230 C. Argyris et al. scalar measure. This enables us to formulate the problem for the general case of prediction inference, and recover parameter inference as a special case. The posterior covariance matrix is found using two different approaches, one global that uses Monte Carlo sampling [9, 10] and one local that uses sensitivities. First we present the general formulation, then develop the two approaches, and finally we verify the methodology using a simple linear example. 26.2 OED Formulation for Response Predictions Throughout the text all involved quantities are real-valued vectors whose size is denoted when they are first introduced. For clarity purposes, no specific vector notation is used. Let M=M(θ) represent a parametrized model of a structure, which depends on parameters θ ∈RNθ whose values are uncertain. Model M(θ) can be used to calculate two different quantities, namely the measured and predicted quantities, both of which depend on parameters θ. These two quantities represent the forward problem. The former is the quantity that is going to be measured when the experiment is performed, and the latter is the quantity that we would like to predict accurately after having collected the experimental data. Let g =g(θ) ∈ RNg represent the predicted quantity of interest. It depends on the uncertain model parameters θ and therefore it is also uncertain. Our goal is to perform the experiment in such a way that the estimate of g(θ) is as accurate as possible. Knowledge about g(θ) comes through knowledge of parameters θ, which in turn is obtained through the measured data collected from the experiment. The inference of parameters θ from measured data constitutes the inverse problem. Then the updated uncertainty inθ is propagated to uncertainty in the predicted quantityg(θ) (uncertainty propagation). We assume that the experiment can be performed for ND different designs. Each design, denoted by m = 1, . . . ,ND, can be conceived as a different sensor setup with Nm sensors which would lead to a different measured data set. However, this data set is not known in advance before the experiment takes place. In order to model the process of data collection we consider the two main reasons for mismatch between measured data and model predictions for the measured data: (1) imperfect knowledge about model parameters θ and (2) imperfect model coupled with random measurement error (prediction error). Both sources of uncertainty are taken into account in the prediction error equation which models the measured data set under design mas: dm ∼dm(θ) +em (26.1) where dm(θ) ∈ RNm is the model prediction for the measured data under design mfor parameters θ, and em ∈ RNm is the prediction error term, considered to be a multivariate zero-mean Gaussian random vector with covariance matrix em. Following the Bayesian framework for uncertainty quantification, parameter uncertainty is modelled by assigning a prior PDF to the parameters, namely p(θ). Then the posterior is given by Bayes’ rule as: p(θ| d) = p( d|θ)p(θ) p( d) (26.2) where p( d|θ) is the Gaussian likelihood defined through the prediction error equation (26.1), p(θ) is the prior and p( d) is the evidence. The flow of information from measured data to parameters and to predictions can be seen schematically in Fig. 26.1. The goal is to find which of the ND possible experimental designs will result in the most accurate posterior prediction about g(θ). This is formulated as a discrete optimization problem: mopt =argmin Um, m=1, . . . ,ND (26.3) where the objective function Um reflects the uncertainty in the estimate of g(θ) resulting from data collected from experimental design m. Note that if we set g(θ) = θ the problem reduces from inference of predictions to inference of model parameters. Hence, the problem of optimizing the design for parameter inference can be seen as a special case of optimizing for prediction inference, where the predictions are set equal to the parameters. However, the choice of objective function Um must also reflect that possibility.
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