixed time to perform a simulation of a given number of realizations, the corresponding time for a single realization of the random process can be specified. The temporal discretizations required to achieve the desired simulation time follow directly from the time required for one cycle of each model combination.
Model combinations with more computational expense must be discretized more coarsely in order for them to run within the specified budget. Once all model combinations are forced to take the same amount of time, they can be compared on the basis of error alone. Given a number of cycles to simulate over total time and a vector of computational times for one evaluation of each model combination , a vector of the temporal discretization for each combination can be computed with Eq. (3.18), and is calculated as the largest value of as in Eq. (3.19).
(3.18)
(3.19)
Then, starting from cycle , the mean output after cycles can be approximated for each of the model combinations as in Eq. (3.14). However, the variance for each model combination is only accumulated for the number of cycles for the particular temporal discretization required.
Therefore, the standard deviation of at cycles after cycle can be approximated for each of the model combinations as
√∑ for (3.20)
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The result of this treatment is that the standard deviations of the predictions of the faster (cheaper) models are smaller than their more expensive counterparts because of the finer temporal discretization. Since time is no longer a consideration (all models are allotted equal computational time), only the discrepancy needs to be considered in the decision, and the optimal combination c can be selected in a similar manner to Section 3.4.1 as
( ) (3.21)
If none of the alternative model combinations can meet this tolerance criterion with a high confidence (e.g. 95%), then the benchmark model combination itself should be executed. The simulation then proceeds exactly as shown in the previous section.