@article {NegriManzoniRozza2015,
	title = {Reduced basis approximation of parametrized optimal flow control problems for the Stokes equations},
	journal = {Computers and Mathematics with Applications},
	volume = {69},
	number = {4},
	year = {2015},
	pages = {319{\textendash}336},
	abstract = {<p>This paper extends the reduced basis method for the solution of parametrized optimal control problems presented in Negri et al. (2013) to the case of noncoercive (elliptic) equations, such as the Stokes equations. We discuss both the theoretical properties-with particular emphasis on the stability of the resulting double nested saddle-point problems and on aggregated error estimates-and the computational aspects of the method. Then, we apply it to solve a benchmark vorticity minimization problem for a parametrized bluff body immersed in a two or a three-dimensional flow through boundary control, demonstrating the effectivity of the methodology.</p>},
	doi = {10.1016/j.camwa.2014.12.010},
	author = {Federico Negri and Andrea Manzoni and Gianluigi Rozza}
}
@article {2015,
	title = {Reduced Basis Isogeometric Methods (RB-IGA) for the real-time simulation of potential flows about parametrized NACA airfoils},
	journal = {Comput Methods Appl Mech Eng. 2015;284:1147{\textendash}1180},
	number = {;284},
	year = {2015},
	abstract = {We present a Reduced Basis (RB) method based on Isogeometric Analysis (IGA) for the rapid and reliable evaluation of PDE systems characterized by complex geometrical features. At the current state of the art, this is the first case of coupling between RB and IGA methods. The construction of the RB method relies on an Isogeometric Boundary Element Method (IGA-BEM) as the high-fidelity technique, allowing a direct interface with Computer Aided Design (CAD) tools. A suitable Empirical Interpolation Method (EIM) ensures an efficient offline/online decomposition between the construction and the evaluation of the RB method. We consider the real-time simulation of potential flows past airfoils, parametrized with respect to the angle of attack and the NACA number identifying their shape, and we provide a validation of our methodology with respect to experimental data and reference numerical codes, showing in both cases a very good agreement.We present a Reduced Basis (RB) method based on Isogeometric Analysis (IGA) for the rapid and reliable evaluation of PDE systems characterized by complex geometrical features. At the current state of the art, this is the first case of coupling between RB and IGA methods. The construction of the RB method relies on an Isogeometric Boundary Element Method (IGA-BEM) as the high-fidelity technique, allowing a direct interface with Computer Aided Design (CAD) tools. A suitable Empirical Interpolation Method (EIM) ensures an efficient offline/online decomposition between the construction and the evaluation of the RB method. We consider the real-time simulation of potential flows past airfoils, parametrized with respect to the angle of attack and the NACA number identifying their shape, and we provide a validation of our methodology with respect to experimental data and reference numerical codes, showing in both cases a very good agreement.},
	doi = {10.1016/j.cma.2014.11.037},
	author = {Andrea Manzoni and Filippo Salmoiraghi and Luca Heltai}
}
@article {2013,
	title = {Reduced basis approximation and a posteriori error estimation for Stokes  flows in parametrized geometries: roles of the inf-sup stability constants},
	journal = {Numerische Mathematik, 2013},
	number = {SISSA Preprint;43/2012/M},
	year = {2013},
	publisher = {Springer},
	abstract = {In this paper we review and we extend the reduced basis approximation and a posteriori error estimation for steady Stokes flows in a ffinely parametrized geometries, focusing on the role played by the Brezzi\\\'s and Babu ska\\\'s stability constants. The crucial ingredients of the methodology are a Galerkin projection onto a low-dimensional space of basis functions properly selected, an a ne parametric dependence enabling to perform competitive Off ine-Online splitting in the computational\\r\\nprocedure and a rigorous a posteriori error estimation on  eld variables.\\r\\nThe combination of these three factors yields substantial computational savings which are at the basis of an e fficient model order reduction, ideally suited for real-time simulation and many-query contexts (e.g. optimization, control or parameter identi cation). In particular, in this work we focus on i) the stability of the reduced basis approximation based on the Brezzi\\\'s saddle point theory and the introduction of a supremizer operator on the pressure terms, ii) a rigorous a posteriori error estimation procedure for velocity and pressure  elds based on the Babu ska\\\'s inf-sup constant (including residuals calculations), iii) the computation of a lower bound of the stability constant, and iv) di erent options for the reduced basis spaces construction. We present some illustrative results for both\\r\\ninterior and external steady Stokes flows in parametrized geometries representing two parametrized classical Poiseuille and Couette \\r\\nflows, a channel contraction and a simple flow control problem around a curved obstacle.},
