Publications

2018

  1. S. Hijazi, S. Ali, G. Stabile, F. Ballarin, and G. Rozza, “The Effort of Increasing Reynolds Number in Projection-Based Reduced Order Methods: from Laminar to Turbulent Flows”, 2018.
    [BibTeX] [Abstract] [Download preprint]
    We present two different reduced order strategies for incompressible parameterized Navier-Stokes equations characterized by varying Reynolds numbers. The first strategy deals with low Reynolds number (laminar flow) and is based on a stabilized finite element method during the offline stage followed by a Galerkin projection on reduced basis spaces generated by a greedy algorithm. The second methodology is based on a full order finite volume discretization. The latter methodology will be used for flows with moderate to high Reynolds number characterized by turbulent patterns. For the treatment of the mentioned turbulent flows at the reduced order level, a new POD-Galerkin approach is proposed. The new approach takes into consideration the contribution of the eddy viscosity also during the online stage and is based on the use of interpolation. The two methodologies are tested on classic benchmark test cases.

    @unpublished{HijaziAliStabileBallarinRozza2018,
    author = {Hijazi, Saddam and Ali, Shafqat and Stabile, Giovanni and Ballarin, Francesco and Rozza, Gianluigi},
    title = {The Effort of Increasing Reynolds Number in Projection-Based Reduced Order Methods: from Laminar to Turbulent Flows},
    year = {2018},
    preprint = {https://arxiv.org/abs/1807.11370},
    abstract = { We present two different reduced order strategies for incompressible parameterized Navier-Stokes equations characterized by varying Reynolds numbers. The first strategy deals with low Reynolds number (laminar flow) and is based on a stabilized finite element method during the offline stage followed by a Galerkin projection on reduced basis spaces generated by a greedy algorithm. The second methodology is based on a full order finite volume discretization. The latter methodology will be used for flows with moderate to high Reynolds number characterized by turbulent patterns. For the treatment of the mentioned turbulent flows at the reduced order level, a new POD-Galerkin approach is proposed. The new approach takes into consideration the contribution of the eddy viscosity also during the online stage and is based on the use of interpolation. The two methodologies are tested on classic benchmark test cases.}
    }

  2. G. Stabile, F. Ballarin, G. Zuccarino, and G. Rozza, “A reduced order variational multiscale approach for turbulent flows”, 2018.
    [BibTeX] [Abstract] [Download preprint]
    The purpose of this work is to present a reduced order modeling framework for parametrized turbulent flows with moderately high Reynolds numbers within the variational multiscale (VMS) method. The Reduced Order Models (ROMs) presented in this manuscript are based on a POD-Galerkin approach with a VMS stabilization technique. Two different reduced order models are presented, which differ on the stabilization used during the Galerkin projection. In the first case the VMS stabilization method is used at both the full order and the reduced order level. In the second case, the VMS stabilization is used only at the full order level, while the projection of the standard Navier-Stokes equations is performed instead at the reduced order level. The former method is denoted as consistent ROM, while the latter is named non-consistent ROM, in order to underline the different choices made at the two levels. Particular attention is also devoted to the role of inf-sup stabilization by means of supremizers in ROMs based on a VMS formulation. Finally, the developed methods are tested on a numerical benchmark.

    @unpublished{StabileBallarinZuccarinoRozza2018,
    author = {Stabile, Giovanni and Ballarin, Francesco and Zuccarino, Giacomo and Rozza, Gianluigi},
    title = {A reduced order variational multiscale approach for turbulent flows},
    year = {2018},
    preprint = {https://arxiv.org/abs/1809.11101},
    abstract = {The purpose of this work is to present a reduced order modeling framework for parametrized turbulent flows with moderately high Reynolds numbers within the variational multiscale (VMS) method. The Reduced Order Models (ROMs) presented in this manuscript are based on a POD-Galerkin approach with a VMS stabilization technique. Two different reduced order models are presented, which differ on the stabilization used during the Galerkin projection. In the first case the VMS stabilization method is used at both the full order and the reduced order level. In the second case, the VMS stabilization is used only at the full order level, while the projection of the standard Navier-Stokes equations is performed instead at the reduced order level. The former method is denoted as consistent ROM, while the latter is named non-consistent ROM, in order to underline the different choices made at the two levels. Particular attention is also devoted to the role of inf-sup stabilization by means of supremizers in ROMs based on a VMS formulation. Finally, the developed methods are tested on a numerical benchmark.}
    }

