In the reduced order modeling (ROM) framework, the solution of a parametric partial differential equation is approximated by combining the high-fidelity solutions of the problem at hand for several properly chosen configurations. Examples of the ROM application, in the naval field, can be found in [31, 24]. Mandatory ingredient for the ROM methods is the relation between the high-fidelity solutions and the parameters. Dealing with geometrical parameters, especially in the industrial context, this relation may be unknown and not trivial (simulations over hand morphed geometries) or very complex (high number of parameters or many nested morphing techniques). To overcome these scenarios, we propose in this contribution an efficient and complete data-driven framework involving ROM techniques for shape design and optimization, extending the pipeline presented in [7]. By applying the singular value decomposition (SVD) to the points coordinates defining the hull geometry –- assuming the topology is inaltered by the deformation –-, we are able to compute the optimal space which the deformed geometries belong to, hence using the modal coefficients as the new parameters we can reconstruct the parametric formulation of the domain. Finally the output of interest is approximated using the proper orthogonal decomposition with interpolation technique. To conclude, we apply this framework to a naval shape design problem where the bulbous bow is morphed to reduce the total resistance of the ship advancing in calm water.

%B VIII International Conference on Computational Methods in Marine Engineering %G eng %0 Conference Paper %B VIII International Conference on Computational Methods in Marine Engineering %D 2019 %T Efficient Reduction in Shape Parameter Space Dimension for Ship Propeller Blade Design %A Andrea Mola %A Marco Tezzele %A Mahmoud Gadalla %A Valdenazzi, Federica %A Grassi, Davide %A Padovan, Roberta %A Gianluigi Rozza %XIn this work, we present the results of a ship propeller design optimization campaign carried out in the framework of the research project PRELICA, funded by the Friuli Venezia Giulia regional government. The main idea of this work is to operate on a multidisciplinary level to identify propeller shapes that lead to reduced tip vortex-induced pressure and increased efficiency without altering the thrust. First, a specific tool for the bottom-up construction of parameterized propeller blade geometries has been developed. The algorithm proposed operates with a user defined number of arbitrary shaped or NACA airfoil sections, and employs arbitrary degree NURBS to represent the chord, pitch, skew and rake distribution as a function of the blade radial coordinate. The control points of such curves have been modified to generate, in a fully automated way, a family of blade geometries depending on as many as 20 shape parameters. Such geometries have then been used to carry out potential flow simulations with the Boundary Element Method based software PROCAL. Given the high number of parameters considered, such a preliminary stage allowed for a fast evaluation of the performance of several hundreds of shapes. In addition, the data obtained from the potential flow simulation allowed for the application of a parameter space reduction methodology based on active subspaces (AS) property, which suggested that the main propeller performance indices are, at a first but rather accurate approximation, only depending on a single parameter which is a linear combination of all the original geometric ones. AS analysis has also been used to carry out a constrained optimization exploiting response surface method in the reduced parameter space, and a sensitivity analysis based on such surrogate model. The few selected shapes were finally used to set up high fidelity RANS simulations and select an optimal shape.

%B VIII International Conference on Computational Methods in Marine Engineering %G eng %0 Conference Paper %B VIII International Conference on Computational Methods in Marine Engineering %D 2019 %T Shape optimization through proper orthogonal decomposition with interpolation and dynamic mode decomposition enhanced by active subspaces %A Marco Tezzele %A Nicola Demo %A Gianluigi Rozza %XWe propose a numerical pipeline for shape optimization in naval engineering involving two different non-intrusive reduced order method (ROM) techniques. Such methods are proper orthogonal decomposition with interpolation (PODI) and dynamic mode decomposition (DMD). The ROM proposed will be enhanced by active subspaces (AS) as a pre-processing tool that reduce the parameter space dimension and suggest better sampling of the input space. We will focus on geometrical parameters describing the perturbation of a reference bulbous bow through the free form deformation (FFD) technique. The ROM are based on a finite volume method (FV) to simulate the multi-phase incompressible flow around the deformed hulls. In previous works we studied the reduction of the parameter space in naval engineering through AS [38, 10] focusing on different parts of the hull. PODI and DMD have been employed for the study of fast and reliable shape optimization cycles on a bulbous bow in [9]. The novelty of this work is the simultaneous reduction of both the input parameter space and the output fields of interest. In particular AS will be trained computing the total drag resistance of a hull advancing in calm water and its gradients with respect to the input parameters. DMD will improve the performance of each simulation of the campaign using only few snapshots of the solution fields in order to predict the regime state of the system. Finally PODI will interpolate the coefficients of the POD decomposition of the output fields for a fast approximation of all the fields at new untried parameters given by the optimization algorithm. This will result in a non-intrusive data-driven numerical optimization pipeline completely independent with respect to the full order solver used and it can be easily incorporated into existing numerical pipelines, from the reference CAD to the optimal shape.

%B VIII International Conference on Computational Methods in Marine Engineering %G eng %0 Journal Article %J Journal of Scientific Computing %D 2018 %T Certified Reduced Basis Approximation for the Coupling of Viscous and Inviscid Parametrized Flow Models %A Immanuel Martini %A Bernard Haasdonk %A Gianluigi Rozza %B Journal of Scientific Computing %V 74 %P 197-219 %G eng %U https://www.scopus.com/inward/record.uri?eid=2-s2.0-85017156114&doi=10.1007%2fs10915-017-0430-y&partnerID=40&md5=023ef0bb95713f4442d1fa374c92a964 %R 10.1007/s10915-017-0430-y %0 Book Section %B Mathematical and Numerical Modeling of the Cardiovascular System and Applications %D 2018 %T Combined parameter and model reduction of cardiovascular problems by means of active subspaces and POD-Galerkin methods %A Marco Tezzele %A Francesco Ballarin %A Gianluigi Rozza %B Mathematical and Numerical Modeling of the Cardiovascular System and Applications %I Springer %P 185–207 %G eng %0 Book Section %B Cardiovascular Mechanics %D 2018 %T Computational methods in cardiovascular mechanics %A Auricchio, Ferdinando %A Conti, Michele %A Lefieux, Adrian %A Morganti, Simone %A Alessandro Reali %A Gianluigi Rozza %A Veneziani, Alessandro %E Michel F. Labrosse %XThe introduction of computational models in cardiovascular sciences has been progressively bringing new and unique tools for the investigation of the physiopathology. Together with the dramatic improvement of imaging and measuring devices on one side, and of computational architectures on the other one, mathematical and numerical models have provided a new, clearly noninvasive, approach for understanding not only basic mechanisms but also patient-specific conditions, and for supporting the design and the development of new therapeutic options. The terminology in silico is, nowadays, commonly accepted for indicating this new source of knowledge added to traditional in vitro and in vivo investigations. The advantages of in silico methodologies are basically the low cost in terms of infrastructures and facilities, the reduced invasiveness and, in general, the intrinsic predictive capabilities based on the use of mathematical models. The disadvantages are generally identified in the distance between the real cases and their virtual counterpart required by the conceptual modeling that can be detrimental for the reliability of numerical simulations.

%B Cardiovascular Mechanics %I CRC Press %P 54 %G eng %U https://www.taylorfrancis.com/books/e/9781315280288/chapters/10.1201%2Fb21917-5 %& Computational methods in cardiovascular mechanics %0 Journal Article %J Advanced Modeling and Simulation in Engineering Sciences %D 2018 %T Dimension reduction in heterogeneous parametric spaces with application to naval engineering shape design problems %A Marco Tezzele %A Filippo Salmoiraghi %A Andrea Mola %A Gianluigi Rozza %XWe present the results of the first application in the naval architecture field of a methodology based on active subspaces properties for parameters space reduction. The physical problem considered is the one of the simulation of the hydrodynamic flow past the hull of a ship advancing in calm water. Such problem is extremely relevant at the preliminary stages of the ship design, when several flow simulations are typically carried out by the engineers to assess the dependence of the hull total resistance on the geometrical parameters of the hull, and others related with flows and hull properties. Given the high number of geometric and physical parameters which might affect the total ship drag, the main idea of this work is to employ the active subspaces properties to identify possible lower dimensional structures in the parameter space. Thus, a fully automated procedure has been implemented to produce several small shape perturbations of an original hull CAD geometry, in order to exploit the resulting shapes to run high fidelity flow simulations with different structural and physical parameters as well, and then collect data for the active subspaces analysis. The free form deformation procedure used to morph the hull shapes, the high fidelity solver based on potential flow theory with fully nonlinear free surface treatment, and the active subspaces analysis tool employed in this work have all been developed and integrated within SISSA mathLab as open source tools. The contribution will also discuss several details of the implementation of such tools, as well as the results of their application to the selected target engineering problem.

