00704nas a2200181   4500008004100000245009900041210006900140300001400209490000700223100001700230700002000247700002000267700002200287700002100309700002000330700002200350856015000372       2017                        eng d00aNumerical modeling of hemodynamics scenarios of patient-specific coronary artery bypass grafts0 aNumerical modeling of hemodynamics scenarios of patientspecific   a1373-13990 v161 aBallarin, F.1 aFaggiano, Elena1 aManzoni, Andrea1 aQuarteroni, Alfio1 aRozza, Gianluigi1 aIppolito, Sonia1 aScrofani, Roberto  uhttps://www.scopus.com/inward/record.uri?eid=2-s2.0-85015065851&doi=10.1007%2fs10237-017-0893-7&partnerID=40&md5=c388f20bd5de14187bad9ed7d9affbd001710nas a2200193   4500008004100000245011900041210006900160260001400229520106200243100001701305700002001322700002001342700002101362700002201383700002001405700002201425700001801447856005101465       2016                        en  d00aA fast virtual surgery platform for many scenarios haemodynamics of patient-specific coronary artery bypass grafts0 afast virtual surgery platform for many scenarios haemodynamics o  bSubmitted3 aA 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.1 aBallarin, F.1 aFaggiano, Elena1 aManzoni, Andrea1 aRozza, Gianluigi1 aQuarteroni, Alfio1 aIppolito, Sonia1 aScrofani, Roberto1 aAntona, Carlo  uhttp://urania.sissa.it/xmlui/handle/1963/3524001813nas a2200169   4500008004100000245015600041210006900197520118400266100001701450700002001467700002001487700002001507700002201527700002101549700002201570856005101592       2015                        en  d00aFast simulations of patient-specific haemodynamics of coronary artery bypass grafts based on a POD-Galerkin method and a vascular shape parametrization0 aFast simulations of patientspecific haemodynamics of coronary ar3 aIn 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–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.1 aBallarin, F.1 aFaggiano, Elena1 aIppolito, Sonia1 aManzoni, Andrea1 aQuarteroni, Alfio1 aRozza, Gianluigi1 aScrofani, Roberto  uhttp://urania.sissa.it/xmlui/handle/1963/3462301235nas a2200145   4500008004100000245010300041210006900144300001400213490000700227520066400234100002000898700002000918700002100938856013000959       2015                        eng d00aReduced basis approximation of parametrized optimal flow control problems for the Stokes equations0 aReduced basis approximation of parametrized optimal flow control  a319–3360 v693 a<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>1 aNegri, Federico1 aManzoni, Andrea1 aRozza, Gianluigi  uhttps://www.math.sissa.it/publication/reduced-basis-approximation-parametrized-optimal-flow-control-problems-stokes-equations02449nas a2200121   4500008004100000245012900041210006900170520189900239100002002138700002502158700001702183856012702200       2015                        en  d00aReduced Basis Isogeometric Methods (RB-IGA) for the real-time simulation of potential flows about parametrized NACA airfoils0 aReduced Basis Isogeometric Methods RBIGA for the realtime simula3 aWe 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.1 aManzoni, Andrea1 aSalmoiraghi, Filippo1 aHeltai, Luca  uhttps://www.math.sissa.it/publication/reduced-basis-isogeometric-methods-rb-iga-real-time-simulation-potential-flows-about00682nas a2200145   4500008004100000245009900041210006900140260001000209520018600219100001700405700002000422700002200442700002100464856005100485       2015                        en  d00aSupremizer stabilization of POD-Galerkin approximation of parametrized Navier-Stokes equations0 aSupremizer stabilization of PODGalerkin approximation of paramet  bWiley3 aIn this work, we present a stable proper orthogonal decomposition–Galerkin approximation for parametrized steady incompressible Navier–Stokes equations with low Reynolds number.1 aBallarin, F.1 aManzoni, Andrea1 aQuarteroni, Alfio1 aRozza, Gianluigi  uhttp://urania.sissa.it/xmlui/handle/1963/3470101997nas a2200109   4500008004100000245014300041210006900184520134000253653014901593100002001742856012501762       2014                        en  d00aAn efficient computational framework for reduced basis approximation and a posteriori error estimation of parametrized Navier-Stokes flows0 aefficient computational framework for reduced basis approximatio3 aWe 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.10aReduced Basis Method, parametrized Navier-Stokes equations, steady incompressible fluids, a posteriori error estimation, approximation stability1 aManzoni, Andrea  uhttps://www.math.sissa.it/publication/efficient-computational-framework-reduced-basis-approximation-and-posteriori-error01650nas a2200145   4500008004100000245007300041210006900114260001300183520112000196100001801316700002001334700002201354700002101376856010701397       2014                        en  d00aModel Order Reduction in Fluid Dynamics: Challenges and Perspectives0 aModel Order Reduction in Fluid Dynamics Challenges and Perspecti  bSpringer3 aThis 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.1 aLassila, Toni1 aManzoni, Andrea1 aQuarteroni, Alfio1 aRozza, Gianluigi  uhttps://www.math.sissa.it/publication/model-order-reduction-fluid-dynamics-challenges-and-perspectives01619nas a2200145   4500008004100000245010700041210006900148260001300217520111600230100001701346700002001363700002101383700001801404856005101422       