  3. M. Strazzullo, F. Ballarin, R. Mosetti, and G. Rozza, “Model Reduction for Parametrized Optimal Control Problems in Environmental Marine Sciences and Engineering”, SIAM Journal on Scientific Computing, 40(4), p. pp. B1055-B1079, 2018.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]
    We propose reduced order methods as a suitable approach to face parametrized optimal control problems governed by partial differential equations, with applications in en- vironmental marine sciences and engineering. Environmental parametrized optimal control problems are usually studied for different configurations described by several physical and/or geometrical parameters representing different phenomena and structures. The solution of parametrized problems requires a demanding computational effort. In order to save com- putational time, we rely on reduced basis techniques as a reliable and rapid tool to solve parametrized problems. We introduce general parametrized linear quadratic optimal control problems, and the saddle-point structure of their optimality system. Then, we propose a POD-Galerkin reduction of the optimality system. Finally, we test the resulting method on two environmental applications: a pollutant control in the Gulf of Trieste, Italy and a solution tracking governed by quasi-geostrophic equations describing North Atlantic Ocean dynamic. The two experiments underline how reduced order methods are a reliable and convenient tool to manage several environmental optimal control problems, for different mathematical models, geographical scale as well as physical meaning.

    @article{StrazzulloBallarinMosettiRozza2017,
    author = {Strazzullo, Maria and Ballarin, Francesco and Mosetti, Renzo and Rozza, Gianluigi},
    title = {Model Reduction for Parametrized Optimal Control Problems in Environmental Marine Sciences and Engineering},
    journal = {SIAM Journal on Scientific Computing},
    volume = {40},
    number = {4},
    pages = {B1055-B1079},
    year = {2018},
    preprint = {https://arxiv.org/abs/1710.01640},
    doi = {10.1137/17M1150591},
    abstract = {We propose reduced order methods as a suitable approach to face parametrized optimal control problems governed by partial differential equations, with applications in en- vironmental marine sciences and engineering. Environmental parametrized optimal control problems are usually studied for different configurations described by several physical and/or geometrical parameters representing different phenomena and structures. The solution of parametrized problems requires a demanding computational effort. In order to save com- putational time, we rely on reduced basis techniques as a reliable and rapid tool to solve parametrized problems. We introduce general parametrized linear quadratic optimal control problems, and the saddle-point structure of their optimality system. Then, we propose a POD-Galerkin reduction of the optimality system. Finally, we test the resulting method on two environmental applications: a pollutant control in the Gulf of Trieste, Italy and a solution tracking governed by quasi-geostrophic equations describing North Atlantic Ocean dynamic. The two experiments underline how reduced order methods are a reliable and convenient tool to manage several environmental optimal control problems, for different mathematical models, geographical scale as well as physical meaning.}
    }

  4. L. Venturi, D. Torlo, F. Ballarin, and G. Rozza, “Weighted reduced order methods for parametrized partial differential equations with random inputs”, 2018.
    [BibTeX] [Abstract] [Download preprint]
    In this manuscript we discuss weighted reduced order methods for stochastic partial differential equations. Random inputs (such as forcing terms, equation coefficients, boundary conditions) are considered as parameters of the equations. We take advantage of the resulting parametrized formulation to propose an efficient reduced order model; we also profit by the underlying stochastic assumption in the definition of suitable weights to drive to reduction process. Two viable strategies are discussed, namely the weighted reduced basis method and the weighted proper orthogonal decomposition method. A numerical example on a parametrized elasticity problem is shown.