%B Advanced Modeling and Simulation in Engineering Sciences %V 5 %P 25 %8 Sep %G eng %R 10.1186/s40323-018-0118-3 %0 Conference Proceedings %B The 28th International Ocean and Polar Engineering Conference %D 2018 %T An efficient shape parametrisation by free-form deformation enhanced by active subspace for hull hydrodynamic ship design problems in open source environment %A Nicola Demo %A Marco Tezzele %A Andrea Mola %A Gianluigi Rozza %K Active subspaces %K Boundary element method %K Dynamic mode decomposition %K Fluid structure interaction %K Free form deformation %K Fully nonlinear potential %K Numerical towing tank %X In this contribution, we present the results of the application of a parameter space reduction methodology based on active subspaces to the hull hydrodynamic design problem. Several parametric deformations of an initial hull shape are considered to assess the influence of the shape parameters considered on the hull total drag. The hull resistance is typically computed by means of numerical simulations of the hydrodynamic flow past the ship. Given the high number of parameters involved - which might result in a high number of time consuming hydrodynamic simulations - assessing whether the parameters space can be reduced would lead to considerable computational cost reduction. Thus, the main idea of this work is to employ the active subspaces to identify possible lower dimensional structures in the parameter space, or to verify the parameter distribution in the position of the control points. To this end, a fully automated procedure has been implemented to produce several small shape perturbations of an original hull CAD geometry which are then used to carry out high-fidelity flow simulations and collect data for the active subspaces analysis. To achieve full automation of the open source pipeline described, both the free form deformation methodology employed for the hull perturbations and the solver based on unsteady potential flow theory, with fully nonlinear free surface treatment, are directly interfaced with CAD data structures and operate using IGES vendor-neutral file formats as input files. The computational cost of the fluid dynamic simulations is further reduced through the application of dynamic mode decomposition to reconstruct the steady state total drag value given only few initial snapshots of the simulation. The active subspaces analysis is here applied to the geometry of the DTMB-5415 naval combatant hull, which is which is a common benchmark in ship hydrodynamics simulations. %B The 28th International Ocean and Polar Engineering Conference %I International Society of Offshore and Polar Engineers %C Sapporo, Japan %G eng %U https://www.onepetro.org/conference-paper/ISOPE-I-18-481 %0 Generic %D 2018 %T The Effort of Increasing Reynolds Number in Projection-Based Reduced Order Methods: from Laminar to Turbulent Flows %A Saddam Hijazi %A Shafqat Ali %A Giovanni Stabile %A Francesco Ballarin %A Gianluigi Rozza %G eng %0 Journal Article %J The Journal of Open Source Software %D 2018 %T EZyRB: Easy Reduced Basis method %A Nicola Demo %A Marco Tezzele %A Gianluigi Rozza %B The Journal of Open Source Software %V 3 %P 661 %G eng %U https://joss.theoj.org/papers/10.21105/joss.00661 %R 10.21105/joss.00661 %0 Journal Article %J Computers & Fluids %D 2018 %T Finite volume POD-Galerkin stabilised reduced order methods for the parametrised incompressible Navier-Stokes equations %A Giovanni Stabile %A Gianluigi Rozza %B Computers & Fluids %I Elsevier {BV} %8 feb %G eng %U https://doi.org/10.1016/j.compfluid.2018.01.035 %R 10.1016/j.compfluid.2018.01.035 %0 Journal Article %J International Journal of Computational Fluid Dynamics %D 2018 %T Free-form deformation, mesh morphing and reduced-order methods: enablers for efficient aerodynamic shape optimisation %A Filippo Salmoiraghi %A Scardigli, Angela %A Telib, Haysam %A Gianluigi Rozza %XIn this work, we provide an integrated pipeline for the model-order reduction of turbulent flows around parametrised geometries in aerodynamics. In particular, free-form deformation is applied for geometry parametrisation, whereas two different reduced-order models based on proper orthogonal decomposition (POD) are employed in order to speed-up the full-order simulations: the first method exploits POD with interpolation, while the second one is based on domain decomposition. For the sampling of the parameter space, we adopt a Greedy strategy coupled with Constrained Centroidal Voronoi Tessellations, in order to guarantee a good compromise between space exploration and exploitation. The proposed framework is tested on an industrially relevant application, i.e. the front-bumper morphing of the DrivAer car model, using the finite-volume method for the full-order resolution of the Reynolds-Averaged Navier–Stokes equations.

%B International Journal of Computational Fluid Dynamics %I Taylor & Francis %V 32 %P 233-247 %G eng %R 10.1080/10618562.2018.1514115 %0 Journal Article %J ArXiv e-prints %D 2018 %T A Localized Reduced-Order Modeling Approach for PDEs with Bifurcating Solutions %A Martin W. Hess %A A. Alla %A Annalisa Quaini %A Gianluigi Rozza %A M. Gunzburger %K Mathematics - Numerical Analysis %B ArXiv e-prints %G eng %0 Conference Paper %B Technology and Science for the Ships of the Future: Proceedings of NAV 2018: 19th International Conference on Ship & Maritime Research %D 2018 %T Model Order Reduction by means of Active Subspaces and Dynamic Mode Decomposition for Parametric Hull Shape Design Hydrodynamics %A Marco Tezzele %A Nicola Demo %A Mahmoud Gadalla %A Andrea Mola %A Gianluigi Rozza %X We present the results of the application of a parameter space reduction methodology based on active subspaces (AS) to the hull hydrodynamic design problem. Several parametric deformations of an initial hull shape are considered to assess the influence of the shape parameters on the hull wave resistance. Such problem is relevant at the preliminary stages of the ship design, when several flow simulations are carried out by the engineers to establish a certain sensibility with respect to the parameters, which might result in a high number of time consuming hydrodynamic simulations. The main idea of this work is to employ the AS to identify possible lower dimensional structures in the parameter space. The complete pipeline involves the use of free form deformation to parametrize and deform the hull shape, the full order solver based on unsteady potential flow theory with fully nonlinear free surface treatment directly interfaced with CAD, the use of dynamic mode decomposition to reconstruct the final steady state given only few snapshots of the simulation, and the reduction of the parameter space by AS, and shared subspace. Response surface method is used to minimize the total drag. %B Technology and Science for the Ships of the Future: Proceedings of NAV 2018: 19th International Conference on Ship & Maritime Research %I IOS Press %C Trieste, Italy %G eng %U http://ebooks.iospress.nl/publication/49270 %R 10.3233/978-1-61499-870-9-569 %0 Journal Article %J SIAM Journal on Scientific Computing %D 2018 %T Model Reduction for Parametrized Optimal Control Problems in Environmental Marine Sciences and Engineering %A Maria Strazzullo %A Francesco Ballarin %A Mosetti, R. %A Gianluigi Rozza %B SIAM Journal on Scientific Computing %V 40 %P B1055-B1079 %G eng %U https://doi.org/10.1137/17M1150591 %R 10.1137/17M1150591 %0 Journal Article %J The Journal of Open Source Software %D 2018 %T PyDMD: Python Dynamic Mode Decomposition %A Nicola Demo %A Marco Tezzele %A Gianluigi Rozza %B The Journal of Open Source Software %V 3 %P 530 %G eng %U https://joss.theoj.org/papers/734e4326edd5062c6e8ee98d03df9e1d %R 10.21105/joss.00530 %0 Generic %D 2018 %T A Reduced Basis approach for PDEs on parametrized geometries based on the Shifted Boundary Finite Element Method and application to fluid dynamics %A Efthymios N. Karatzas %A Giovanni Stabile %A Leo Nouveau %A Guglielmo Scovazzi %A Gianluigi Rozza %G eng %0 Generic %D 2018 %T A Reduced Order Approach for the Embedded Shifted Boundary FEM and a Heat Exchange System on Parametrized Geometries %A Efthymios N. Karatzas %A Giovanni Stabile %A N. Atallah %A Guglielmo Scovazzi %A Gianluigi Rozza %G eng %0 Conference Paper %B Technology and Science for the Ships of the Future: Proceedings of NAV 2018: 19th International Conference on Ship & Maritime Research %D 2018 %T Shape Optimization by means of Proper Orthogonal Decomposition and Dynamic Mode Decomposition %A Nicola Demo %A Marco Tezzele %A Gianluca Gustin %A Gianpiero Lavini %A Gianluigi Rozza %X Shape optimization is a challenging task in many engineering fields, since the numerical solutions of parametric system may be computationally expensive. This work presents a novel optimization procedure based on reduced order modeling, applied to a naval hull design problem. The advantage introduced by this method is that the solution for a specific parameter can be expressed as the combination of few numerical solutions computed at properly chosen parametric points. The reduced model is built using the proper orthogonal decomposition with interpolation (PODI) method. We use the free form deformation (FFD) for an automated perturbation of the shape, and the finite volume method to simulate the multiphase incompressible flow around the deformed hulls. Further computational reduction is done by the dynamic mode decomposition (DMD) technique: from few high dimensional snapshots, the system evolution is reconstructed and the final state of the simulation is faithfully approximated. Finally the global optimization algorithm iterates over the reduced space: the approximated drag and lift coefficients are projected to the hull surface, hence the resistance is evaluated for the new hulls until the convergence to the optimal shape is achieved. We will present the results obtained applying the described procedure to a typical Fincantieri cruise ship. %B Technology and Science for the Ships of the Future: Proceedings of NAV 2018: 19th International Conference on Ship & Maritime Research %I IOS Press %C Trieste, Italy %G eng %U http://ebooks.iospress.nl/publication/49229 %& 212 %R 10.3233/978-1-61499-870-9-212 %0 Journal Article %J Communication in Applied Industrial Mathematics %D 2017 %T Advances in Reduced order modelling for CFD: vortex shedding around a circular cylinder using a POD-Galerkin method %A Giovanni Stabile %A Saddam Hijazi %A Stefano Lorenzi %A Andrea Mola %A Gianluigi Rozza %K finite volume, CFD %K Reduced order methods %XVortex shedding around circular cylinders is a well known and studied phenomenon that appears in many engineering fields. In this work a Reduced Order Model (ROM) of the incompressible flow around a circular cylinder, built performing a Galerkin projection of the governing equations onto a lower dimensional space is presented. The reduced basis space is generated using a Proper Orthogonal Decomposition (POD) approach. In particular the focus is into (i) the correct reproduction of the pressure field, that in case of the vortex shedding phenomenon, is of primary importance for the calculation of the drag and lift coefficients; (ii) for this purpose the projection of the Governing equations (momentum equation and Poisson equation for pressure) is performed onto different reduced basis space for velocity and pressure, respectively; (iii) all the relevant modifications necessary to adapt standard finite element POD-Galerkin methods to a finite volume framework are presented. The accuracy of the reduced order model is assessed against full order results.

