236 C. Argyris et al. 26.5.2 Sensitivity-Based Method Substituting the linear measured quantity from Eq. (26.15) to (26.13) we obtain: σ−2 θ| dx =∇θ (θx)σ −2 e ∇θ (θx) +σ−2 p = x 2 σ 2 e + 1 σ2 p = x 2 σ 2 p σ 2 e σ 2 p + σ 2 e σ 2 e σ 2 p = x 2 σ 2 p +σ 2 e σ 2 e σ 2 p σ 2 θ| dx = σ 2 e σ 2 p σ 2 e +σ 2 px 2 (26.26) which has exactly the same form as the one obtained analytically with the global approach in Eq. (26.23), and therefore the same analysis of the results holds here as well. The reason why the sensitivity-based approach gives the exact solution is because the measured quantity is linear with respect to the parameter and this makes the method exact and not approximate. 26.6 Conclusions A model-based Bayesian optimal sensor placement framework was presented where the interest lies in using the experimental data for making predictions using the model. The posterior covariance matrix of the predictions was used as a measure of uncertainty. Two methods were presented for evaluating the covariance matrix, one global and one local. The global method uses Monte Carlo sampling to evaluate the required integrals while the local method uses Gaussian approximation of the posterior and is based on the sensitivities of the measured and predicted quantities with respect to parameters. A simple linear numerical application was used where the analytical solution is known, in order to demonstrate how the two methods can be applied. Acknowledgement This work was performed within the frame of the project C16/17/008 “Efficient methods for large-scale PDE-constrained optimization in the presence of uncertainty and complex technological constraints” funded by KU Leuven. References 1. Chaloner, K., Verdinelli, I.: Bayesian experimental design: a review. Stat. Sci. 10, 273–304 (1995) 2. Beck, J.L., Katafygiotis, L.S.: Updating models and their uncertainties. I: Bayesian statistical framework. J. Eng. Mech. 124, 463–467 (1998) 3. Lindley, D.V.: On a measure of the information provided by an experiment. Ann. Math. Stat. 27, 986–1005 (1956) 4. Shah, P.C., Udwadia, F.E.: A methodology for optimal sensor locations for identification of dynamic systems. J. Appl. Mech. 45, 188–196 (1978) 5. Udwadia, F.E.: Methodology for optimum sensor locations for parameter identification in dynamic systems. J. Eng. Mech. 120, 368–390 (1994) 6. Papadimitriou, C., Beck, J.L., Au, S.K.: Entropy-based optimal sensor location for structural model updating. J. Vib. Control 6, 781–800 (2000) 7. Yuen, K.-V., Katafygiotis, L.S., Papadimitriou, C., Mickleborough, N.C.: Optimal sensor placement methodology for identification with unmeasured excitation. J. Dyn. Syst. Meas. Control 123, 677 (2001) 8. Ye, S.Q., Ni, Y.Q.: Information entropy based algorithm of sensor placement optimization for structural damage detection. Smart Struct. Syst. 10, 443–458 (2012) 9. Ryan, K.J.: Estimating expected information gains for experimental designs with application to the random fatigue-limit model. J. Comput. Graph. Stat. 12, 585–603 (2003) 10. Huan, X., Marzouk, Y.M.: Simulation-based optimal Bayesian experimental design for nonlinear systems. J. Comput. Phys. 232, 288–317 (2013)
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