Publications

2022

  1. F. Ballarin, G. Rozza, and M. Strazzullo, “Space-time POD-Galerkin approach for parametric flow control”, in Numerical Control: Part A, E. Trélat and E. Zuazua (eds.), Elsevier, vol. 23, pp. 307-338, 2022.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]
    In this contribution we propose reduced order methods to fast and reliably solve parametrized optimal control problems governed by time dependent nonlinear partial differential equations. Our goal is to provide a general tool to deal with the time evolution of several nonlinear optimality systems in many-query context, where a system must be analysed for various physical and geometrical features. Optimal control is a tool which can be used in order to fill the gab between collected data and mathematical model and it is usually related to very time consuming activities: inverse problems, statistics, etc. Standard discretization techniques may lead to unbearable simulations for real applications. We aim at showing how reduced order modelling can solve this issue. We rely on a space-time POD-Galerkin reduction in order to solve the optimal control problem in a low dimensional reduced space in a fast way for several parametric instances. The generality of the proposed algorithm is validated with a numerical test based on environmental sciences: a reduced optimal control problem governed by Shallow Waters Equations parametrized not only in the physics features, but also in the geometrical ones. We will show how the reduced model can be useful in order to recover desired velocity and height profiles more rapidly with respect to the standard simulation, not loosing in accuracy.
    @incollection{BallarinRozzaStrazzullo2020,
    author = {Francesco Ballarin and Gianluigi Rozza and Maria Strazzullo},
    title = {Space-time POD-Galerkin approach for parametric flow control},
    year = {2022},
    preprint = {https://arxiv.org/abs/2011.12101},
    doi = {10.1016/bs.hna.2021.12.009},
    editor = {Emmanuel Trélat and Enrique Zuazua},
    series = {Handbook of Numerical Analysis},
    publisher = {Elsevier},
    volume = {23},
    pages = {307-338},
    year = {2022},
    booktitle = {Numerical Control: Part A},
    abstract = {In this contribution we propose reduced order methods to fast and reliably solve parametrized optimal control problems governed by time dependent nonlinear partial differential equations. Our goal is to provide a general tool to deal with the time evolution of several nonlinear optimality systems in many-query context, where a system must be analysed for various physical and geometrical features. Optimal control is a tool which can be used in order to fill the gab between collected data and mathematical model and it is usually related to very time consuming activities: inverse problems, statistics, etc. Standard discretization techniques may lead to unbearable simulations for real applications. We aim at showing how reduced order modelling can solve this issue. We rely on a space-time POD-Galerkin reduction in order to solve the optimal control problem in a low dimensional reduced space in a fast way for several parametric instances. The generality of the proposed algorithm is validated with a numerical test based on environmental sciences: a reduced optimal control problem governed by Shallow Waters Equations parametrized not only in the physics features, but also in the geometrical ones. We will show how the reduced model can be useful in order to recover desired velocity and height profiles more rapidly with respect to the standard simulation, not loosing in accuracy.}
    }
  2. T. Kadeethum, F. Ballarin, D. O’Malley, Y. Choi, N. Bouklas, and H. Yoon, “Reduced order modeling with Barlow Twins self-supervised learning: Navigating the space between linear and nonlinear solution manifolds”, 2022.
    [BibTeX] [Abstract] [Download preprint]
    We propose a unified data-driven reduced order model (ROM) that bridges the performance gap between linear and nonlinear manifold approaches. Deep learning ROM (DL-ROM) using deep-convolutional autoencoders (DC-AE) has been shown to capture nonlinear solution manifolds but fails to perform adequately when linear subspace approaches such as proper orthogonal decomposition (POD) would be optimal. Besides, most DL-ROM models rely on convolutional layers, which might limit its application to only a structured mesh. The proposed framework in this study relies on the combination of an autoencoder (AE) and Barlow Twins (BT) self-supervised learning, where BT maximizes the information content of the embedding with the latent space through a joint embedding architecture. Through a series of benchmark problems of natural convection in porous media, BT-AE performs better than the previous DL-ROM framework by providing comparable results to POD-based approaches for problems where the solution lies within a linear subspace as well as DL-ROM autoencoder-based techniques where the solution lies on a nonlinear manifold; consequently, bridges the gap between linear and nonlinear reduced manifolds. Furthermore, this BT-AE framework can operate on unstructured meshes, which provides flexibility in its application to standard numerical solvers, on-site measurements, experimental data, or a combination of these sources.
