Bayesian Optimal Experimental Design Using Asymptotic Approximations

Bayesian optimal experimental design (OED) tools for model parameter estimation and response predictions in structural dynamics include sampling (Huan and Marzouk, J. Comput. Phys., 232:288–317, 2013) and asymptotic techniques (Papadimitriou et al., J. Vib. Control., 6:781–800, 2000). This work compares the two techniques and discusses the theoretical and computational advantages of asymptotic techniques. It is shown that the OED based on maximizing the expected Kullback-Leibler divergence between the prior and posterior distribution of the model parameters is equivalent, asymptotically for large number of data and small model prediction error, to minimizing asymptotic estimates of the robust information entropy measure introduced in the past (Papadimitriou et al., J. Vib. Control., 6:781–800, 2000; Papadimitriou, J. Sound Vib., 278:923–947, 2004; Papadimitriou and Lombaert, Mech. Syst. Signal Process., 28:105–127, 2012) for structural dynamics applications. Based on the asymptotic approximations, techniques are proposed to overcome the sensor clustering. In addition, an insightful analysis is presented that clarifies the effect of the variances of Bayesian priors on the optimal design. Finally the importance of uncertainties in nuisance model parameters is pointed out and the expected utility functions are extended to take into account such uncertainties. A heuristic forward sequential sensor placement algorithm (Papadimitriou, J. Sound Vib., 278:923–947, 2004) is effective in solving the optimization problem in the continuous physical domain of variation of the sensor locations, bypassing the problem of multiple local/global optima manifested in optimal experimental designs and providing near optima solutions in a fraction of the computational effort required in expensive stochastic optimization algorithms. The theoretical and computational developments are demonstrated for optimal sensor placement designs for applications taken from structural mechanics and dynamics areas. Examples covering the optimal sensor placement design for parameter estimation and response predictions are covered.