Reduced order models for coupled systems

Efficient numerical simulations of coupled multi-component systems can be particularly challenging. This is mostly due to the complexity of their solutions, as mutual interactions may cause emergent behaviors, including synchronization and instabilities. Variations in the physical parameters and the multi-query simulations arising from, e.g., control problems, pose additional challenges. In parallel, to reduce the computational complexity while preserving the accuracy of the underlying numerical discretization scheme, model order reduction techniques have proven to be effective for a plethora of models and problems. The goal of this thesis is to design model order reduction methods specifically targeted to coupled systems. This allows one to detect (and possibly discover) emergent behaviors in multi-component systems without the need to simulate the original, possibly expensive model. This work is divided into three main parts. In the first part, we assume that the coupled system is known and the full model can be exploited to construct projection-based surrogate models. Variations in the constitutive parameters lead to qualitatively different system behaviors, that can be explored at a reduced level. In this setting, we propose and numerically validate an efficient reduction method for systems exhibiting synchronization, including phase dynamics equations and a model for circadian oscillators. In the second part, we consider scenarios in which the full coupled model cannot be used to construct the surrogate. This is the case when one does not know how the components will be assembled at a later stage or repeated simulations of the full system are prohibitive due to the high degree of complexity of the problem. The reduced models are constructed by simulating the system components separately with a suitable (artificial) parametrization of the boundary conditions and projecting the local discretization operators. The surrogates can subsequently be used to recover the main features of the coupled system of interest under parameter variations. We first apply these techniques to an oscillatory mechanical system consisting of pendulum clocks hanging on a wooden structure (the Huygens' experiment), and we subsequently extend them to more general cases, including diffusion-reaction models and fluid-structure interaction problems. In the third part, we aim to construct reduced models for systems whose solvers are available only as black boxes, i.e., with no access to the (local) discretization operators. We use purely data-driven interpolation methods combined with data obtained from the coupled model or an artificial parametrization, in the same spirit of the first two parts of this thesis. We show the efficiency of our method in a variety of problems, including highly heterogeneous and multi-physics models.