    @unpublished{KadeethumBallarinOMalleyChoiBouklasYoon2022,
    author = {T. Kadeethum and F. Ballarin and D. O'Malley and Y. Choi and N. Bouklas and H. Yoon},
    title = {Reduced order modeling with Barlow Twins self-supervised learning: Navigating the space between linear and nonlinear solution manifolds},
    year = {2022},
    preprint = {https://arxiv.org/abs/2202.05460},
    abstract = {We propose a unified data-driven reduced order model (ROM) that bridges the performance gap between linear and nonlinear manifold approaches. Deep learning ROM (DL-ROM) using deep-convolutional autoencoders (DC-AE) has been shown to capture nonlinear solution manifolds but fails to perform adequately when linear subspace approaches such as proper orthogonal decomposition (POD) would be optimal. Besides, most DL-ROM models rely on convolutional layers, which might limit its application to only a structured mesh. The proposed framework in this study relies on the combination of an autoencoder (AE) and Barlow Twins (BT) self-supervised learning, where BT maximizes the information content of the embedding with the latent space through a joint embedding architecture. Through a series of benchmark problems of natural convection in porous media, BT-AE performs better than the previous DL-ROM framework by providing comparable results to POD-based approaches for problems where the solution lies within a linear subspace as well as DL-ROM autoencoder-based techniques where the solution lies on a nonlinear manifold; consequently, bridges the gap between linear and nonlinear reduced manifolds. Furthermore, this BT-AE framework can operate on unstructured meshes, which provides flexibility in its application to standard numerical solvers, on-site measurements, experimental data, or a combination of these sources.}
    }
  3. T. Kadeethum, F. Ballarin, Y. Choi, D. O’Malley, H. Yoon, and N. Bouklas, “Non-intrusive reduced order modeling of natural convection in porous media using convolutional autoencoders: Comparison with linear subspace techniques”, 2022.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]
    Natural convection in porous media is a highly nonlinear multiphysical problem relevant to many engineering applications (e.g., the process of CO2 sequestration). Here, we present a non-intrusive reduced order model of natural convection in porous media employing deep convolutional autoencoders for the compression and reconstruction and either radial basis function (RBF) interpolation or artificial neural networks (ANNs) for mapping parameters of partial differential equations (PDEs) on the corresponding nonlinear manifolds. To benchmark our approach, we also describe linear compression and reconstruction processes relying on proper orthogonal decomposition (POD) and ANNs. We present comprehensive comparisons among different models through three benchmark problems. The reduced order models, linear and nonlinear approaches, are much faster than the finite element model, obtaining a maximum speed-up of 7 * 10^6 because our framework is not bound by the Courant-Friedrichs-Lewy condition; hence, it could deliver quantities of interest at any given time contrary to the finite element model. Our model's accuracy still lies within a mean squared error of 0.07 (two-order of magnitude lower than the maximum value of the finite element results) in the worst-case scenario. We illustrate that, in specific settings, the nonlinear approach outperforms its linear counterpart and vice versa. We hypothesize that a visual comparison between principal component analysis (PCA) or t-Distributed Stochastic Neighbor Embedding (t-SNE) could indicate which method will perform better prior to employing any specific compression strategy.
