26 Optimal Sensor Placement for Response Predictions Using Local and Global Methods 235 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Number of samples 0.45 0.46 0.47 0.48 0.49 0.5 0.51 0.52 Objective function e = 1, p = 1, x = 1 Monte Carlo Analytic (a) 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Number of samples 4.7 4.75 4.8 4.85 4.9 4.95 5 5.05 5.1 5.15 5.2 Objective function 10-3 e = 0.1, p = 0.1, x = 1 Monte Carlo Analytic (b) 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Number of samples 4.6 4.7 4.8 4.9 5 5.1 5.2 5.3 Objective function 10-5 e = 0.01, p = 0.01, x = 1 Monte Carlo Analytic (c) 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Number of samples 8.8 9 9.2 9.4 9.6 9.8 10 Objective function 10-7 e = 0.01, p = 0.01, x = 10 Monte Carlo Analytic (d) Fig. 26.2 Comparison between Monte Carlo (blue circles) and analytical solutions (black line) for different values of σ e, σp and x. Monte Carlo objective function values are normalized by the analytical value for each case of σ e, σp andx.Case (a): σ e =1, σp =1, x =1, Case (b): σ e =0.1, σp =0.1, x =1, Case (c): σ e =0.01, σp =0.01, x =1, Case (d): σ e =0.01, σp =0.01, x =10 We can observe how the Monte Carlo solution converges to the true analytical solution as the number of samples is increased (unbiased estimator). We can also see the associated noise (variance) when moving from one number of samples to the next, which is introduced due to the random sampling. The variance is due to both the random sampling of parameter values from the prior and due to the random data samples from the likelihood. For each objective function evaluation a new batch of parameter and data samples is drawn, which leads to slightly different results. The average error in the first three sub-figures (x =1) is about 2% while in the fourth (x =10) is about 0.5%. Also in the case of x =10 the Monte Carlo estimator stabilizes much more quickly compared with x =1. This is due to the fact that there is a much greater signal-tonoise ratio for x =10 and the uncertainty introduced in the estimator through data noise has a much smaller contribution thanwithx =1. This leads to a better convergence behaviour. Next we evaluate the objective function with the sensitivity-based method, using the derivative of the measured quantity.
RkJQdWJsaXNoZXIy MTMzNzEzMQ==