Data assimilation and sensor selection for configurable forward models : challenges and opportunities for model order reduction methods

• Datenassimilierung und Sensorwahl für konfigurierbare Vorwärtsmodelle : Herausforderungen und Gelegenheiten für Modellreduktionsmethoden

Aretz, Nicole; Reusken, Arnold (Thesis advisor); Veroy-Grepl, Karen (Thesis advisor); Ghattas, Omar (Thesis advisor)

Aachen : RWTH Aachen University (2022, 2023)
Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2022

Abstract

Models are fundamental for numerical approximations since they introduce physical laws to drive the simulations. However, models are imperfect, causing uncertainty in the prediction. This uncertainty can be decreased by incorporating observational data of the described physical system through an inverse problem. With measurement data driving the quality of the inverse solution, obtaining informative observations at restricted experimental cost is essential. This task is complicated further when the model is contingent on flexible model configurations, each giving rise to a separate inverse problem. In this thesis, we address challenges and opportunities for model order reduction in the inference of linear model uncertainties in partial differential equations that are additionally characterized by variable configuration parameters. For the inverse solution in deterministic settings, we employ the 3D-VAR and 4D-VAR data assimilation methods to weigh model deviations against data misfits. In probabilistic settings, we use linear Bayesian inversion to obtain configuration-dependent posterior probability distributions as data-driven updates of prior information. Analyzing the numerical stability of the inverse solutions, we derive an observability coefficient measuring the ratio between model modifications and induced observational changes. In both the deterministic and probabilistic settings we propose iterative sensor selection algorithms which exploit their respective observability coefficients to choose sensor combinations with uniform observation properties over all admissible configurations. The algorithms are suitable for correlated noise models and large-scale forward models, achieving computational efficiency through model order reduction. For the 3D-VAR and 4D-VAR data assimilation methods, we apply reduced basis (RB) model order reduction techniques to facilitate their real-time approximation for varying configurations. After a preparatory offline phase, the presented RB-3D-VAR and RB-4D-VAR methods are computable at significantly reduced cost, while the approximation error can be monitored through rigorous and certified a posteriori error bounds. In particular, we prove new results for space-time Continuous-Galerkin RB approximations of parabolic equations that are applicable beyond data assimilation. We verify the sensor selection and RB approximation results of this thesis on a steady-state heat conduction problem over a thermal block, a contaminant-dispersion problem over a Taylor-Green vortex velocity field, and a geothermal model over a section of the Perth Basin in Western Australia.