Chapter 26 Optimal Sensor Placement for Response Predictions Using Local and Global Methods Costas Argyris, Costas Papadimitriou, and Geert Lombaert Abstract A Bayesian framework for model-based optimal sensor placement for response predictions is presented. Our interest lies in determining the parameters of the model in order to make predictions about a particular response quantity of interest. This problem is not adequately explored since the majority of currently available literature is focused on parameter inference, rather than prediction inference. The model parameters are inferred by collecting experimental data which depends on the chosen sensor locations. The parameter values are uncertain and their uncertainty is described by a prior probability density function. The measured quantity, or data, is a quantity that can be predicted by the model which depends on both parameters and sensor locations. A prediction error equation is used to describe the discrepancy between the model-predicted measured quantity and the actual data collected from the experiment. The sensor locations are optimized with respect to prediction inference, while the case of parameter inference is derived as a special case under a more general framework. The posterior covariance matrix is used as a measure of uncertainty in the predictions. Two approaches are developed for its calculation, one global and one local. The local approach is based on sensitivities at a fixed value of the parameters, while the global approach uses Monte Carlo sampling and explores the full range of uncertainty in the parameters. A simple numerical example is presented in order to illustrate and verify the two approaches. Keywords Optimal sensor placement · Bayesian inference · Robust predictions · Uncertainty quantification · Monte Carlo integration 26.1 Introduction Model-based optimal sensor placement is concerned with finding which is the best way to perform an experiment such that a specific purpose is achieved, using a model of the system as a guide. Common purposes include parameter inference and making predictions using the model [1]. Herein we are interested in optimizing the design for prediction inference, and recover parameter inference as a special case. The parameters are uncertain, as are the experimental data, since no experiment has taken place at the time of design. These uncertainties are treated within the Bayesian framework for uncertainty quantification [2], by assigning a prior probability density function (PDF) for the parameters and a probabilistic model for the difference between model predictions and data, known as the prediction error. Following the Bayesian method, the posterior represents our updated state of uncertainty about the parameters or predictions given the data. In order to formulate the objective function one needs to chose a scalar measure to describe posterior uncertainty. The seminal paper of Lindley [3] suggests using the expected gain in Shannon information from prior to posterior as a measure of the information provided by an experiment. Information theory measures, based on scalar measures of the Fisher Information Matrix (FIM) [4, 5] and on information entropy [6–8] have been proposed in the past for structural parameter estimation problems. Much of the currently available literature is focused on designing the experiment for parameter inference. However, when trying to derive both parameter and prediction inference under the same framework, information-theoretic quantities such as the entropy can be troublesome to work with due to the complicated relation between the posterior PDF of parameters and predictions. In this work we use the posterior covariance matrix to describe uncertainty and specifically its determinant as a C. Argyris ( ) · G. Lombaert Department of Civil Engineering, KU Leuven, Leuven, Belgium e-mail: costas.argyris@kuleuven.be C. Papadimitriou Department of Mechanical Engineering, University of Thessaly, Volos, Greece © Society for Experimental Mechanics, Inc. 2020 R. Barthorpe (ed.), Model Validation and Uncertainty Quantification, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-12075-7_26 229
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