%B Communication in Applied Industrial Mathematics %G eng %U https://arxiv.org/abs/1701.03424 %9 reviewed %0 Journal Article %J Journal of Scientific Computing %D 2017 %T On the Application of Reduced Basis Methods to Bifurcation Problems in Incompressible Fluid Dynamics %A Giuseppe Pitton %A Gianluigi Rozza %XIn this paper we apply a reduced basis framework for the computation of flow bifurcation (and stability) problems in fluid dynamics. The proposed method aims at reducing the complexity and the computational time required for the construction of bifurcation and stability diagrams. The method is quite general since it can in principle be specialized to a wide class of nonlinear problems, but in this work we focus on an application in incompressible fluid dynamics at low Reynolds numbers. The validation of the reduced order model with the full order computation for a benchmark cavity flow problem is promising.

%B Journal of Scientific Computing %G eng %R 10.1007/s10915-017-0419-6 %0 Book Section %B Spectral and High Order Methods for Partial Differential Equations %D 2017 %T Certified Reduced Basis Method for Affinely Parametric Isogeometric Analysis NURBS Approximation %A Denis Devaud %A Gianluigi Rozza %XIn this work we apply reduced basis methods for parametric PDEs to an isogeometric formulation based on

NURBS. The motivation for this work is an integrated and complete work pipeline from CAD to parametrization

of domain geometry, then from full order to certified reduced basis solution. IsoGeometric Analysis

(IGA) is a growing research theme in scientic computing and computational mechanics, as well as reduced

basis methods for parametric PDEs. Their combination enhances the solution of some class of problems,

especially the ones characterized by parametrized geometries we introduced in this work. For a general

overview on Reduced Basis (RB) methods we recall [7, 15] and on IGA [3]. This work wants to demonstrate

that it is also possible for some class of problems to deal with ane geometrical parametrization combined

with a NURBS IGA formulation. This is what this work brings as original ingredients with respect to other

works dealing with reduced order methods and IGA (set in a non-affine formulation, and using a POD [2]

sampling without certication: see for example for potential flows [12] and for Stokes flows [17]). In this work

we show a certication of accuracy and a complete integration between IGA formulation and parametric

certified greedy RB formulation. Section 2 recalls the abstract setting for parametrized PDEs, Section 3

recalls IGA setting, Section 4 deals with RB formulation, and Section 5 illustrates two numerical examples in heat transfer with different parametrization.

%B Spectral and High Order Methods for Partial Differential Equations %7 Bittencourt, Dumont, Hesthaven. (Eds). %I Springer %C Heildeberg %V 119 %@ 978-3-319-65869-8 %G eng %0 Journal Article %J SIAM Journal on Numerical Analysis %D 2017 %T On a certified smagorinsky reduced basis turbulence model %A Rebollo, T.C. %A E.D. Ávila %A Marmol, M.G. %A Francesco Ballarin %A Gianluigi Rozza %B SIAM Journal on Numerical Analysis %V 55 %P 3047-3067 %G eng %U https://www.scopus.com/inward/record.uri?eid=2-s2.0-85039928218&doi=10.1137%2f17M1118233&partnerID=40&md5=221d9cd2bcc74121fcef93efd9d3d76c %R 10.1137/17M1118233 %0 Journal Article %J Journal of Computational Physics %D 2017 %T Computational reduction strategies for the detection of steady bifurcations in incompressible fluid-dynamics: Applications to Coanda effect in cardiology %A Giuseppe Pitton %A Annalisa Quaini %A Gianluigi Rozza %K Parametrized Navier–Stokes equations %K Reduced basis method %K Stability of flows %K Symmetry breaking bifurcation %XWe focus on reducing the computational costs associated with the hydrodynamic stability of solutions of the incompressible Navier–Stokes equations for a Newtonian and viscous fluid in contraction–expansion channels. In particular, we are interested in studying steady bifurcations, occurring when non-unique stable solutions appear as physical and/or geometric control parameters are varied. The formulation of the stability problem requires solving an eigenvalue problem for a partial differential operator. An alternative to this approach is the direct simulation of the flow to characterize the asymptotic behavior of the solution. Both approaches can be extremely expensive in terms of computational time. We propose to apply Reduced Order Modeling (ROM) techniques to reduce the demanding computational costs associated with the detection of a type of steady bifurcations in fluid dynamics. The application that motivated the present study is the onset of asymmetries (i.e., symmetry breaking bifurcation) in blood flow through a regurgitant mitral valve, depending on the Reynolds number and the regurgitant mitral valve orifice shape.

This chapter presents an overview of model order reduction – a new paradigm in the field of simulation-based engineering sciences, and one that can tackle the challenges and leverage the opportunities of modern ICT technologies. Despite the impressive progress attained by simulation capabilities and techniques, a number of challenging problems remain intractable. These problems are of different nature, but are common to many branches of science and engineering. Among them are those related to high-dimensional problems, problems involving very different time scales, models defined in degenerate domains with at least one of the characteristic dimensions much smaller than the others, model requiring real-time simulation, and parametric models. All these problems represent a challenge for standard mesh-based discretization techniques; yet the ability to solve these problems efficiently would open unexplored routes for real-time simulation, inverse analysis, uncertainty quantification and propagation, real-time optimization, and simulation-based control – critical needs in many branches of science and engineering. Model order reduction offers new simulation alternatives by circumventing, or at least alleviating, otherwise intractable computational challenges. In the present chapter, we revisit three of these model reduction techniques: proper orthogonal decomposition, proper generalized decomposition, and reduced basis methodologies.} preprint = {http://preprints.sissa.it/xmlui/bitstream/handle/1963/35194/ECM_MOR.pdf?sequence=1&isAllowed=y

%B Encyclopedia of Computational Mechanics Second Edition %I John Wiley & Sons %P 1-36 %G eng %& Model Reduction Methods %R 10.1002/9781119176817.ecm2110 %0 Journal Article %J Biomechanics and Modeling in Mechanobiology %D 2017 %T Numerical modeling of hemodynamics scenarios of patient-specific coronary artery bypass grafts %A Francesco Ballarin %A Elena Faggiano %A Andrea Manzoni %A Alfio Quarteroni %A Gianluigi Rozza %A Sonia Ippolito %A Roberto Scrofani %B Biomechanics and Modeling in Mechanobiology %V 16 %P 1373-1399 %G eng %U https://www.scopus.com/inward/record.uri?eid=2-s2.0-85015065851&doi=10.1007%2fs10237-017-0893-7&partnerID=40&md5=c388f20bd5de14187bad9ed7d9affbd0 %R 10.1007/s10237-017-0893-7 %0 Journal Article %J Communications in Applied and Industrial Mathematics %D 2017 %T POD-Galerkin reduced order methods for CFD using Finite Volume Discretisation: vortex shedding around a circular cylinder %A Giovanni Stabile %A Saddam Hijazi %A Andrea Mola %A Stefano Lorenzi %A Gianluigi Rozza %B Communications in Applied and Industrial Mathematics %I Walter de Gruyter {GmbH} %V 8 %P 210–236 %8 dec %G eng %U https://doi.org/10.1515/caim-2017-0011 %R 10.1515/caim-2017-0011 %0 Journal Article %J SIAM/ASA Journal on Uncertainty Quantification %D 2017 %T Reduced Basis Methods for Uncertainty Quantification %A Peng Chen %A Alfio Quarteroni %A Gianluigi Rozza %XIn this work we review a reduced basis method for the solution of uncertainty quantification problems. Based on the basic setting of an elliptic partial differential equation with random input, we introduce the key ingredients of the reduced basis method, including proper orthogonal decomposition and greedy algorithms for the construction of the reduced basis functions, a priori and a posteriori error estimates for the reduced basis approximations, as well as its computational advantages and weaknesses in comparison with a stochastic collocation method [I. Babuška, F. Nobile, and R. Tempone, *SIAM Rev.*, 52 (2010), pp. 317--355]. We demonstrate its computational efficiency and accuracy for a benchmark problem with parameters ranging from a few to a few hundred dimensions. Generalizations to more complex models and applications to uncertainty quantification problems in risk prediction, evaluation of statistical moments, Bayesian inversion, and optimal control under uncertainty are also presented to illustrate how to use the reduced basis method in practice. Further challenges, advancements, and research opportunities are outlined.

Read More: http://epubs.siam.org/doi/abs/10.1137/151004550

POD–Galerkin reduced-order models (ROMs) for fluid-structure interaction problems (incompressible fluid and thin structure) are proposed in this paper. Both the high-fidelity and reduced-order methods are based on a Chorin-Temam operator-splitting approach. Two different reduced-order methods are proposed, which differ on velocity continuity condition, imposed weakly or strongly, respectively. The resulting ROMs are tested and compared on a representative haemodynamics test case characterized by wave propagation, in order to assess the capabilities of the proposed strategies.

%B Model Reduction of Parametrized Systems %I Springer International Publishing %P 149–167 %G eng %& Reduced-order semi-implicit schemes for fluid-structure interaction problems %R 10.1007/978-3-319-58786-8_10 %0 Book Section %B Numerical Mathematics and Advanced Applications - ENUMATH 2017 %D 2017 %T A Spectral Element Reduced Basis Method in Parametric CFD %A Martin W. Hess %A Gianluigi Rozza %XWe consider the Navier-Stokes equations in a channel with varying Reynolds numbers. The model is discretized with high-order spectral element ansatz functions, resulting in 14 259 degrees of freedom. The steady-state snapshot solu- tions define a reduced order space, which allows to accurately evaluate the steady- state solutions for varying Reynolds number with a reduced order model within a fixed-point iteration. In particular, we compare different aspects of implementing the reduced order model with respect to the use of a spectral element discretization. It is shown, how a multilevel static condensation in the pressure and velocity boundary degrees of freedom can be combined with a reduced order modelling approach to enhance computational times in parametric many-query scenarios.