    @unpublished{VenturiTorloBallarinRozza2018,
    author = {Venturi, Luca and Torlo, Davide and Ballarin, Francesco and Rozza, Gianluigi},
    title = {Weighted reduced order methods for parametrized partial differential equations with random inputs},
    year = {2018},
    preprint = {https://arxiv.org/abs/1805.00828},
    abstract = {In this manuscript we discuss weighted reduced order methods for stochastic partial differential equations. Random inputs (such as forcing terms, equation coefficients, boundary conditions) are considered as parameters of the equations. We take advantage of the resulting parametrized formulation to propose an efficient reduced order model; we also profit by the underlying stochastic assumption in the definition of suitable weights to drive to reduction process. Two viable strategies are discussed, namely the weighted reduced basis method and the weighted proper orthogonal decomposition method. A numerical example on a parametrized elasticity problem is shown.}
    }

  5. L. Venturi, F. Ballarin, and G. Rozza, “A weighted POD method for elliptic PDEs with random inputs”, 2018.
    [BibTeX] [Abstract] [Download preprint]
    In this work we propose and analyze a weighted proper orthogonal decomposition method to solve elliptic partial differential equations depending on random input data, for stochastic problems that can be transformed into parametric systems. The algorithm is introduced alongside the weighted greedy method. Our proposed method aims to minimize the error in a L2 norm and, in contrast to the weighted greedy approach, it does not require the availability of an error bound. Moreover, we consider sparse discretization of the input space in the construction of the reduced model; for high-dimensional problems, provided the sampling is done accordingly to the parameters distribution, this enables a sensible reduction of computational costs, while keeping a very good accuracy with respect to high fidelity solutions. We provide many numerical tests to asses the performance of the proposed method compared to an equivalent reduced order model without weighting, as well as to the weighted greedy approach, in both low and higher dimensional problems.

    @unpublished{VenturiBallarinRozza2018,
    author = {Venturi, Luca and Ballarin, Francesco and Rozza, Gianluigi},
    title = {A weighted POD method for elliptic PDEs with random inputs},
    year = {2018},
    preprint = {https://arxiv.org/abs/1802.08724},
    abstract = {In this work we propose and analyze a weighted proper orthogonal decomposition method to solve elliptic partial differential equations depending on random input data, for stochastic problems that can be transformed into parametric systems. The algorithm is introduced alongside the weighted greedy method. Our proposed method aims to minimize the error in a L2 norm and, in contrast to the weighted greedy approach, it does not require the availability of an error bound. Moreover, we consider sparse discretization of the input space in the construction of the reduced model; for high-dimensional problems, provided the sampling is done accordingly to the parameters distribution, this enables a sensible reduction of computational costs, while keeping a very good accuracy with respect to high fidelity solutions.
    We provide many numerical tests to asses the performance of the proposed method compared to an equivalent reduced order model without weighting, as well as to the weighted greedy approach, in both low and higher dimensional problems.}
    }

2017

  1. F. Ballarin, E. Faggiano, A. Manzoni, A. Quarteroni, G. Rozza, S. Ippolito, C. Antona, and R. Scrofani, “Numerical modeling of hemodynamics scenarios of patient-specific coronary artery bypass grafts”, Biomechanics and Modeling in Mechanobiology, 16(4), p. pp. 1373–1399, 2017.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]
    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.

