Optimum Experimental Design with PDE Constraints for Identification of Model Uncertainty in Load-bearing Systems

In this thesis we introduce a novel algorithm to identify model uncertainty based on methods from optimum experimental design with partial differential equation (PDE) constraints and statistical hypothesis testing. We first introduce five different approaches based on a frequentist and a Bayesian probabilistic perspective to estimate the parameter‘s a posteriori probability distribution from noisy data. Furthermore, we examine modern approaches to optimal sensor placement and make an extension to optimal input configuration. In so doing, we introduce a PDE-constrained optimization problem, which adds a cost term to sparsify the number of used sensors and a smooth regularization for the inputs to the objective function, and solve it with an adjoint approach.
The data which are collected in an optimally designed experiment is used to infer parameter estimates that have a small variance. We construct a hypothesis test to falsify the assumption that repeated calibration and validation procedures should yield parameter values in the same small confidence region. If a new set of data leads to estimates that lie outside of this confidence region, then model uncertainty is detected with a given small threshold to the Type I error probability. We prove that for linear models the probability that our algorithm falsely identifies model uncertainty is identical to the small test level. We also prove that the smaller the confidence region the better the rejection of false linear models.
Finally, we apply our algorithm to detect model uncertainty in mathematical models of a forming machine and in the linear-elastic model of vibrations in a truss. We conclude this thesis with an evaluation of the numerical results and we give an outlook on large-scale problems in optimal input configuration with PDE constraints.