%B Numerical Mathematics and Advanced Applications - ENUMATH 2017 %I Springer %V 126 %G eng %& A Spectral Element Reduced Basis Method in Parametric CFD %0 Conference Paper %B Numerical Mathematics and Advanced Applications - ENUMATH 2017, Springer, in press %D 2017 %T A Spectral Element Reduced Basis Method in Parametric CFD %A Martin W. Hess %A Gianluigi Rozza %B Numerical Mathematics and Advanced Applications - ENUMATH 2017, Springer, in press %G eng %0 Conference Paper %B Proceedings of the ECCOMAS Congress 2016, VII European Conference on Computational Methods in Applied Sciences and Engineering, %D 2016 %T Advances in geometrical parametrization and reduced order models and methods for computational fluid dynamics problems in applied sciences and engineering: overview and perspectives %A Filippo Salmoiraghi %A Francesco Ballarin %A Giovanni Corsi %A Andrea Mola %A Marco Tezzele %A Gianluigi Rozza %E Papadrakakis, M. %E Papadopoulos, V. %E Stefanou, G. %E Plevris, V. %XSeveral problems in applied sciences and engineering require reduction techniques in order to allow computational tools to be employed in the daily practice, especially in iterative procedures such as optimization or sensitivity analysis. Reduced order methods need to face increasingly complex problems in computational mechanics, especially into a multiphysics setting. Several issues should be faced: stability of the approximation, efficient treatment of nonlinearities, uniqueness or possible bifurcations of the state solutions, proper coupling between fields, as well as offline-online computing, computational savings and certification of errors as measure of accuracy. Moreover, efficient geometrical parametrization techniques should be devised to efficiently face shape optimization problems, as well as shape reconstruction and shape assimilation problems. A related aspect deals with the management of parametrized interfaces in multiphysics problems, such as fluid-structure interaction problems, and also a domain decomposition based approach for complex parametrized networks. We present some illustrative industrial and biomedical problems as examples of recent advances on methodological developments.

%B Proceedings of the ECCOMAS Congress 2016, VII European Conference on Computational Methods in Applied Sciences and Engineering, %I ECCOMAS %C Crete, Greece %8 06/2016 %G en %1 35466 %2 Mathematics %4 1 %# MAT/08 %$ Submitted by Gianluigi Rozza (grozza@sissa.it) on 2016-05-13T00:28:01Z No. of bitstreams: 1 eccomas2016_AROMA.pdf: 1846196 bytes, checksum: 9636e713df80de178d87fd2feff76f91 (MD5) %0 Report %D 2016 %T A fast virtual surgery platform for many scenarios haemodynamics of patient-specific coronary artery bypass grafts %A Francesco Ballarin %A Elena Faggiano %A Andrea Manzoni %A Gianluigi Rozza %A Alfio Quarteroni %A Sonia Ippolito %A Roberto Scrofani %A Carlo Antona %X 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. %I Submitted %G en %U http://urania.sissa.it/xmlui/handle/1963/35240 %1 35545 %2 Mathematics %4 1 %# MAT/08 %$ Submitted by Gianluigi Rozza (grozza@sissa.it) on 2016-10-14T23:07:40Z No. of bitstreams: 1 BMMB_SISSA_report.pdf: 16374062 bytes, checksum: 7ee82fd9d989ed91bf9cc721ae2114a0 (MD5) %0 Report %D 2016 %T Isogeometric analysis-based reduced order modelling for incompressible linear viscous flows in parametrized shapes %A Filippo Salmoiraghi %A Francesco Ballarin %A Luca Heltai %A Gianluigi Rozza %X In this work we provide a combination of isogeometric analysis with reduced order modelling techniques, based on proper orthogonal decomposition, to guarantee computational reduction for the numerical model, and with free-form deformation, for versatile geometrical parametrization. We apply it to computational fluid dynamics problems considering a Stokes flow model. The proposed reduced order model combines efficient shape deformation and accurate and stable velocity and pressure approximation for incompressible viscous flows, computed with a reduced order method. Efficient offine-online computational decomposition is guaranteed in view of repetitive calculations for parametric design and optimization problems. Numerical test cases show the efficiency and accuracy of the proposed reduced order model. %I Springer, AMOS Advanced Modelling and Simulation in Engineering Sciences %G en %U http://urania.sissa.it/xmlui/handle/1963/35199 %1 35493 %2 Mathematics %4 1 %# MAT/08 %$ Submitted by Gianluigi Rozza (grozza@sissa.it) on 2016-06-20T23:55:25Z No. of bitstreams: 1 MAIN_IGA_FFD_ROM.pdf: 1631282 bytes, checksum: b050e472f11943b95d8307915a50b6da (MD5) %0 Book Section %B Wiley Encyclopedia of Computational Mechanics, 2016 %D 2016 %T Model Order Reduction: a survey %A Francisco Chinesta %A Antonio Huerta %A Gianluigi Rozza %A Karen Willcox %B Wiley Encyclopedia of Computational Mechanics, 2016 %I Wiley %G en %U http://urania.sissa.it/xmlui/handle/1963/35194 %1 35470 %2 Mathematics %4 1 %# MAT/08 %$ Submitted by Gianluigi Rozza (grozza@sissa.it) on 2016-05-27T00:41:12Z No. of bitstreams: 1 ECM_MOR.pdf: 746542 bytes, checksum: 93d6252fe8a175c4378b96bd4192712c (MD5) %0 Journal Article %J Annals of Nuclear Energy, 87, 2 (2016): pp. 198-208 %D 2016 %T A multi-physics reduced order model for the analysis of Lead Fast Reactor single channel %A Alberto Sartori %A Antonio Cammi %A Lelio Luzzi %A Gianluigi Rozza %X In this work, a Reduced Basis method, with basis functions sampled by a Proper Orthogonal Decomposition technique, has been employed to develop a reduced order model of a multi-physics parametrized Lead-cooled Fast Reactor single-channel. Being the first time that a reduced order model is developed in this context, the work focused on a methodological approach and the coupling between the neutronics and the heat transfer, where the thermal feedbacks on neutronics are explicitly taken into account, in time-invariant settings. In order to address the potential of such approach, two different kinds of varying parameters have been considered, namely one related to a geometric quantity (i.e., the inner radius of the fuel pellet) and one related to a physical quantity (i.e., the inlet lead velocity). The capabilities of the presented reduced order model (ROM) have been tested and compared with a high-fidelity finite element model (upon which the ROM has been constructed) on different aspects. In particular, the comparison focused on the system reactivity prediction (with and without thermal feedbacks on neutronics), the neutron flux and temperature field reconstruction, and on the computational time. The outcomes provided by the reduced order model are in good agreement with the high-fidelity finite element ones, and a computational speed-up of at least three orders of magnitude is achieved as well. %B Annals of Nuclear Energy, 87, 2 (2016): pp. 198-208 %I Elsevier %V 87 %P 208 %G en %U http://urania.sissa.it/xmlui/handle/1963/35191 %1 35471 %2 Mathematics %4 1 %# MAT/08 %$ Approved for entry into archive by Maria Pia Calandra (calapia@sissa.it) on 2016-06-16T11:08:05Z (GMT) No. of bitstreams: 0 %& 198 %R doi:10.1016/j.anucene.2015.09.002 %0 Journal Article %D 2016 %T POD-Galerkin Method for Finite Volume Approximation of Navier-Stokes and RANS Equations %A Stefano Lorenzi %A Antonio Cammi %A Lelio Luzzi %A Gianluigi Rozza %X Numerical simulation of fluid flows requires important computational efforts but it is essential in engineering applications. Reduced Order Model (ROM) can be employed whenever fast simulations are required, or in general, whenever a trade-off between computational cost and solution accuracy is a preeminent issue as in process optimization and control. In this work, the efforts have been put to develop a ROM for Computational Fluid Dynamics (CFD) application based on Finite Volume approximation, starting from the results available in turbulent Reynold-Averaged Navier Stokes simulations in order to enlarge the application field of Proper Orthogonal Decomposition – Reduced Order Model (POD – ROM) technique to more industrial fields. The approach is tested in the classic benchmark of the numerical simulation of the 2D lid-driven cavity. In particular, two simulations at Re = 103 and Re = 105 have been considered in order to assess both a laminar and turbulent case. Some quantities have been compared with the Full Order Model in order to assess the performance of the proposed ROM procedure i.e., the kinetic energy of the system and the reconstructed quantities of interest (velocity, pressure and turbulent viscosity). In addition, for the laminar case, the comparison between the ROM steady-state solution and the data available in literature has been presented. The results have turned out to be very satisfactory both for the accuracy and the computational times. As a major outcome, the approach turns out not to be affected by the energy blow up issue characterizing the results obtained by classic turbulent POD-Galerkin methods. %I Computer Methods in Applied Mechanics and Engineering, Elsevier %G en %1 35502 %2 Mathematics %4 1 %# MAT/08 %$ Submitted by Gianluigi Rozza (grozza@sissa.it) on 2016-08-06T00:39:18Z No. of bitstreams: 1 Manuscript_second_rev.pdf: 6749813 bytes, checksum: 08833a58f5485216e08c0597a7c13975 (MD5) %0 Journal Article %J International Journal Numerical Methods for Fluids %D 2016 %T POD–Galerkin monolithic reduced order models for parametrized fluid-structure interaction problems %A Francesco Ballarin %A Gianluigi Rozza %X In this paper we propose a monolithic approach for reduced order modelling of parametrized fluid-structure interaction problems based on a proper orthogonal decomposition (POD)–Galerkin method. Parameters of the problem are related to constitutive properties of the fluid or structural problem, or to geometrical parameters related to the domain configuration at the initial time. We provide a detailed description of the parametrized formulation of the multiphysics problem in its components, together with some insights on how to obtain an offline-online efficient computational procedure through the approximation of parametrized nonlinear tensors. Then, we present the monolithic POD–Galerkin method for the online computation of the global structural displacement, fluid velocity and pressure of the coupled problem. Finally, we show some numerical results to highlight the capabilities of the proposed reduced order method and its computational performances %B International Journal Numerical Methods for Fluids %I Wiley %G en %1 35465 %2 Mathematics %4 1 %# MAT/08 %$ Submitted by Gianluigi Rozza (grozza@sissa.it) on 2016-05-13T00:02:12Z No. of bitstreams: 2 Navon75.pdf: 4121319 bytes, checksum: 70f177ea434e4e289b7df8f7aefe5534 (MD5) img.png: 368575 bytes, checksum: 8bdf24261b0824a8bbd57d499de78f41 (MD5) %R 10.1002/fld.4252 %0 Journal Article %J NERS-14-1062; ASME J of Nuclear Rad Sci, 2, 2 (2016) 021019 %D 2016 %T A Reduced Basis Approach for Modeling the Movement of Nuclear Reactor Control Rods %A Alberto Sartori %A Antonio Cammi %A Lelio Luzzi %A Gianluigi Rozza %X This work presents a reduced order model (ROM) aimed at simulating nuclear reactor control rods movement and featuring fast-running prediction of reactivity and neutron flux distribution as well. In particular, the reduced basis (RB) method (built upon a high-fidelity finite element (FE) approximation) has been employed. The neutronics has been modeled according to a parametrized stationary version of the multigroup neutron diffusion equation, which can be formulated as a generalized eigenvalue problem. Within the RB framework, the centroidal Voronoi tessellation is employed as a sampling technique due to the possibility of a hierarchical parameter space exploration, without relying on a “classical” a posteriori error estimation, and saving an important amount of computational time in the offline phase. Here, the proposed ROM is capable of correctly predicting, with respect to the high-fidelity FE approximation, both the reactivity and neutron flux shape. In this way, a computational speedup of at least three orders of magnitude is achieved. If a higher precision is required, the number of employed basis functions (BFs) must be increased. %B NERS-14-1062; ASME J of Nuclear Rad Sci, 2, 2 (2016) 021019 %I ASME %V 2 %P 8 %8 02/2016 %G en %U http://urania.sissa.it/xmlui/handle/1963/35192 %N 2 %1 35473 %2 Mathematics %4 1 %$ Submitted by Gianluigi Rozza (grozza@sissa.it) on 2016-05-30T00:18:10Z No. of bitstreams: 1 sartori_post-icone.pdf: 1377810 bytes, checksum: 5c09a1296518bbbd4b61b709ced8dbca (MD5) %R 10.1115/1.4031945 %0 Journal Article %J Communications in Computational Physics %D 2016 %T Reduced basis approaches in time-dependent noncoercive settings for modelling the movement of nuclear reactor control rods %A Alberto Sartori %A Antonio Cammi %A Lelio Luzzi %A Gianluigi Rozza %XIn this work, two approaches, based on the certified Reduced Basis method, have been developed for simulating the movement of nuclear reactor control rods, in time-dependent non-coercive settings featuring a 3D geometrical framework. In particular, in a first approach, a piece-wise affine transformation based on subdomains division has been implemented for modelling the movement of one control rod. In the second approach, a “staircase” strategy has been adopted for simulating the movement of all the three rods featured by the nuclear reactor chosen as case study. The neutron kinetics has been modelled according to the so-called multi-group neutron diffusion, which, in the present case, is a set of ten coupled parametrized parabolic equations (two energy groups for the neutron flux, and eight for the precursors). Both the reduced order models, developed according to the two approaches, provided a very good accuracy compared with high-fidelity results, assumed as “truth” solutions. At the same time, the computational speed-up in the Online phase, with respect to the fine “truth” finite element discretization, achievable by both the proposed approaches is at least of three orders of magnitude, allowing a real-time simulation of the rod movement and control.