	keywords = {parametrized Stokes equations},
	url = {http://hdl.handle.net/1963/6339},
	author = {Gianluigi Rozza and Phuong Huynh and Andrea Manzoni}
}
@article {NegriRozzaManzoniQuarteroni2013,
	title = {Reduced basis method for parametrized elliptic optimal control problems},
	journal = {SIAM Journal on Scientific Computing},
	volume = {35},
	number = {5},
	year = {2013},
	pages = {A2316{\textendash}A2340},
	abstract = {We propose a suitable model reduction paradigm-the certified reduced basis method (RB)-for the rapid and reliable solution of parametrized optimal control problems governed by partial differential equations. In particular, we develop the methodology for parametrized quadratic optimization problems with elliptic equations as a constraint and infinite-dimensional control variable. First, we recast the optimal control problem in the framework of saddle-point problems in order to take advantage of the already developed RB theory for Stokes-type problems. Then, the usual ingredients of the RB methodology are called into play: a Galerkin projection onto a low-dimensional space of basis functions properly selected by an adaptive procedure; an affine parametric dependence enabling one to perform competitive offline-online splitting in the computational procedure; and an efficient and rigorous a posteriori error estimate on the state, control, and adjoint variables as well as on the cost functional. Finally, we address some numerical tests that confirm our theoretical results and show the efficiency of the proposed technique.},
	doi = {10.1137/120894737},
	author = {Federico Negri and Gianluigi Rozza and Andrea Manzoni and Alfio Quarteroni}
}
@article {2013,
	title = {A Reduced Computational and Geometrical Framework for Inverse Problems in Haemodynamics},
	number = {SISSA preprint;15/2013/MATE},
	year = {2013},
	institution = {SISSA},
	author = {Toni Lassila and Andrea Manzoni and Alfio Quarteroni and Gianluigi Rozza}
}
@article {2013,
	title = {A reduced-order strategy for solving inverse Bayesian identification problems in physiological flows},
	number = {SISSA preprint;14/2013/MATE},
	year = {2013},
	institution = {SISSA},
	author = {Toni Lassila and Andrea Manzoni and Alfio Quarteroni and Gianluigi Rozza}
}
@article {2013,
	title = {Reduction Strategies for Shape Dependent Inverse Problems in Haemodynamics},
	number = {SISSA preprint;13/2013/MATE},
	year = {2013},
	institution = {SISSA},
	author = {Toni Lassila and Andrea Manzoni and Gianluigi Rozza}
}
@proceedings {2012,
	title = {Reduction strategies for PDE-constrained oprimization problems in Haemodynamics},
	number = {SISSA;41/2012/M},
	year = {2012},
	abstract = {Solving optimal control problems for many different scenarios obtained by varying a set of parameters in the state system is a computationally extensive task. In this paper we present a new reduced framework for the formulation, the analysis and the numerical solution of parametrized PDE-constrained optimization problems. This framework is based on a suitable saddle-point formulation of the optimal control problem and exploits the reduced basis method for the rapid and reliable solution of parametrized PDEs, leading to a relevant computational reduction with respect to traditional discretization techniques such as the finite element method. This allows a very efficient evaluation of state solutions and cost functionals, leading to an effective solution of repeated optimal control problems, even on domains of variable shape, for which a further (geometrical) reduction is pursued, relying on flexible shape parametrization techniques. This setting is applied to the solution of two problems arising from haemodynamics, dealing with both data reconstruction and data assimilation over domains of variable shape,\\r\\nwhich can be recast in a common PDE-constrained optimization formulation.},
	keywords = {inverse problems},
	url = {http://hdl.handle.net/1963/6338},
	author = {Gianluigi Rozza and Andrea Manzoni and Federico Negri}
}
