Hybrid data assimilation methods, Part I: Numerical comparison between GEIM and PBDW

Abstract

Hybrid Data Assimilation (HDA) methods are a class of numerical methods that aim at integrating Model Order Reduction (MOR) techniques into a Data Assimilation (DA) framework, thus combining mathematical models and experimental data. The objective is to reduce the solution time using MOR algorithms whilst keeping the accuracy of the models at the desired level using observations, which serve as an update to the a priori prediction of the model. This two-part work investigated HDA techniques by applying them to two classes of problems: numerical benchmark cases (part 1) and experimental facilities (part 2). In particular, this paper discusses the former, focusing on the numerical formulation of the methodologies and on the effect of noisy data. Indeed, real-world experimental data are always polluted by errors and uncertainties; therefore, it is critical to first assess the performance of these techniques on numerical benchmark cases with the artificial introduction of random noise before applying them to real-world experimental facilities. As such, this paper applies the Generalised Empirical Interpolation Method (GEIM) and the Parameterised-Background Data-Weak (PBDW) formulation to a non-adiabatic airflow over the classical computational fluid-dynamics benchmark of the 3D Backward Facing Step (BFS). Results show how both algorithms are valuable tools to reconstruct the state of the system when measurements are available, whilst assessing the effect of noise on the available data; in particular, the GEIM is a bit better than the PBDW since a lower reconstruction error is achieved with fewer sensors.