Efficient Reduction Techniques for the Simulation and Optimization of Parametrized Systems

This thesis is concerned with the development, analysis and implementation of efficient reduced order models (ROMs) for the simulation and optimization of parametrized partial differential equations (PDEs). Indeed, since the high-fidelity approximation of many complex models easily leads to solve large-scale problems, the need to perform multiple simulations to explore different scenarios, as well as to achieve rapid responses, often requires unaffordable computational resources. Alleviating this extreme computational effort represents the main motivation for developing ROMs, i.e. low-dimensional approximations of the underlying high-fidelity problem. Among a wide range of model order reduction approaches, here we focus on the so-called projection-based methods, in particular Galerkin and Petrov-Galerkin reduced basis methods. In this context, the goal is to generate low cost and fast, but still sufficiently accurate ROMs which characterize the system response for the whole range of input parameters we are interested in. In particular, several challenges have to be faced to ensure reliability and computational efficiency. As regards the former, this thesis presents some heuristic approaches to approximate the stability factor of parameterized nonlinear PDEs, a key ingredient of any a posteriori error estimate. Concerning computational efficiency, we propose different strategies to combine the `Matrix Discrete Empirical Interpolation Method' (MDEIM) with a state approximation resulting either from a proper orthogonal decomposition or a greedy approach. Specifically, we exploit the MDEIM to develop fast and efficient ROMs for nonaffinely parametrized elliptic and parabolic PDEs, as well as for the time-dependent Navier-Stokes equations. The efficacy of the proposed methods is demonstrated on a variety of computationally-intensive applications, such as the shape optimization of an acoustic device, the simulation of blood flow in cerebral aneurysms and the simulation of solute dynamics in blood flow and arterial walls. %and coupled blood flow and mass transport in human arteries. Furthermore, the above-mentioned techniques have been exploited to develop a model order reduction framework for parametrized optimization problems constrained by either linear or nonlinear stationary PDEs. In particular, among this wide class of problems, here we focus on those featuring high-dimensional control variables. To cope with this high dimensionality and complexity, we propose an all-at-once optimize-then-reduce paradigm, where a simultaneous state and control reduction is performed. This methodology is applied first to a data reconstruction problem arising in hemodynamics, and then to several optimal flow control problems.