@article {2023,
	title = {An optimisation{\textendash}based domain{\textendash}decomposition reduced order model for the incompressible Navier-Stokes equations},
	volume = {151},
	year = {2023},
	month = {2023/12/01/},
	pages = {172 - 189},
	abstract = {<p>The aim of this work is to present a model reduction technique in the framework of optimal control problems for partial differential equations. We combine two approaches used for reducing the computational cost of the mathematical numerical models: domain{\textendash}decomposition (DD) methods and reduced{\textendash}order modelling (ROM). In particular, we consider an optimisation{\textendash}based domain{\textendash}decomposition algorithm for the parameter{\textendash}dependent stationary incompressible Navier{\textendash}Stokes equations. Firstly, the problem is described on the subdomains coupled at the interface and solved through an optimal control problem, which leads to the complete separation of the subdomain problems in the DD method. On top of that, a reduced model for the obtained optimal{\textendash}control problem is built; the procedure is based on the Proper Orthogonal Decomposition technique and a further Galerkin projection. The presented methodology is tested on two fluid dynamics benchmarks: the stationary backward{\textendash}facing step and lid-driven cavity flow. The numerical tests show a significant reduction of the computational costs in terms of both the problem dimensions and the number of optimisation iterations in the domain{\textendash}decomposition algorithm.</p>},
	keywords = {Computational fluid dynamics, Domain decomposition, Optimal control, Proper orthogonal decomposition, Reduced order modelling},
	isbn = {0898-1221},
	url = {https://www.sciencedirect.com/science/article/pii/S0898122123004248},
	author = {Ivan Prusak and Monica Nonino and Davide Torlo and Francesco Ballarin and Gianluigi Rozza}
}
@article {2015,
	title = {Reduced basis approximation and a-posteriori error estimation for the coupled Stokes-Darcy system},
	journal = {Advances in Computational Mathematics},
	volume = {special issue for MoRePaS 2012},
	year = {2015},
	abstract = {<p>The coupling of a free flow with a flow through porous media has many potential applications in several fields related with computational science and engineering, such as blood flows, environmental problems or food technologies. We present a reduced basis method for such coupled problems. The reduced basis method is a model order reduction method applied in the context of parametrized systems. Our approach is based on a heterogeneous domain decomposition formulation, namely the Stokes-Darcy problem. Thanks to an offline/online-decomposition, computational times can be drastically reduced. At the same time the induced error can be bounded by fast evaluable a-posteriori error bounds. In the offline-phase the proposed algorithms make use of the decomposed problem structure. Rigorous a-posteriori error bounds are developed, indicating the accuracy of certain lifting operators used in the offline-phase as well as the accuracy of the reduced coupled system. Also, a strategy separately bounding pressure and velocity errors is extended. Numerical experiments dealing with groundwater flow scenarios demonstrate the efficiency of the approach as well as the limitations regarding a-posteriori error estimation.</p>},
	keywords = {Domain decomposition, Error estimation, Non-coercive problem, Porous medium equation, Reduced basis method, Stokes flow},
	issn = {1019-7168},
	doi = {     10.1007/s10444-014-9396-6},
	author = {Immanuel Martini and Gianluigi Rozza and Bernard Haasdonk}
}
