Statistical reduced order modelling for the parametric Helmholtz equation

Predictive modelling combining both numerical simulations and real-world measurement data, obtained from sensors, is gaining importance in computational science and engineering. Even with large-scale finite element models, a mismatch to the sensor data often remains, which can be attributed to different sources of uncertainty. For such a scenario, the statistical finite element method (statFEM) can be used to condition a simulated field on given sensor data. This yields a posterior solution which resembles the data much better and additionally provides consistent estimates of uncertainty, including model misspecification. For frequency or parameter dependent problems, occurring, e.g. in acoustics or electromagnetism, solving the full order model across the frequency range of interest and conditioning it on data quickly results in a prohibitive computational cost. In this case, the introduction of a surrogate in the form of a reduced order model (ROM) yields much smaller systems of equations. In this paper, we propose a reduced order statFEM framework relying on Krylov-based moment matching. We introduce a data model which explicitly includes the bias induced by the reduced approximation, which is estimated by an error indicator. The results of the new statistical reduced order method are compared to the standard statFEM procedure applied to a ROM prior without explicitly accounting for reduced order bias. The proposed method achieves better accuracy and faster convergence throughout a given frequency range for a variety of numerical examples and artificial (i.e. simulated) sensor data.