    @ARTICLE{BallarinFaggianoManzoniQuarteroniRozzaIppolitoAntonaScrofani2016,
    title = {Numerical modeling of hemodynamics scenarios of patient-specific coronary artery bypass grafts},
    journal = {Biomechanics and Modeling in Mechanobiology},
    abstract = {A fast computational framework is devised to the study of several configurations of patient-specific coronary artery bypass grafts. This is especially useful to perform a sensitivity analysis of the haemodynamics for different flow conditions occurring in native coronary arteries and bypass grafts, the investigation of the progression of the coronary artery disease and the choice of the most appropriate surgical procedure. A complete pipeline, from the acquisition of patientspecific medical images to fast parametrized computational simulations, is proposed. Complex surgical configurations employed in the clinical practice, such as Y-grafts and sequential grafts, are studied. A virtual surgery platform based on model reduction of unsteady Navier Stokes equations for blood dynamics is proposed to carry out sensitivity analyses in a very rapid and reliable way. A specialized geometrical parametrization is employed to compare the effect of stenosis and anastomosis variation on the outcome of the surgery in several relevant cases.},
    author = {Francesco Ballarin and Elena Faggiano and Andrea Manzoni and Alfio Quarteroni and Gianluigi Rozza and Sonia Ippolito and Carlo Antona and Roberto Scrofani},
    doi = {10.1007/s10237-017-0893-7},
    year = {2017},
    volume = {16},
    number = {4},
    pages = {1373--1399},
    preprint = {https://urania.sissa.it/xmlui/bitstream/handle/1963/35240/BMMB_SISSA_report.pdf?sequence=1&isAllowed=y}
    }

  2. F. Ballarin, G. Rozza, and Y. Maday, “Reduced-order semi-implicit schemes for fluid-structure interaction problems”, in Model Reduction of Parametrized Systems, P. Benner, M. Ohlberger, A. Patera, G. Rozza, and K. Urban (eds.), Springer International Publishing, p. pp. 149–167, 2017.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]
    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.

    @INBOOK{BallarinRozzaMaday2017,
    chapter = {Reduced-order semi-implicit schemes for fluid-structure interaction problems},
    year = {2017},
    author = {Ballarin, Francesco and Rozza, Gianluigi and Maday, Yvon},
    editor = {Benner, Peter and Ohlberger, Mario and Patera, Anthony and Rozza, Gianluigi and Urban, Karsten},
    booktitle = {Model Reduction of Parametrized Systems},
    publisher = {Springer International Publishing},
    pages = {149--167},
    abstract = {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.},
    doi = {10.1007/978-3-319-58786-8_10},
    preprint = {https://arxiv.org/abs/1711.10829}
    }

  3. F. Ballarin, A. D’Amario, S. Perotto, and G. Rozza, “A POD-Selective Inverse Distance Weighting method for fast parametrized shape morphing”, 2017.
    [BibTeX] [Abstract] [Download preprint]
    Efficient shape morphing techniques play a crucial role in the approximation of partial differential equations defined in parametrized domains, such as for fluid-structure interaction or shape optimization problems. In this paper, we focus on Inverse Distance Weighting (IDW) interpolation techniques, where a reference domain is morphed into a deformed one via the displacement of a set of control points. We aim at reducing the computational burden characterizing a standard IDW approach without compromising the accuracy. To this aim, first we propose an improvement of IDW based on a geometric criterion which automatically selects a subset of the original set of control points. Then, we combine this new approach with a model reduction technique based on a Proper Orthogonal Decomposition of the set of admissible displacements. This choice further reduces computational costs. We verify the performances of the new IDW techniques on several tests by investigating the trade-off reached in terms of accuracy and efficiency.

    @unpublished{BallarinDAmarioPerottoRozza2017,
    author = {Ballarin, Francesco and D'Amario, Alessandro and Perotto, Simona and Rozza, Gianluigi},
    title = {A POD-Selective Inverse Distance Weighting method for fast parametrized shape morphing},
    year = {2017},
    preprint = {https://arxiv.org/abs/1710.09243},
    abstract = {Efficient shape morphing techniques play a crucial role in the approximation of partial differential equations defined in parametrized domains, such as for fluid-structure interaction or shape optimization problems. In this paper, we focus on Inverse Distance Weighting (IDW) interpolation techniques, where a reference domain is morphed into a deformed one via the displacement of a set of control points. We aim at reducing the computational burden characterizing a standard IDW approach without compromising the accuracy. To this aim, first we propose an improvement of IDW based on a geometric criterion which automatically selects a subset of the original set of control points. Then, we combine this new approach with a model reduction technique based on a Proper Orthogonal Decomposition of the set of admissible displacements. This choice further reduces computational costs. We verify the performances of the new IDW techniques on several tests by investigating the trade-off reached in terms of accuracy and efficiency.}
    }