%B Communications in Computational Physics %I SISSA %8 2016 %G en %U http://urania.sissa.it/xmlui/handle/1963/34963 %N in press %1 35188 %2 Mathematics %$ Submitted by Gianluigi Rozza (grozza@sissa.it) on 2015-11-05T15:06:51Z No. of bitstreams: 1 manuscript.pdf: 2056729 bytes, checksum: 2a2fad22dc931e8c4094642e21c479cd (MD5) %0 Journal Article %J Computers and Mathematics with Applications %D 2016 %T Reduced basis method and domain decomposition for elliptic problems in networks and complex parametrized geometries %A Laura Iapichino %A Alfio Quarteroni %A Gianluigi Rozza %X The aim of this work is to solve parametrized partial differential equations in computational domains represented by networks of repetitive geometries by combining reduced basis and domain decomposition techniques. The main idea behind this approach is to compute once, locally and for few reference shapes, some representative finite element solutions for different values of the parameters and with a set of different suitable boundary conditions on the boundaries: these functions will represent the basis of a reduced space where the global solution is sought for. The continuity of the latter is assured by a classical domain decomposition approach. Test results on Poisson problem show the flexibility of the proposed method in which accuracy and computational time may be tuned by varying the number of reduced basis functions employed, or the set of boundary conditions used for defining locally the basis functions. The proposed approach simplifies the pre-computation of the reduced basis space by splitting the global problem into smaller local subproblems. Thanks to this feature, it allows dealing with arbitrarily complex network and features more flexibility than a classical global reduced basis approximation where the topology of the geometry is fixed. %B Computers and Mathematics with Applications %I Elsevier %V 71 %P 430 %8 01/2016 %G en %N 1 %1 35187 %2 Mathematics %4 1 %# MAT/08 %$ Submitted by Gianluigi Rozza (grozza@sissa.it) on 2015-11-05T14:59:46Z No. of bitstreams: 1 RDF2_reduced.pdf: 3254431 bytes, checksum: 508b8775e27dfb868f1b39c270cb4a6f (MD5) %& 408 %0 Book %B Springer Briefs in Mathematics %D 2015 %T Certified Reduced Basis Methods for Parametrized Partial Differential Equations %A Jan S Hesthaven %A Gianluigi Rozza %A Benjamin Stamm %K a posteriori error bounds %K empirical interpolation %K parametrized partial differential equations %K reduced basis methods, greedy algorithms %XThis book provides a thorough introduction to the mathematical and algorithmic aspects of certified reduced basis methods for parametrized partial differential equations. Central aspects ranging from model construction, error estimation and computational efficiency to empirical interpolation methods are discussed in detail for coercive problems. More advanced aspects associated with time-dependent problems, non-compliant and non-coercive problems and applications with geometric variation are also discussed as examples.

%B Springer Briefs in Mathematics %7 1 %I Springer %C Switzerland %P 135 %@ 978-3-319-22469-5 %G eng %6 1 %0 Report %D 2015 %T Fast simulations of patient-specific haemodynamics of coronary artery bypass grafts based on a POD-Galerkin method and a vascular shape parametrization %A Francesco Ballarin %A Elena Faggiano %A Sonia Ippolito %A Andrea Manzoni %A Alfio Quarteroni %A Gianluigi Rozza %A Roberto Scrofani %X 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–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. %G en %U http://urania.sissa.it/xmlui/handle/1963/34623 %1 34824 %2 Mathematics %4 1 %# MAT/08 %$ Submitted by Gianluigi Rozza (grozza@sissa.it) on 2015-10-07T00:30:27Z No. of bitstreams: 1 REPORT.pdf: 10426315 bytes, checksum: 6e5ddf4eb4cacdc7e803c2db1a540fc9 (MD5) %0 Journal Article %J Advances in Computational Mathematics %D 2015 %T Model order reduction of parameterized systems (MoRePaS): Preface to the special issue of advances in computational mathematics %A Peter Benner %A Mario Ohlberger %A Anthony Patera %A Gianluigi Rozza %A Sorensen, D.C. %A Karsten Urban %B Advances in Computational Mathematics %V 41 %P 955–960 %G eng %R 10.1007/s10444-015-9443-y %0 Journal Article %J Numerische Mathematik, (2015), 36 p. Article in Press %D 2015 %T Multilevel and weighted reduced basis method for stochastic optimal control problems constrained by Stokes equations %A Gianluigi Rozza %A Peng Chen %A Alfio Quarteroni %X In this paper we develop and analyze a multilevel weighted reduced basis method for solving stochastic optimal control problems constrained by Stokes equations. We prove the analytic regularity of the optimal solution in the probability space under certain assumptions on the random input data. The finite element method and the stochastic collocation method are employed for the numerical approximation of the problem in the deterministic space and the probability space, respectively, resulting in many large-scale optimality systems to solve. In order to reduce the unaffordable computational effort, we propose a reduced basis method using a multilevel greedy algorithm in combination with isotropic and anisotropic sparse-grid techniques. A weighted a posteriori error bound highlights the contribution stemming from each method. Numerical tests on stochastic dimensions ranging from 10 to 100 demonstrate that our method is very efficient, especially for solving high-dimensional and large-scale optimization problems. %B Numerische Mathematik, (2015), 36 p. Article in Press %I Springer %G en %U http://urania.sissa.it/xmlui/handle/1963/34491 %1 34680 %2 Mathematics %4 1 %# MAT/08 %$ Submitted by Gianluigi Rozza (grozza@sissa.it) on 2015-08-25T23:32:48Z No. of bitstreams: 1 06August2014_wRBM4SOC.pdf: 531409 bytes, checksum: d1a2f18b0de17872919c430779f7180c (MD5) %R 10.1007/s00211-015-0743-4 %0 Journal Article %J Advances in Computational Mathematics %D 2015 %T Reduced basis approximation and a-posteriori error estimation for the coupled Stokes-Darcy system %A Immanuel Martini %A Gianluigi Rozza %A Bernard Haasdonk %K Domain decomposition %K Error estimation %K Non-coercive problem %K Porous medium equation %K Reduced basis method %K Stokes flow %XThe 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.

