@article {2017,
	title = {Numerical modeling of hemodynamics scenarios of patient-specific coronary artery bypass grafts},
	journal = {Biomechanics and Modeling in Mechanobiology},
	volume = {16},
	year = {2017},
	pages = {1373-1399},
	doi = {10.1007/s10237-017-0893-7},
	url = {https://www.scopus.com/inward/record.uri?eid=2-s2.0-85015065851\&doi=10.1007\%2fs10237-017-0893-7\&partnerID=40\&md5=c388f20bd5de14187bad9ed7d9affbd0},
	author = {F. Ballarin and Elena Faggiano and Andrea Manzoni and Alfio Quarteroni and Gianluigi Rozza and Sonia Ippolito and Roberto Scrofani}
}
@article {2016,
	title = {A fast virtual surgery platform for many scenarios haemodynamics of patient-specific coronary artery bypass grafts},
	year = {2016},
	institution = {Submitted},
	abstract = {A fast computational framework is devised to the study of
several configurations of patient-specific coronary artery bypass
grafts. This is especially useful to perform a sensitivity
analysis of the haemodynamics for different flow conditions
occurring in native coronary arteries and bypass grafts, the
investigation of the progression of the coronary artery disease and the choice of the most appropriate surgical procedure.
A complete pipeline, from the acquisition of patientspecific
medical images to fast parametrized computational
simulations, is proposed. Complex surgical configurations
employed in the clinical practice, such as Y-grafts and sequential
grafts, are studied. A virtual surgery platform based
on model reduction of unsteady Navier Stokes equations for
blood dynamics is proposed to carry out sensitivity analyses
in a very rapid and reliable way. A specialized geometrical
parametrization is employed to compare the effect of stenosis
and anastomosis variation on the outcome of the surgery
in several relevant cases.},
	url = {http://urania.sissa.it/xmlui/handle/1963/35240},
	author = {F. Ballarin and Elena Faggiano and Andrea Manzoni and Gianluigi Rozza and Alfio Quarteroni and Sonia Ippolito and Roberto Scrofani and Carlo Antona}
}
@article {2015,
	title = {Fast simulations of patient-specific haemodynamics of coronary artery bypass grafts based on a POD-Galerkin method and a vascular shape parametrization},
	year = {2015},
	abstract = {In this work a reduced-order computational framework for the study of haemodynamics
in three-dimensional patient-specific configurations of coronary artery bypass grafts
dealing with a wide range of scenarios is proposed. We combine several efficient algorithms to face at the same time both the geometrical complexity involved in the description of the vascular network and the huge computational cost entailed by time dependent patient-specific flow simulations. Medical imaging procedures allow to reconstruct patient-specific configurations from clinical data. A centerlines-based parametrization is proposed to efficiently handle geometrical variations. POD{\textendash}Galerkin reduced-order models are employed to cut down large computational costs. This computational framework allows to characterize blood flows for different physical and geometrical variations relevant in the clinical practice, such as stenosis factors and anastomosis variations, in a rapid and reliable way. Several numerical results are
discussed, highlighting the computational performance of the proposed framework, as well as its capability to perform sensitivity analysis studies, so far out of reach.},
	url = {http://urania.sissa.it/xmlui/handle/1963/34623},
	author = {F. Ballarin and Elena Faggiano and Sonia Ippolito and Andrea Manzoni and Alfio Quarteroni and Gianluigi Rozza and Roberto Scrofani}
}
@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 {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 = {An efficient computational framework for reduced basis approximation and a posteriori error estimation of parametrized Navier-Stokes flows},
	year = {2014},
	abstract = {We present the current Reduced Basis framework for the efficient numerical approximation of parametrized steady Navier-Stokes equations. We have extended the existing setting developed in the last decade (see e.g. [Deparis, Veroy \& Patera, Quarteroni \& Rozza] to more general affine and nonaffine parametrizations (such as  volume-based techniques), to a simultaneous velocity-pressure error estimates and to a fully decoupled Offline/Online procedure in order to speedup the solution of the reduced-order problem. This is particularly suitable for real-time and many-query contexts, which are both part of our final goal. Furthermore, we present an efficient numerical implementation for treating nonlinear advection terms in a convenient way. A residual-based a posteriori error estimation with respect to a truth, full-order Finite Element approximation is provided for joint pressure/velocity errors, according to the Brezzi-Rappaz-Raviart stability theory. To do this, we take advantage of an extension of  the Successive Constraint Method for the estimation of stability factors and of a suitable fixed-point algorithm for the approximation of Sobolev embedding constants. Finally, we present some numerical test cases, in order to show both the approximation properties and the computational efficiency of the derived framework.},
	keywords = {Reduced Basis Method, parametrized Navier-Stokes equations, steady incompressible fluids, a posteriori error estimation, approximation stability},
	author = {Andrea Manzoni}
}
@article {2014,
	title = {Model Order Reduction in Fluid Dynamics: Challenges and Perspectives},
	number = {Modeling simulation and applications;volume 9; pages 235-273;},
	year = {2014},
	publisher = {Springer},