  4. T. Chacón Rebollo, E. Delgado Ávila, M. Gómez Mármol, F. Ballarin, and G. Rozza, “On a certified Smagorinsky reduced basis turbulence model”, SIAM Journal on Numerical Analysis, 55(6), p. pp. 3047–3067, 2017.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]
    In this work we present a reduced basis Smagorinsky turbulence model for steady flows. We approximate the non-linear eddy diffusion term using the Empirical Interpolation Method, and the velocity-pressure unknowns by an independent reduced-basis procedure. This model is based upon an a posteriori error estimation for Smagorinsky turbulence model. The theoretical development of the a posteriori error estimation is based on previous works, according to the Brezzi-Rappaz-Raviart stability theory, and adapted for the non-linear eddy diffusion term. We present some numerical tests, programmed in FreeFem++, in which we show an speedup on the computation by factor larger than 1000 in benchmark 2D flows.

    @article{ChaconDelgadoGomezBallarinRozza2017,
    author = {Chacón Rebollo, Tomás and Delgado Ávila, Enrique and Gómez Mármol, Macarena and Ballarin, Francesco and Rozza, Gianluigi},
    title = {On a certified Smagorinsky reduced basis turbulence model},
    year = {2017},
    preprint = {https://arxiv.org/abs/1709.00243},
    abstract = {In this work we present a reduced basis Smagorinsky turbulence model for steady flows. We approximate the non-linear eddy diffusion term using the Empirical Interpolation Method, and the velocity-pressure unknowns by an independent reduced-basis procedure. This model is based upon an a posteriori error estimation for Smagorinsky turbulence model. The theoretical development of the a posteriori error estimation is based on previous works, according to the Brezzi-Rappaz-Raviart stability theory, and adapted for the non-linear eddy diffusion term. We present some numerical tests, programmed in FreeFem++, in which we show an speedup on the computation by factor larger than 1000 in benchmark 2D flows.},
    journal = {SIAM Journal on Numerical Analysis},
    doi = {10.1137/17M1118233},
    year = {2017},
    volume = {55},
    number = {6},
    pages = {3047--3067},
    }

  5. M. Tezzele, F. Ballarin, and G. Rozza, “Combined parameter and model reduction of cardiovascular problems by means of active subspaces and POD-Galerkin methods”, 2017.
    [BibTeX] [Abstract] [Download preprint]
    In this chapter we introduce a combined parameter and model reduction methodology and present its application to the efficient numerical estimation of a pressure drop in a set of deformed carotids. The aim is to simulate a wide range of possible occlusions after the bifurcation of the carotid. A parametric description of the admissible deformations, based on radial basis functions interpolation, is introduced. Since the parameter space may be very large, the first step in the combined reduction technique is to look for active subspaces in order to reduce the parameter space dimension. Then, we rely on model order reduction methods over the lower dimensional parameter subspace, based on a POD-Galerkin approach, to further reduce the required computational effort and enhance computational efficiency.

    @unpublished{TezzeleBallarinRozza2017,
    author = {Tezzele, Marco and Ballarin, Francesco and Rozza, Gianluigi},
    title = {Combined parameter and model reduction of cardiovascular problems by means of active subspaces and POD-Galerkin methods},
    year = {2017},
    preprint = {https://arxiv.org/abs/1711.10884},
    abstract = {In this chapter we introduce a combined parameter and model reduction methodology and present its application to the efficient numerical estimation of a pressure drop in a set of deformed carotids. The aim is to simulate a wide range of possible occlusions after the bifurcation of the carotid. A parametric description of the admissible deformations, based on radial basis functions interpolation, is introduced. Since the parameter space may be very large, the first step in the combined reduction technique is to look for active subspaces in order to reduce the parameter space dimension. Then, we rely on model order reduction methods over the lower dimensional parameter subspace, based on a POD-Galerkin approach, to further reduce the required computational effort and enhance computational efficiency.}
    }