%B Advances in Computational Mathematics %V special issue for MoRePaS 2012 %G eng %N in press %R 10.1007/s10444-014-9396-6 %0 Journal Article %J Lecture Notes in Computational Science and Engineering %D 2015 %T Reduced basis approximation of parametrized advection-diffusion PDEs with high Péclet number %A Pacciarini, P. %A Gianluigi Rozza %XIn this work we show some results about the reduced basis approximation of advection dominated parametrized problems, i.e. advection-diffusion problems with high Péclet number. These problems are of great importance in several engineering applications and it is well known that their numerical approximation can be affected by instability phenomena. In this work we compare two possible stabilization strategies in the framework of the reduced basis method, by showing numerical results obtained for a steady advection-diffusion problem.

%B Lecture Notes in Computational Science and Engineering %V 103 %P 419–426 %G eng %R 10.1007/978-3-319-10705-9__41 %0 Journal Article %J Computers and Mathematics with Applications %D 2015 %T Reduced basis approximation of parametrized optimal flow control problems for the Stokes equations %A Federico Negri %A Andrea Manzoni %A Gianluigi Rozza %XThis 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.

%B Computers and Mathematics with Applications %V 69 %P 319–336 %G eng %R 10.1016/j.camwa.2014.12.010 %0 Journal Article %D 2015 %T Supremizer stabilization of POD-Galerkin approximation of parametrized Navier-Stokes equations %A Francesco Ballarin %A Andrea Manzoni %A Alfio Quarteroni %A Gianluigi Rozza %X In this work, we present a stable proper orthogonal decomposition–Galerkin approximation for parametrized steady incompressible Navier–Stokes equations with low Reynolds number. %I Wiley %G en %U http://urania.sissa.it/xmlui/handle/1963/34701 %1 34915 %2 Mathematics %4 1 %# MAT/08 %$ Submitted by Gianluigi Rozza (grozza@sissa.it) on 2015-10-22T11:39:58Z No. of bitstreams: 1 IJNME_2014.pdf: 6761966 bytes, checksum: ad65f2c4d2dbd30a4a1590ff42ee49a0 (MD5) %R 10.1002/nme.4772 %0 Journal Article %D 2014 %T Comparison between reduced basis and stochastic collocation methods for elliptic problems %A Peng Chen %A Alfio Quarteroni %A Gianluigi Rozza %X The stochastic collocation method (Babuška et al. in SIAM J Numer Anal 45(3):1005-1034, 2007; Nobile et al. in SIAM J Numer Anal 46(5):2411-2442, 2008a; SIAM J Numer Anal 46(5):2309-2345, 2008b; Xiu and Hesthaven in SIAM J Sci Comput 27(3):1118-1139, 2005) has recently been applied to stochastic problems that can be transformed into parametric systems. Meanwhile, the reduced basis method (Maday et al. in Comptes Rendus Mathematique 335(3):289-294, 2002; Patera and Rozza in Reduced basis approximation and a posteriori error estimation for parametrized partial differential equations Version 1.0. Copyright MIT, http://augustine.mit.edu, 2007; Rozza et al. in Arch Comput Methods Eng 15(3):229-275, 2008), primarily developed for solving parametric systems, has been recently used to deal with stochastic problems (Boyaval et al. in Comput Methods Appl Mech Eng 198(41-44):3187-3206, 2009; Arch Comput Methods Eng 17:435-454, 2010). In this work, we aim at comparing the performance of the two methods when applied to the solution of linear stochastic elliptic problems. Two important comparison criteria are considered: (1), convergence results of the approximation error; (2), computational costs for both offline construction and online evaluation. Numerical experiments are performed for problems from low dimensions O (1) to moderate dimensions O (10) and to high dimensions O (100). The main result stemming from our comparison is that the reduced basis method converges better in theory and faster in practice than the stochastic collocation method for smooth problems, and is more suitable for large scale and high dimensional stochastic problems when considering computational costs. %I Springer %G en %U http://urania.sissa.it/xmlui/handle/1963/34727 %1 34916 %2 Mathematics %4 1 %$ Approved for entry into archive by Maria Pia Calandra (calapia@sissa.it) on 2015-10-26T11:39:14Z (GMT) No. of bitstreams: 0 %R 10.1007/s10915-013-9764-2 %0 Journal Article %J Annals of Nuclear Energy %D 2014 %T Comparison of a Modal Method and a Proper Orthogonal Decomposition approach for multi-group time-dependent reactor spatial kinetics %A Alberto Sartori %A Davide Baroli %A Antonio Cammi %A Davide Chiesa %A Lelio Luzzi %A Roberto R. Ponciroli %A Ezio Previtali %A Marco E. Ricotti %A Gianluigi Rozza %A Monica Sisti %XIn this paper, two modelling approaches based on a Modal Method (MM) and on the Proper Orthogonal Decomposition (POD) technique, for developing a control-oriented model of nuclear reactor spatial kinetics, are presented and compared. Both these methods allow developing neutronics description by means of a set of ordinary differential equations. The comparison of the outcomes provided by the two approaches focuses on the capability of evaluating the reactivity and the neutron flux shape in different reactor configurations, with reference to a TRIGA Mark II reactor. The results given by the POD-based approach are higher-fidelity with respect to the reference solution than those computed according to the MM-based approach, in particular when the perturbation concerns a reduced region of the core. If the perturbation is homogeneous throughout the core, the two approaches allow obtaining comparable accuracy results on the quantities of interest. As far as the computational burden is concerned, the POD approach ensures a better efficiency rather than direct Modal Method, thanks to the ability of performing a longer computation in the preprocessing that leads to a faster evaluation during the on-line phase.

%B Annals of Nuclear Energy %I Elsevier %V 71 %P 229 %8 09/2014 %G en %U http://urania.sissa.it/xmlui/handle/1963/35039 %1 35270 %2 Physics %4 1 %$ Approved for entry into archive by Maria Pia Calandra (calapia@sissa.it) on 2015-11-18T12:07:02Z (GMT) No. of bitstreams: 0 %& 217 %R 10.1016/j.anucene.2014.03.043 %0 Journal Article %J International Journal of Computational Fluid Dynamics %D 2014 %T Efficient geometrical parametrisation techniques of interfaces for reduced-order modelling: application to fluid–structure interaction coupling problems %A Forti, D. %A Gianluigi Rozza %X We present some recent advances and improvements in shape parametrisation techniques of interfaces for reduced-order modelling with special attention to fluid–structure interaction problems and the management of structural deformations, namely, to represent them into a low-dimensional space (by control points). This allows to reduce the computational effort, and to significantly simplify the (geometrical) deformation procedure, leading to more efficient and fast reduced-order modelling applications in this kind of problems. We propose an efficient methodology to select the geometrical control points for the radial basis functions based on a modal greedy algorithm to improve the computational efficiency in view of more complex fluid–structure applications in several fields. The examples provided deal with aeronautics and wind engineering. %B International Journal of Computational Fluid Dynamics %V 28 %P 158–169 %G eng %R 10.1080/10618562.2014.932352 %0 Book Section %B Separated representations and PGD-based model reduction : fundamentals and applications %D 2014 %T Fundamentals of Reduced Basis Method for problems governed by parametrized PDEs and applications %A Gianluigi Rozza %K reduced basis method, linear elasticity, heat transfer, error bounds, parametrized PDEs %XIn this chapter we consider Reduced Basis (RB) approximations of parametrized Partial Differential Equations (PDEs). The the idea behind RB is to decouple the generation and projection stages (Offline/Online computational procedures) of the approximation process in order to solve parametrized PDEs in a fast, inexpensive and reliable way. The RB method, especially applied to 3D problems, allows great computational savings with respect to the classical Galerkin Finite Element (FE) Method. The standard FE method is typically ill suited to (i) iterative contexts like optimization, sensitivity analysis and many-queries in general, and (ii) real time evaluation. We consider for simplicity coercive PDEs. We discuss all the steps to set up a RB approximation, either from an analytical and a numerical point of view. Then we present an application of the RB method to a steady thermal conductivity problem in heat transfer with emphasis on geometrical and physical parameters.