	abstract = {This chapter reviews techniques of model reduction of fluid dynamics systems. Fluid systems are known to be difficult to reduce efficiently due to several reasons. First of all, they exhibit strong nonlinearities - which are mainly related either to nonlinear convection terms and/or some geometric variability - that often cannot be treated by simple linearization. Additional difficulties arise when attempting model reduction of unsteady flows, especially when long-term transient behavior needs to be accurately predicted using reduced order models and more complex features, such as turbulence or multiphysics phenomena, have to be taken into consideration. We first discuss some general principles that apply to many parametric model order reduction problems, then we apply them on steady and unsteady viscous flows modelled by the incompressible Navier-Stokes equations. We address questions of inf-sup stability, certification through error estimation, computational issues and-in the unsteady case - long-time stability of the reduced model. Moreover, we provide an extensive list of literature references.},
	doi = {10.1007/978-3-319-02090-7_9},
	author = {Toni Lassila 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}
}
@article {2013,
	title = {A combination between the reduced basis method and the ANOVA expansion: On the computation of sensitivity indices},
	journal = {Comptes Rendus Mathematique. Volume 351, Issue 15-16, August 2013, Pages 593-598},
	year = {2013},
	publisher = {Elsevier},
	abstract = {<p>We consider a method to efficiently evaluate in a real-time context an output based on the numerical solution of a partial differential equation depending on a large number of parameters. We state a result allowing to improve the computational performance of a three-step RB-ANOVA-RB method. This is a combination of the reduced basis (RB) method and the analysis of variations (ANOVA) expansion, aiming at compressing the parameter space without affecting the accuracy of the output. The idea of this method is to compute a first (coarse) RB approximation of the output of interest involving all the parameter components, but with a large tolerance on the a posteriori error estimate; then, we evaluate the ANOVA expansion of the output and freeze the least important parameter components; finally, considering a restricted model involving just the retained parameter components, we compute a second (fine) RB approximation with a smaller tolerance on the a posteriori error estimate. The fine RB approximation entails lower computational costs than the coarse one, because of the reduction of parameter dimensionality. Our result provides a criterion to avoid the computation of those terms in the ANOVA expansion that are related to the interaction between parameters in the bilinear form, thus making the RB-ANOVA-RB procedure computationally more feasible.</p>},
	keywords = {Partial differential equations},
	doi = {10.1016/j.crma.2013.07.023},
	url = {http://hdl.handle.net/1963/7389},
	author = {Denis Devaud and Andrea Manzoni and Gianluigi Rozza}
}
@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}
}
@article {2012,
	title = {Boundary control and shape optimization for the robust design of bypass anastomoses under uncertainty},
	journal = {Mathematical Modelling and Numerical Analysis, in press, 2012-13},
	number = {SISSA;42/2012/M},
	year = {2012},
	publisher = {Cambridge University Press},
	abstract = {We review the optimal design of an arterial bypass graft following either a (i) boundary optimal control approach, or a (ii) shape optimization formulation. The main focus is quantifying and treating the uncertainty in the residual flow when the hosting artery is not completely occluded,\\r\\nfor which the worst-case in terms of recirculation e ffects is inferred to correspond to a strong ori fice flow through near-complete occlusion. A worst-case optimal control approach is applied to the steady\\r\\nNavier-Stokes equations in 2D to identify an anastomosis angle and a cu ed shape that are robust with respect to a possible range of residual \\r\\nflows. We also consider a reduced order modelling framework\\r\\nbased on reduced basis methods in order to make the robust design problem computationally feasible. The results obtained in 2D are compared with simulations in a 3D geometry but without model\\r\\nreduction or the robust framework.},
	keywords = {shape optimization},
	url = {http://hdl.handle.net/1963/6337},
	author = {Toni Lassila and Andrea Manzoni and Alfio Quarteroni and Gianluigi Rozza}
}
@inbook {2012,
	title = {Generalized reduced basis methods and n-width estimates for the approximation of the solution manifold of parametric PDEs},
	booktitle = {Springer, Indam Series, Vol. 4, 2012},
	number = {SISSA Preprint;44/2012/M},
	year = {2012},
	publisher = {Springer},
	organization = {Springer},
	abstract = {The set of solutions of a parameter-dependent linear partial di fferential equation with smooth coe fficients typically forms a compact manifold in a Hilbert space. In this paper we review the generalized reduced basis method as a fast computational tool for the uniform approximation of the solution manifold. We focus on operators showing an affi ne parametric dependence, expressed as a linear combination of parameter-independent operators through some smooth, parameter-dependent scalar functions. In the case that the parameter-dependent operator has a dominant term in its affi ne expansion, one can prove the existence of exponentially convergent uniform approximation spaces for the entire solution manifold. These spaces can be constructed without any assumptions on the parametric regularity of the manifold \\r\\nonly spatial regularity of the solutions is required. The exponential convergence rate is then inherited by the generalized reduced basis method. We provide a numerical example related to parametrized elliptic\\r\\nequations con rming the predicted convergence rates.},
	keywords = {solution manifold},
	url = {http://hdl.handle.net/1963/6340},
	author = {Toni Lassila and Andrea Manzoni and Alfio Quarteroni 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}
}