  6. D. Torlo, F. Ballarin, and G. Rozza, “Stabilized weighted reduced basis methods for parametrized advection dominated problems with random inputs”, 2017.
    [BibTeX] [Abstract] [Download preprint]
    In this work, we propose viable and efficient strategies for stabilized parametrized advection dominated problems, with random inputs. In particular, we investigate the combination of wRB (weighted reduced basis) method for stochastic parametrized problems with stabilized reduced basis method, which is the integration of classical stabilization methods (SUPG, in our case) in the Offline-Online structure of the RB method. Moreover, we introduce a reduction method that selectively enables online stabilization; this leads to a sensible reduction of computational costs, while keeping a very good accuracy with respect to high fidelity solutions. We present numerical test cases to assess the performance of the proposed methods in steady and unsteady problems related to heat transfer phenomena.

    @unpublished{TorloBallarinRozza2017,
    author = {Torlo, Davide and Ballarin, Francesco and Rozza, Gianluigi},
    title = {Stabilized weighted reduced basis methods for parametrized advection dominated problems with random inputs},
    year = {2017},
    preprint = {https://arxiv.org/abs/1711.11275},
    abstract = {In this work, we propose viable and efficient strategies for stabilized parametrized advection dominated problems, with random inputs. In particular, we investigate the combination of wRB (weighted reduced basis) method for stochastic parametrized problems with stabilized reduced basis method, which is the integration of classical stabilization methods (SUPG, in our case) in the Offline-Online structure of the RB method. Moreover, we introduce a reduction method that selectively enables online stabilization; this leads to a sensible reduction of computational costs, while keeping a very good accuracy with respect to high fidelity solutions. We present numerical test cases to assess the performance of the proposed methods in steady and unsteady problems related to heat transfer phenomena.}
    }

2016

  1. F. Ballarin, E. Faggiano, S. Ippolito, A. Manzoni, A. Quarteroni, G. Rozza, and R. Scrofani, “Fast simulations of patient-specific haemodynamics of coronary artery bypass grafts based on a POD-Galerkin method and a vascular shape parametrization”, Journal of Computational Physics, 315, p. pp. 609–628, 2016.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]
    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 carry out sensitivity analysis studies, so far out of reach. In particular, a reduced-order simulation takes only a few minutes to run, resulting in computational savings of 99% of CPU time with respect to the full-order discretization. Moreover, the error between full-order and reduced-order solutions is also studied, and it is numerically found to be less than 1% for reduced-order solutions obtained with just O(100) online degrees of freedom.

    @ARTICLE{BallarinFaggianoIppolitoManzoniQuarteroniRozzaScrofani2015,
    author = {Ballarin, F. and Faggiano, E. and Ippolito, S. and Manzoni, A. and Quarteroni, A. and Rozza, G. and Scrofani, R.},
    title = {Fast simulations of patient-specific haemodynamics of coronary artery bypass grafts based on a {POD}-{G}alerkin method and a vascular shape parametrization},
    year = {2016},
    journal = {Journal of Computational Physics},
    volume = {315},
    pages = {609--628},
    abstract = {In this work a reduced-order computational framework for the study of haemodynamics in three-dimensional patient-specific configurations of coronary artery bypass grafts dealing with a wide range of scenarios is proposed. We combine several efficient algorithms to face at the same time both the geometrical complexity involved in the description of the vascular network and the huge computational cost entailed by time dependent patient-specific flow simulations. Medical imaging procedures allow to reconstruct patient-specific configurations from clinical data. A centerlines-based parametrization is proposed to efficiently handle geometrical variations. POD--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 carry out sensitivity analysis studies, so far out of reach. In particular, a reduced-order simulation takes only a few minutes to run, resulting in computational savings of 99% of CPU time with respect to the full-order discretization. Moreover, the error between full-order and reduced-order solutions is also studied, and it is numerically found to be less than 1% for reduced-order solutions obtained with just O(100) online degrees of freedom.},
    doi = {10.1016/j.jcp.2016.03.065},
    preprint = {https://urania.sissa.it/xmlui/bitstream/handle/1963/34623/REPORT.pdf?sequence=1&isAllowed=y}
    }