%B Separated representations and PGD-based model reduction : fundamentals and applications %S CISM International Centre for Mechanical Sciences %I Springer %C Wien %V 554 %G eng %& 4 %R 10.1007/978-3-7091-1794-1_4 %0 Journal Article %J Comptes Rendus Mathematique %D 2014 %T An improvement on geometrical parameterizations by transfinite maps %A Jäggli, C. %A Laura Iapichino %A Gianluigi Rozza %X We present a method to generate a non-affine transfinite map from a given reference domain to a family of deformed domains. The map is a generalization of the Gordon-Hall transfinite interpolation approach. It is defined globally over the reference domain. Once we have computed some functions over the reference domain, the map can be generated by knowing the parametric expressions of the boundaries of the deformed domain. Being able to define a suitable map from a reference domain to a desired deformation is useful for the management of parameterized geometries. %B Comptes Rendus Mathematique %V 352 %P 263–268 %G eng %R 10.1016/j.crma.2013.12.017 %0 Journal Article %D 2014 %T Model Order Reduction in Fluid Dynamics: Challenges and Perspectives %A Toni Lassila %A Andrea Manzoni %A Alfio Quarteroni %A Gianluigi Rozza %X 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. %I Springer %G en %1 34923 %2 Mathematics %4 1 %$ Submitted by gfeltrin@sissa.it (gfeltrin@sissa.it) on 2015-10-22T15:11:51Z No. of bitstreams: 1 preprint2014.pdf: 287014 bytes, checksum: b195410aa3f63643829ed25f1adb6520 (MD5) %R 10.1007/978-3-319-02090-7_9 %0 Conference Paper %B ECMI 2014 proceedings %D 2014 %T Reduced basis method for the Stokes equations in decomposable domains using greedy optimization %A Laura Iapichino %A Alfio Quarteroni %A Gianluigi Rozza %A Volkwein, Stefan %B ECMI 2014 proceedings %P 1–7 %G eng %0 Book %B MS&A %D 2014 %T Reduced Order Methods for Modeling and Computational Reduction %A Alfio Quarteroni %A Gianluigi Rozza %K reduced order methods, MOR, ROM, POD, RB, greedy, CFD, Numerical Analysis %XThis monograph addresses the state of the art of reduced order methods for modeling and computational reduction of complex parametrized systems, governed by ordinary and/or partial differential equations, with a special emphasis on real time computing techniques and applications in computational mechanics, bioengineering and computer graphics.

Several topics are covered, including: design, optimization, and control theory in real-time with applications in engineering; data assimilation, geometry registration, and parameter estimation with special attention to real-time computing in biomedical engineering and computational physics; real-time visualization of physics-based simulations in computer science; the treatment of high-dimensional problems in state space, physical space, or parameter space; the interactions between different model reduction and dimensionality reduction approaches; the development of general error estimation frameworks which take into account both model and discretization effects.

This book is primarily addressed to computational scientists interested in computational reduction techniques for large scale differential problems.

%B MS&A %7 1 %I Springer %C Milano %V 9 %P 334 %G eng %6 1 %R 10.1007/978-3-319-02090-7 %0 Conference Proceedings %B 22nd International Conference on Nuclear Engineering ICONE22 %D 2014 %T A reduced order model for multi-group time-dependent parametrized reactor spatial kinetics %A Alberto Sartori %A Davide Baroli %A Antonio Cammi %A Lelio Luzzi %A Gianluigi Rozza %X

In this work, a Reduced Order Model (ROM) for multigroup time-dependent parametrized reactor spatial kinetics is presented. The Reduced Basis method (built upon a high-fidelity "truth" finite element approximation) has been applied to model the neutronics behavior of a parametrized system composed by a control rod surrounded by fissile material. The neutron kinetics has been described by means of a parametrized multi-group diffusion equation where the height of the control rod (i.e., how much the rod is inserted) plays the role of the varying parameter. In order to model a continuous movement of the rod, a piecewise affine transformation based on subdomain division has been implemented. The proposed ROM is capable to efficiently reproduce the neutron flux distribution allowing to take into account the spatial effects induced by the movement of the control rod with a computational speed-up of 30000 times, with respect to the "truth" model.

%B 22nd International Conference on Nuclear Engineering ICONE22 %7 American Society Mechanical Engineering %I American Society of Mechanical Engineers (ASME) %C Prague, Czech Republic %P V005T17A048-V005T17A048 %8 07/2014 %@ 978-079184595-0 %G en %U http://urania.sissa.it/xmlui/handle/1963/35123 %1 35360 %2 Mathematics %4 1 %$ Approved for entry into archive by Maria Pia Calandra (calapia@sissa.it) on 2015-12-03T10:06:02Z (GMT) No. of bitstreams: 0 %R 10.1115/ICONE22-30707 %0 Journal Article %D 2014 %T Shape Optimization by Free-Form Deformation: Existence Results and Numerical Solution for Stokes Flows %A Francesco Ballarin %A Andrea Manzoni %A Gianluigi Rozza %A Sandro Salsa %X 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. %I Springer %G en %U http://urania.sissa.it/xmlui/handle/1963/34698 %1 34914 %2 Mathematics %4 1 %$ Approved for entry into archive by Maria Pia Calandra (calapia@sissa.it) on 2015-10-22T11:24:27Z (GMT) No. of bitstreams: 0 %R 10.1007/s10915-013-9807-8 %0 Journal Article %J Computer Methods in Applied Mechanics and Engineering %D 2014 %T Stabilized reduced basis method for parametrized advection-diffusion PDEs %A Pacciarini, P. %A Gianluigi Rozza %XIn this work, we propose viable and efficient strategies for the stabilization of the reduced basis approximation of an advection dominated problem. In particular, we investigate the combination of a classic stabilization method (SUPG) with the Offline-Online structure of the RB method. We explain why the stabilization is needed in both stages and we identify, analytically and numerically, which are the drawbacks of a stabilization performed only during the construction of the reduced basis (i.e. only in the Offline stage). We carry out numerical tests to assess the performances of the ``double'' stabilization both in steady and unsteady problems, also related to heat transfer phenomena.

%B Computer Methods in Applied Mechanics and Engineering %V 274 %P 1–18 %G eng %R 10.1016/j.cma.2014.02.005 %0 Conference Paper %B 11th World Congress on Computational Mechanics, WCCM 2014, 5th European Conference on Computational Mechanics, ECCM 2014 and 6th European Conference on Computational Fluid Dynamics, ECFD 2014 %D 2014 %T Stabilized reduced basis method for parametrized scalar advection-diffusion problems at higher Péclet number: Roles of the boundary layers and inner fronts %A Pacciarini, P. %A Gianluigi Rozza %XAdvection-dominated problems, which arise in many engineering situations, often require a fast and reliable approximation of the solution given some parameters as inputs. In this work we want to investigate the coupling of the reduced basis method - which guarantees rapidity and reliability - with some classical stabilization techiques to deal with the advection-dominated condition. We provide a numerical extension of the results presented in [1], focusing in particular on problems with curved boundary layers and inner fronts whose direction depends on the parameter.

%B 11th World Congress on Computational Mechanics, WCCM 2014, 5th European Conference on Computational Mechanics, ECCM 2014 and 6th European Conference on Computational Fluid Dynamics, ECFD 2014 %P 5614–5624 %G eng %U https://infoscience.epfl.ch/record/203327/files/ECCOMAS_PP_GR.pdf %0 Journal Article %D 2014 %T A weighted empirical interpolation method: A priori convergence analysis and applications %A Peng Chen %A Alfio Quarteroni %A Gianluigi Rozza %X We extend the classical empirical interpolation method [M. Barrault, Y. Maday, N.C. Nguyen and A.T. Patera, An empirical interpolation method: application to efficient reduced-basis discretization of partial differential equations. Compt. Rend. Math. Anal. Num. 339 (2004) 667-672] to a weighted empirical interpolation method in order to approximate nonlinear parametric functions with weighted parameters, e.g. random variables obeying various probability distributions. A priori convergence analysis is provided for the proposed method and the error bound by Kolmogorov N-width is improved from the recent work [Y. Maday, N.C. Nguyen, A.T. Patera and G.S.H. Pau, A general, multipurpose interpolation procedure: the magic points. Commun. Pure Appl. Anal. 8 (2009) 383-404]. We apply our method to geometric Brownian motion, exponential Karhunen-Loève expansion and reduced basis approximation of non-affine stochastic elliptic equations. We demonstrate its improved accuracy and efficiency over the empirical interpolation method, as well as sparse grid stochastic collocation method. %I EDP Sciences %G en %U http://urania.sissa.it/xmlui/handle/1963/35021 %1 35253 %2 Mathematics %4 1 %# MAT/05 %$ Approved for entry into archive by Lucio Lubiana (lubiana@sissa.it) on 2015-11-17T10:26:06Z (GMT) No. of bitstreams: 0 %R 10.1051/m2an/2013128 %0 Journal Article %J Comptes Rendus Mathematique. Volume 351, Issue 15-16, August 2013, Pages 593-598 %D 2013 %T A combination between the reduced basis method and the ANOVA expansion: On the computation of sensitivity indices %A Denis Devaud %A Andrea Manzoni %A Gianluigi Rozza %K Partial differential equations %XWe 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.

