@article {2020,
	title = {Stabilized reduced basis methods for parametrized steady Stokes and Navier{\textendash}Stokes equations},
	journal = {Computers and Mathematics with Applications},
	volume = {80},
	year = {2020},
	pages = {2399-2416},
	abstract = {<p>It is well known in the Reduced Basis approximation of saddle point problems that the Galerkin projection on the reduced space does not guarantee the inf{\textendash}sup approximation stability even if a stable high fidelity method was used to generate snapshots. For problems in computational fluid dynamics, the lack of inf{\textendash}sup stability is reflected by the inability to accurately approximate the pressure field. In this context, inf{\textendash}sup stability is usually recovered through the enrichment of the velocity space with suitable supremizer functions. The main goal of this work is to propose an alternative approach, which relies on the residual based stabilization techniques customarily employed in the Finite Element literature, such as Brezzi{\textendash}Pitkaranta, Franca{\textendash}Hughes, streamline upwind Petrov{\textendash}Galerkin, Galerkin Least Square. In the spirit of offline{\textendash}online reduced basis computational splitting, two such options are proposed, namely offline-only stabilization and offline{\textendash}online stabilization. These approaches are then compared to (and combined with) the state of the art supremizer enrichment approach. Numerical results are discussed, highlighting that the proposed methodology allows to obtain smaller reduced basis spaces (i.e., neglecting supremizer enrichment) for which a modified inf{\textendash}sup stability is still preserved at the reduced order level.</p>},
	doi = {10.1016/j.camwa.2020.03.019},
	url = {https://www.scopus.com/inward/record.uri?eid=2-s2.0-85083340115\&doi=10.1016\%2fj.camwa.2020.03.019\&partnerID=40\&md5=7ace96eee080701acb04d8155008dd7d},
	author = {Shafqat Ali and F. Ballarin and Gianluigi Rozza}
}
@article {2018,
	title = {Stabilized weighted reduced basis methods for parametrized advection dominated problems with random inputs},
	journal = {SIAM-ASA Journal on Uncertainty Quantification},
	volume = {6},
	year = {2018},
	pages = {1475-1502},
	abstract = {<p>In this work, we propose viable and eficient strategies for stabilized parametrized advection dominated problems, with random inputs. In particular, we investigate the combination of the wRB (weighted reduced basis) method for stochastic parametrized problems with the stabilized RB (reduced basis) method, which is the integration of classical stabilization methods (streamline/upwind Petrov-Galerkin (SUPG) in our case) in the ofine-online structure of the RB method. Moreover, we introduce a reduction method that selectively enables online stabilization; this leads to a sensible reduction of computational costs, while keeping a very good accuracy with respect to high-fdelity solutions. We present numerical test cases to assess the performance of the proposed methods in steady and unsteady problems related to heat transfer phenomena.</p>},
	doi = {10.1137/17M1163517},
	url = {https://www.scopus.com/inward/record.uri?eid=2-s2.0-85058246502\&doi=10.1137\%2f17M1163517\&partnerID=40\&md5=6c54e2f0eb727cb85060e988486b8ac8},
	author = {D. Torlo and F. Ballarin and Gianluigi Rozza}
}
@article {2015,
	title = {Supremizer stabilization of POD-Galerkin approximation of parametrized Navier-Stokes equations},
	number = {International Journal for Numerical Methods in Engineering;Volume 102, issue 5; pp.1136-1161;},
	year = {2015},
	publisher = {Wiley},
	abstract = {In this work, we present a stable proper orthogonal decomposition{\textendash}Galerkin approximation for parametrized steady incompressible Navier{\textendash}Stokes equations with low Reynolds number.},
	doi = {10.1002/nme.4772},
	url = {http://urania.sissa.it/xmlui/handle/1963/34701},
	author = {F. Ballarin and Andrea Manzoni and Alfio Quarteroni and Gianluigi Rozza}
}
@article {2014,
	title = {Shape Optimization by Free-Form Deformation: Existence Results and Numerical Solution for Stokes Flows},
	number = {Journal of scientific computing;volume 60; issue 3; pages 537-563;},
	year = {2014},
	publisher = {Springer},
	abstract = {Shape optimization problems governed by PDEs result from many applications in computational fluid dynamics. These problems usually entail very large computational costs and require also a suitable approach for representing and deforming efficiently the shape of the underlying geometry, as well as for computing the shape gradient of the cost functional to be minimized. Several approaches based on the displacement of a set of control points have been developed in the last decades, such as the so-called free-form deformations. In this paper we present a new theoretical result which allows to recast free-form deformations into the general class of perturbation of identity maps, and to guarantee the compactness of the set of admissible shapes. Moreover, we address both a general optimization framework based on the continuous shape gradient and a numerical procedure for solving efficiently three-dimensional optimal design problems. This framework is applied to the optimal design of immersed bodies in Stokes flows, for which we consider the numerical solution of a benchmark case study from literature.},
	doi = {10.1007/s10915-013-9807-8},
	url = {http://urania.sissa.it/xmlui/handle/1963/34698},
	author = {F. Ballarin and Andrea Manzoni and Gianluigi Rozza and Sandro Salsa}
}