  2. F. Ballarin and G. Rozza, “POD–Galerkin monolithic reduced order models for parametrized fluid-structure interaction problems”, International Journal for Numerical Methods in Fluids, 82(12), p. pp. 1010–1034, 2016.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]
    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

    @ARTICLE{BallarinRozza2016,
    author = {Francesco Ballarin and Gianluigi Rozza},
    title = {{POD}--{G}alerkin monolithic reduced order models for parametrized fluid-structure interaction problems},
    journal = {International Journal for Numerical Methods in Fluids},
    volume = {82},
    number = {12},
    pages = {1010--1034},
    abstract = {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},
    year = {2016},
    doi = {10.1002/fld.4252},
    preprint = {http://preprints.sissa.it/xmlui/bitstream/handle/1963/35180/Navon75.pdf?sequence=1&isAllowed=y}
    }

  3. F. Salmoiraghi, F. Ballarin, L. Heltai, and G. Rozza, “Isogeometric analysis-based reduced order modelling for incompressible linear viscous flows in parametrized shapes”, Advanced Modeling and Simulation in Engineering Sciences, 3(1), p. pp. 21, 2016.
    [BibTeX] [Abstract] [View on publisher website]
    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.

    @ARTICLE{SalmoiraghiBallarinHeltaiRozza2016,
    title = {Isogeometric analysis-based reduced order modelling for incompressible linear viscous flows in parametrized shapes},
    abstract = {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.},
    author = {Filippo Salmoiraghi and Francesco Ballarin and Luca Heltai and Gianluigi Rozza},
    journal={Advanced Modeling and Simulation in Engineering Sciences},
    year={2016},
    volume={3},
    number={1},
    pages={21},
    doi={10.1186/s40323-016-0076-6},
    }

  4. F. Salmoiraghi, F. Ballarin, G. Corsi, A. Mola, M. Tezzele, and G. Rozza, “Advances in geometrical parametrization and reduced order models and methods for computational fluid dynamics problems in applied sciences and engineering: overview and perspectives”, in Proceedings of the ECCOMAS Congress 2016, VII European Conference on Computational Methods in Applied Sciences and Engineering, 2016.
    [BibTeX] [Abstract] [View on publisher website]
    Several 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.

    @INPROCEEDINGS{SalmoiraghiBallarinCorsiMolaTezzeleRozza2016,
    author = {Salmoiraghi, F. and Ballarin, F. and Corsi, G. and Mola, A. and Tezzele, M. and Rozza, G.},
    title = {Advances in geometrical parametrization and reduced order models and methods for computational fluid dynamics problems in applied sciences and engineering: overview and perspectives},
    booktitle = {Proceedings of the {ECCOMAS} {Congress} 2016, {VII} {E}uropean {C}onference on {C}omputational {M}ethods in {A}pplied {S}ciences and {E}ngineering},
    year = {2016},
    editor = {Papadrakakis, M. and Papadopoulos, V. and Stefanou, G. and Plevris, V.},
    abstract = {Several 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.},
    url = {http://www.eccomas.org/cvdata/cntr1/spc7/dtos/img/mdia/eccomas-2016-vol-1.pdf}
    }

2015

  1. F. Ballarin, “Reduced order models for patient-specific haemodynamics of coronary artery bypass grafts”, PhD Thesis, Politecnico di Milano, 2015.
    [BibTeX] [View on publisher website]
    @PHDTHESIS{Ballarin2015,
    author = {Francesco Ballarin},
    title = {Reduced order models for patient-specific haemodynamics of coronary artery bypass grafts},
    school = {Politecnico di Milano},
    year = {2015},
    url = {https://www.politesi.polimi.it/handle/10589/102804}
    }