%B Comptes Rendus Mathematique. Volume 351, Issue 15-16, August 2013, Pages 593-598 %I Elsevier %G en %U http://hdl.handle.net/1963/7389 %1 7434 %2 Mathematics %4 1 %# MAT/05 ANALISI MATEMATICA %$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2014-06-19T08:56:09Z No. of bitstreams: 1 Devaud_Manzoni_Rozza_2013.pdf: 564002 bytes, checksum: 4c93e74468534915513e6805d440dee9 (MD5) %R 10.1016/j.crma.2013.07.023 %0 Journal Article %J Communications in Applied and Industrial Mathematics %D 2013 %T Free Form Deformation Techniques Applied to 3D Shape Optimization Problems %A Anwar Koshakji %A Alfio Quarteroni %A Gianluigi Rozza %X The purpose of this work is to analyse and study an efficient parametrization technique for a 3D shape optimization problem. After a brief review of the techniques and approaches already available in literature, we recall the Free Form Deformation parametrization, a technique which proved to be efficient and at the same time versatile, allowing to manage complex shapes even with few parameters. We tested and studied the FFD technique by establishing a path, from the geometry definition, to the method implementation, and finally to the simulation and to the optimization of the shape. In particular, we have studied a bulb and a rudder of a race sailing boat as model applications, where we have tested a complete procedure from Computer-Aided-Design to build the geometrical model to discretization and mesh generation. %B Communications in Applied and Industrial Mathematics %G eng %R 10.1685/journal.caim.452 %0 Journal Article %J Numerische Mathematik, 2013 %D 2013 %T Reduced basis approximation and a posteriori error estimation for Stokes flows in parametrized geometries: roles of the inf-sup stability constants %A Gianluigi Rozza %A Phuong Huynh %A Andrea Manzoni %K parametrized Stokes equations %X 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. %B Numerische Mathematik, 2013 %I Springer %G en %U http://hdl.handle.net/1963/6339 %1 6269 %2 Mathematics %4 1 %# MAT/08 ANALISI NUMERICA %$ Submitted by Gianluigi Rozza (grozza@sissa.it) on 2012-12-13T17:19:05Z\\nNo. of bitstreams: 1\\nStokes_HMR10_SISSA_report.pdf: 4641267 bytes, checksum: 82a163163a62d0b9deef8d5d6590f8d7 (MD5) %0 Book Section %B CEMRACS 2013 - Modelling and simulation of complex systems: stochastic and deterministic approaches %D 2013 %T Reduced Basis Approximation for the Structural-Acoustic Design based on Energy Finite Element Analysis (RB-EFEA) %A Denis Devaud %A Gianluigi Rozza %B CEMRACS 2013 - Modelling and simulation of complex systems: stochastic and deterministic approaches %V 48 %P 98-115 %G eng %R http://dx.doi.org/10.1051/proc/201448004 %0 Journal Article %J SIAM Journal on Scientific Computing %D 2013 %T Reduced basis method for parametrized elliptic optimal control problems %A Federico Negri %A Gianluigi Rozza %A Andrea Manzoni %A Alfio Quarteroni %X 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. %B SIAM Journal on Scientific Computing %V 35 %P A2316–A2340 %G eng %R 10.1137/120894737 %0 Report %D 2013 %T A Reduced Computational and Geometrical Framework for Inverse Problems in Haemodynamics %A Toni Lassila %A Andrea Manzoni %A Alfio Quarteroni %A Gianluigi Rozza %I SISSA %G en %1 6571 %2 Mathematics %4 1 %# MAT/08 ANALISI NUMERICA %$ Submitted by Gianluigi Rozza (grozza@sissa.it) on 2013-04-30T14:47:07Z No. of bitstreams: 1 LMQR_inverse_problems_Haemo.pdf: 2913600 bytes, checksum: 9bff594072e6c7b4a664bdade361a1a9 (MD5) %0 Report %D 2013 %T A reduced-order strategy for solving inverse Bayesian identification problems in physiological flows %A Toni Lassila %A Andrea Manzoni %A Alfio Quarteroni %A Gianluigi Rozza %I SISSA %G en %1 6555 %2 Mathematics %4 1 %# MAT/08 ANALISI NUMERICA %$ Submitted by Gianluigi Rozza (grozza@sissa.it) on 2013-04-12T10:16:42Z\nNo. of bitstreams: 1\nHanoi_RQML_2012_REVISED.pdf: 1360222 bytes, checksum: 4cb9ffaaf9fd67e3eacf4d9c060d0940 (MD5) %0 Report %D 2013 %T Reduction Strategies for Shape Dependent Inverse Problems in Haemodynamics %A Toni Lassila %A Andrea Manzoni %A Gianluigi Rozza %I SISSA %G en %1 6554 %2 Mathematics %4 1 %# MAT/08 ANALISI NUMERICA %$ Submitted by Gianluigi Rozza (grozza@sissa.it) on 2013-04-12T09:59:54Z\nNo. of bitstreams: 1\n03910397-1.pdf: 531448 bytes, checksum: 0ec611490748bed35d10f612e8296f27 (MD5) %0 Journal Article %J SIAM Journal on Numerical Analysis %D 2013 %T Stochastic optimal robin boundary control problems of advection-dominated elliptic equations %A Peng Chen %A Alfio Quarteroni %A Gianluigi Rozza %X In this work we deal with a stochastic optimal Robin boundary control problem constrained by an advection-diffusion-reaction elliptic equation with advection-dominated term. We assume that the uncertainty comes from the advection field and consider a stochastic Robin boundary condition as control function. A stochastic saddle point system is formulated and proved to be equivalent to the first order optimality system for the optimal control problem, based on which we provide the existence and uniqueness of the optimal solution as well as some results on stochastic regularity with respect to the random variables. Stabilized finite element approximations in physical space and collocation approximations in stochastic space are applied to discretize the optimality system. A global error estimate in the product of physical space and stochastic space for the numerical approximation is derived. Illustrative numerical experiments are provided. %B SIAM Journal on Numerical Analysis %V 51 %P 2700–2722 %G eng %R 10.1137/120884158 %0 Journal Article %J SIAM Journal on Numerical Analysis %D 2013 %T A weighted reduced basis method for elliptic partial differential equations with random input data %A Peng Chen %A Alfio Quarteroni %A Gianluigi Rozza %X In this work we propose and analyze a weighted reduced basis method to solve elliptic partial differential equations (PDEs) with random input data. The PDEs are first transformed into a weighted parametric elliptic problem depending on a finite number of parameters. Distinctive importance of the solution at different values of the parameters is taken into account by assigning different weights to the samples in the greedy sampling procedure. A priori convergence analysis is carried out by constructive approximation of the exact solution with respect to the weighted parameters. Numerical examples are provided for the assessment of the advantages of the proposed method over the reduced basis method and the stochastic collocation method in both univariate and multivariate stochastic problems. %B SIAM Journal on Numerical Analysis %V 51 %P 3163–3185 %G eng %R 10.1137/130905253 %0 Journal Article %J Mathematical Modelling and Numerical Analysis, in press, 2012-13 %D 2012 %T Boundary control and shape optimization for the robust design of bypass anastomoses under uncertainty %A Toni Lassila %A Andrea Manzoni %A Alfio Quarteroni %A Gianluigi Rozza %K shape optimization %X 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. %B Mathematical Modelling and Numerical Analysis, in press, 2012-13 %I Cambridge University Press %G en %U http://hdl.handle.net/1963/6337 %1 6267 %2 Mathematics %4 1 %# MAT/08 ANALISI NUMERICA %$ Submitted by Gianluigi Rozza (grozza@sissa.it) on 2012-12-13T15:38:41Z\\nNo. of bitstreams: 1\\nLMQR_M2AN_Special_SISSAreport.pdf: 5702019 bytes, checksum: 037e51bde713582eff1ee9766b1b4559 (MD5) %0 Book Section %B Springer, Indam Series, Vol. 4, 2012 %D 2012 %T Generalized reduced basis methods and n-width estimates for the approximation of the solution manifold of parametric PDEs %A Toni Lassila %A Andrea Manzoni %A Alfio Quarteroni %A Gianluigi Rozza %K solution manifold %X 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. %B Springer, Indam Series, Vol. 4, 2012 %I Springer %G en %U http://hdl.handle.net/1963/6340 %1 6270 %2 Mathematics %4 1 %# MAT/08 ANALISI NUMERICA %$ Submitted by Gianluigi Rozza (grozza@sissa.it) on 2012-12-13T17:47:45Z\\nNo. of bitstreams: 1\\nqlmr-bumi_FINAL_SISSAreport.pdf: 377397 bytes, checksum: ecaf5713afa6a6f68992f6331631aff4 (MD5) %0 Conference Proceedings %B European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS 2012) J. Eberhardsteiner et.al. (eds.), Vienna, Austria, 10-14 sept. 2012 %D 2012 %T Reduction strategies for PDE-constrained oprimization problems in Haemodynamics %A Gianluigi Rozza %A Andrea Manzoni %A Federico Negri %K inverse problems %X 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. %B European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS 2012) J. Eberhardsteiner et.al. (eds.), Vienna, Austria, 10-14 sept. 2012 %G en %U http://hdl.handle.net/1963/6338 %1 6268 %2 Mathematics %4 1 %# MAT/08 ANALISI NUMERICA %$ Submitted by Gianluigi Rozza (grozza@sissa.it) on 2012-12-13T16:30:48Z\\r\\nNo. of bitstreams: 1\\r\\npaper_ECCOMAS2012_AM_SISSAreport.pdf: 4599005 bytes, checksum: 014a5a01276f25177386ad374a5917c8 (MD5) %0 Journal Article %J International Journal Numerical Methods Biomedical Engineering %D 2012 %T Simulation-based uncertainty quantification of human arterial network hemodynamics %A Peng Chen %A Alfio Quarteroni %A Gianluigi Rozza %K uncertainty quantification, mathematical modelling of the cardiovascular system, fluid-structure interaction %X This work aims at identifying and quantifying uncertainties from various sources in human cardiovascular\r\nsystem based on stochastic simulation of a one dimensional arterial network. A general analysis of\r\ndifferent uncertainties and probability characterization with log-normal distribution of these uncertainties\r\nis introduced. Deriving from a deterministic one dimensional fluid structure interaction model, we establish\r\nthe stochastic model as a coupled hyperbolic system incorporated with parametric uncertainties to describe\r\nthe blood flow and pressure wave propagation in the arterial network. By applying a stochastic collocation\r\nmethod with sparse grid technique, we study systemically the statistics and sensitivity of the solution with\r\nrespect to many different uncertainties in a relatively complete arterial network with potential physiological\r\nand pathological implications for the first time. %B International Journal Numerical Methods Biomedical Engineering %I Wiley %G en %1 6467 %2 Mathematics %4 1 %# MAT/08 ANALISI NUMERICA %$ Submitted by Gianluigi Rozza (grozza@sissa.it) on 2013-03-07T12:33:02Z\nNo. of bitstreams: 1\nreport.pdf: 1605405 bytes, checksum: bb12cf074ce32a80567a0cde0c0861db (MD5)