2014                        en  d00aShape Optimization by Free-Form Deformation: Existence Results and Numerical Solution for Stokes Flows0 aShape Optimization by FreeForm Deformation Existence Results and  bSpringer3 aShape 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.1 aBallarin, F.1 aManzoni, Andrea1 aRozza, Gianluigi1 aSalsa, Sandro  uhttp://urania.sissa.it/xmlui/handle/1963/3469801890nas a2200145   4500008004100000245011800041210006900159260001300228520137300241653003501614100001801649700002001667700002101687856003601708       2013                        en  d00aA combination between the reduced basis method and the ANOVA expansion: On the computation of sensitivity indices0 acombination between the reduced basis method and the ANOVA expan  bElsevier3 a<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>10aPartial differential equations1 aDevaud, Denis1 aManzoni, Andrea1 aRozza, Gianluigi  uhttp://hdl.handle.net/1963/738902183nas a2200145   4500008004100000245015300041210006900194260001300263520163200276653003401908100002101942700001801963700002001981856003602001       2013                        en  d00aReduced basis approximation and a posteriori error estimation for Stokes  flows in parametrized geometries: roles of the inf-sup stability constants0 aReduced basis approximation and a posteriori error estimation fo  bSpringer3 aIn 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.10aparametrized Stokes equations1 aRozza, Gianluigi1 aHuynh, Phuong1 aManzoni, Andrea  uhttp://hdl.handle.net/1963/633901701nas a2200157   4500008004100000245007600041210006900117300001800186490000700204520113900211100002001350700002101370700002001391700002201411856011001433       2013                        eng d00aReduced basis method for parametrized elliptic optimal control problems0 aReduced basis method for parametrized elliptic optimal control p  aA2316–A23400 v353 aWe 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.1 aNegri, Federico1 aRozza, Gianluigi1 aManzoni, Andrea1 aQuarteroni, Alfio  uhttps://www.math.sissa.it/publication/reduced-basis-method-parametrized-elliptic-optimal-control-problems00548nas a2200133   4500008004100000245009200041210006900133260001000202100001800212700002000230700002200250700002100272856012100293       2013                        en  d00aA Reduced Computational and Geometrical Framework for Inverse Problems in Haemodynamics0 aReduced Computational and Geometrical Framework for Inverse Prob  bSISSA1 aLassila, Toni1 aManzoni, Andrea1 aQuarteroni, Alfio1 aRozza, Gianluigi  uhttps://www.math.sissa.it/publication/reduced-computational-and-geometrical-framework-inverse-problems-haemodynamics00568nas a2200133   4500008004100000245010500041210006900146260001000215100001800225700002000243700002200263700002100285856012800306       2013                        en  d00aA reduced-order strategy for solving inverse Bayesian identification problems in physiological flows0 areducedorder strategy for solving inverse Bayesian identificatio  bSISSA1 aLassila, Toni1 aManzoni, Andrea1 aQuarteroni, Alfio1 aRozza, Gianluigi  uhttps://www.math.sissa.it/publication/reduced-order-strategy-solving-inverse-bayesian-identification-problems-physiological00490nas a2200121   4500008004100000245007900041210006900120260001000189100001800199700002000217700002100237856011000258       2013                        en  d00aReduction Strategies for Shape Dependent Inverse Problems in Haemodynamics0 aReduction Strategies for Shape Dependent Inverse Problems in Hae  bSISSA1 aLassila, Toni1 aManzoni, Andrea1 aRozza, Gianluigi  uhttps://www.math.sissa.it/publication/reduction-strategies-shape-dependent-inverse-problems-haemodynamics01501nas a2200157   4500008004100000245010600041210006900147260003100216520095600247653002301203100001801226700002001244700002201264700002101286856003601307       2012                        en  d00aBoundary control and shape optimization for the robust design of bypass anastomoses under uncertainty0 aBoundary control and shape optimization for the robust design of  bCambridge University Press3 aWe 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.10ashape optimization1 aLassila, Toni1 aManzoni, Andrea1 aQuarteroni, Alfio1 aRozza, Gianluigi  uhttp://hdl.handle.net/1963/633701643nas a2200157   4500008004100000245012600041210006900167260001300236520109700249653002201346100001801368700002001386700002201406700002101428856003601449       2012                        en  d00aGeneralized reduced basis methods and n-width estimates for the approximation of the solution manifold of parametric PDEs0 aGeneralized reduced basis methods and nwidth estimates for the a  bSpringer3 aThe 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.10asolution manifold1 aLassila, Toni1 aManzoni, Andrea1 aQuarteroni, Alfio1 aRozza, Gianluigi  uhttp://hdl.handle.net/1963/634001651nas a2200133   4500008004100000245008400041210006900125520120500194653002101399100002101420700002001441700002001461856003601481       2012                        en  d00aReduction strategies for PDE-constrained oprimization problems in Haemodynamics0 aReduction strategies for PDEconstrained oprimization problems in3 aSolving 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.10ainverse problems1 aRozza, Gianluigi1 aManzoni, Andrea1 aNegri, Federico  uhttp://hdl.handle.net/1963/6338