  2. F. Ballarin, A. Manzoni, A. Quarteroni, and G. Rozza, “Supremizer stabilization of POD–Galerkin approximation of parametrized steady incompressible Navier–Stokes equations”, International Journal for Numerical Methods in Engineering, 102(5), p. pp. 1136–1161, 2015.
    [BibTeX] [Download preprint] [View on publisher website]
    @ARTICLE{BallarinManzoniQuarteroniRozza2015,
    author = {Ballarin, Francesco and Manzoni, Andrea and Quarteroni, Alfio and Rozza, Gianluigi},
    title = {Supremizer stabilization of {POD}--{G}alerkin approximation of parametrized steady incompressible {N}avier--{S}tokes equations},
    journal = {International Journal for Numerical Methods in Engineering},
    year = {2015},
    volume = {102},
    pages = {1136--1161},
    number = {5},
    doi = {10.1002/nme.4772},
    issn = {1097-0207},
    preprint = {https://www.mate.polimi.it/biblioteca/add/qmox/13-2014.pdf}
    }

2014

  1. F. Ballarin, A. Manzoni, G. Rozza, and S. Salsa, “Shape Optimization by Free-Form Deformation: Existence Results and Numerical Solution for Stokes Flows”, Journal of Scientific Computing, 60(3), p. pp. 537–563, 2014.
    [BibTeX] [Abstract] [View on publisher website]
    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.

    @ARTICLE{BallarinManzoniRozzaSalsa2014,
    author = {Ballarin, F. and Manzoni, A. and Rozza, G. and Salsa, S.},
    title = {Shape Optimization by Free-Form Deformation: Existence Results and Numerical Solution for {S}tokes Flows},
    journal = {Journal of Scientific Computing},
    year = {2014},
    volume = {60},
    pages = {537--563},
    number = {3},
    abstract = {Shape optimization problems governed by PDEs result from many applications in computational fluid dynamics. These problems usually entail very large computational costs and require also a suitable approach for representing and deforming efficiently the shape of the underlying geometry, as well as for computing the shape gradient of the cost functional to be minimized. Several approaches based on the displacement of a set of control points have been developed in the last decades, such as the so-called free-form deformations. In this paper we present a new theoretical result which allows to recast free-form deformations into the general class of perturbation of identity maps, and to guarantee the compactness of the set of admissible shapes. Moreover, we address both a general optimization framework based on the continuous shape gradient and a numerical procedure for solving efficiently three-dimensional optimal design problems. This framework is applied to the optimal design of immersed bodies in Stokes flows, for which we consider the numerical solution of a benchmark case study from literature.},
    doi = {10.1007/s10915-013-9807-8}
    }

2011

  1. F. Ballarin, “Shape optimization for tridimensional viscous flows in cardiovascular geometries”, Master Thesis, Politecnico di Milano, 2011.
    [BibTeX] [View on publisher website]
    @MASTERSTHESIS{Ballarin2011,
    author = {Francesco Ballarin},
    title = {Shape optimization for tridimensional viscous flows in cardiovascular geometries},
    language = {Italian},
    school = {Politecnico di Milano},
    year = {2011},
    url = {https://www.politesi.polimi.it/handle/10589/30601}
    }

2009

  1. F. Ballarin and S. Palamara, “Macroscopic models and numerical simulations for traffic flow on networks”, Bachelor Thesis, Politecnico di Milano, 2009.
    [BibTeX] [View on publisher website]
    @BACHELORSTHESIS{BallarinPalamara2009,
    author = {Francesco Ballarin and Simone Palamara},
    title = {Macroscopic models and numerical simulations for traffic flow on networks},
    language = {Italian},
    school = {Politecnico di Milano},
    year = {2009},
    url = {https://www.mate.polimi.it/biblioteca/?pp=view&id=277&collezione=tesi&L=i}
    }