Adaptive Surrogate Modelling for the Uncertainty Quantification of Large-Scale Nonlinear PDE-based Problems

The quantification and control of uncertainty is paramount for the simulation-based characterization of engineering systems. However, the high costs of uncertainty quantification (UQ) often preclude engineers from developing a comprehensive understanding of the uncertainty inherent in large-scale simulations. This thesis presents work towards scaling UQ to large-scale, engineering-relevant problems via three distinct contributions. First, we develop a goal-oriented, adaptive method for the forward UQ of systems modeled by stochastically parametrized nonlinear conservation laws. The method exploits localized structure in the spatio-parameter approximation spaces to enable efficient and reliable UQ of quantities of interest. The formulation comprises (i) a discontinuous Galerkin method; (ii) element-wise polynomial chaos expansions; (iii) spatio-stochastic dual-weighted residual error estimates; and (iv) a projection-based spatio-parameter anisotropic error indicator and the associated adaptation mechanics. We demonstrate the effectiveness of the formulation for transonic turbulent flows with uncertainties in flow conditions and turbulence model parameters. Second, we develop an efficient multi-fidelity data assimilation method for large-scale nonlinear dynamical systems. The formulation comprises (i) an ensemble Kalman filter (EnKF) to tractably handle high-dimensional, nonlinear dynamical models; (ii) projection-based reduced order models (ROMs), which accelerate the evaluation of ensemble forecast by several orders of magnitude and are constructed on the fly using the ensemble of full-order model (FOM) trajectories; and (iii) multi-fidelity statistical estimates that optimally combine the FOM and ROM forecasts. We demonstrate the effectiveness of the formulation using unsteady aerodynamic flow past an airfoil. Lastly, we develop a data assimilation method for large-scale dynamical systems that exhibit strong nonlinearities, which can render the standard EnKF less effective. The formulation comprises (i) ensemble transport filter (EnTF), which generalizes the EnKF to stronger nonlinarities using the transport map, (ii) ROMs, which reduce the state dimension and accelerate the ensemble forecast to overcome the scaling challenges associated with transport maps, and (iii) a method to mitigate stability issues that arise when constructing transport maps for ROMs. We demonstrate the efficacy of the EnTF relative to EnKF using unsteady aerodynamics flow past an airfoil.

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