Predictive modeling involving simulation and sensor data at the same time, is a growing challenge in computational science. 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 at the frequency grid and conditioning it on data quickly results in a prohibitive computational cost. In this case, the introduction of a surrogate in form of a reduced order model 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 inexpensive error indicator. The results of the new statistical reduced order method are compared to the standard statFEM procedure applied to a ROM prior, i.e. without explicitly accounting for the reduced order bias. The proposed method yields better accuracy and faster convergence throughout a given frequency range for different numerical examples.

翻译：在计算科学中，同时涉及仿真与传感器数据的预测建模正日益成为一项挑战。即使采用大规模有限元模型，仿真结果与传感器数据之间仍常存在不匹配，这可归因于多种不确定性来源。针对此类场景，统计有限元方法可用于将仿真场基于给定的传感器数据进行条件化处理。该方法产生的后验解能显著改善与数据的拟合度，并能同时提供包括模型误设在内的不确定性一致估计。对于频率或参数相关的问题（例如声学或电磁学领域），在频率网格上求解全阶模型并基于数据进行条件化处理会迅速导致难以承受的计算成本。此时，引入降阶模型形式的代理模型可大幅缩减方程组规模。本文提出了一种基于Krylov子空间矩匹配的降阶统计有限元框架。我们构建了显式包含降阶近似所引入偏差的数据模型，该偏差通过低成本误差指示器进行估计。通过不同数值算例，将新统计降阶方法的结果与应用于降阶模型先验的标准统计有限元流程（即未显式考虑降阶偏差）进行比较。结果表明，在给定频率范围内，所提方法具有更优的精度与更快的收敛速度。