    @unpublished{KadeethumBallarinChoiOMalleyYoonBouklas2021,
    author = {T. Kadeethum and F. Ballarin and Y. Choi and D. O'Malley and H. Yoon and N. Bouklas},
    title = {Non-intrusive reduced order modeling of natural convection in porous media using convolutional autoencoders: Comparison with linear subspace techniques},
    preprint = {https://arxiv.org/abs/2107.11460},
    journal = {Advances in Water Resources},
    volume = {160},
    pages = {104098},
    year = {2022},
    doi = {10.1016/j.advwatres.2021.104098},
    abstract = {Natural convection in porous media is a highly nonlinear multiphysical problem relevant to many engineering applications (e.g., the process of CO2 sequestration). Here, we present a non-intrusive reduced order model of natural convection in porous media employing deep convolutional autoencoders for the compression and reconstruction and either radial basis function (RBF) interpolation or artificial neural networks (ANNs) for mapping parameters of partial differential equations (PDEs) on the corresponding nonlinear manifolds. To benchmark our approach, we also describe linear compression and reconstruction processes relying on proper orthogonal decomposition (POD) and ANNs. We present comprehensive comparisons among different models through three benchmark problems. The reduced order models, linear and nonlinear approaches, are much faster than the finite element model, obtaining a maximum speed-up of 7 * 10^6 because our framework is not bound by the Courant-Friedrichs-Lewy condition; hence, it could deliver quantities of interest at any given time contrary to the finite element model. Our model's accuracy still lies within a mean squared error of 0.07 (two-order of magnitude lower than the maximum value of the finite element results) in the worst-case scenario. We illustrate that, in specific settings, the nonlinear approach outperforms its linear counterpart and vice versa. We hypothesize that a visual comparison between principal component analysis (PCA) or t-Distributed Stochastic Neighbor Embedding (t-SNE) could indicate which method will perform better prior to employing any specific compression strategy.}
    }
  4. M. Nonino, F. Ballarin, G. Rozza, and Y. Maday, “Projection based semi–implicit partitioned Reduced Basis Method for non parametrized and parametrized Fluid–Structure Interaction problems”, 2022.
    [BibTeX] [Abstract] [Download preprint]
    The goal of this manuscript is to present a partitioned Model Order Reduction method that is based on a semi–implicit projection scheme to solve multiphysics problems. We implement a Reduced Order Method based on a Proper Orthogonal Decomposition, with the aim of addressing both time–dependent and time–dependent, parametrized Fluid–Structure Interaction problems, where the fluid is incompressible and the structure is thick and two dimensional.
    @unpublished{NoninoBallarinRozzaMaday2022,
    author = {Monica Nonino and Francesco Ballarin and Gianluigi Rozza and Yvon Maday},
    title = {Projection based semi--implicit partitioned Reduced Basis Method for non parametrized and parametrized Fluid--Structure Interaction problems},
    year = {2022},
    preprint = {https://arxiv.org/abs/2201.03236},
    abstract = {The goal of this manuscript is to present a partitioned Model Order Reduction method that is based on a semi--implicit projection scheme to solve multiphysics problems. We implement a Reduced Order Method based on a Proper Orthogonal Decomposition, with the aim of addressing both time--dependent and time--dependent, parametrized Fluid--Structure Interaction problems, where the fluid is incompressible and the structure is thick and two dimensional.}
    }
  5. S. Perotto, G. Bellini, F. Ballarin, K. Calò, V. Mazzi, and U. Morbiducci, “Isogeometric Hierarchical Model Reduction for advection-diffusion process simulation in microchannels”, 2022.
    [BibTeX] [Abstract] [Download preprint]
    Microfluidics proved to be a key technology in various applications, allowing to reproduce large-scale laboratory settings at a more sustainable small-scale. The current effort is focused on enhancing the mixing process of different passive species at the micro-scale, where a laminar flow regime damps turbulence effects. Chaotic advection is often used to improve mixing effects also at very low Reynolds numbers. In particular, we focus on passive micromixers, where chaotic advection is mainly achieved by properly selecting the geometry of microchannels. In such a context, reduced order modeling can play a role, especially in the design of new geometries. In this chapter, we verify the reliability and the computational benefits lead by a Hierarchical Model (HiMod) reduction when modeling the transport of a passive scalar in an S-shaped microchannel. Such a geometric configuration provides an ideal setting where to apply a HiMod approximation, which exploits the presence of a leading dynamics to commute the original three-dimensional model into a system of one-dimensional coupled problems. It can be proved that HiMod reduction guarantees a very good accuracy when compared with a high-fidelity model, despite a drastic reduction in terms of number of unknowns.
    @unpublished{PerottoBelliniBallarinCaloMazziMorbiducci2022,
    author = {S. Perotto and G. Bellini and F. Ballarin and K. Cal\`o and V. Mazzi and U. Morbiducci},
    title = {Isogeometric Hierarchical Model Reduction for advection-diffusion process simulation in microchannels},
    year = {2022},
    preprint = {https://arxiv.org/abs/2205.08127},
    abstract = {Microfluidics proved to be a key technology in various applications, allowing to reproduce large-scale laboratory settings at a more sustainable small-scale. The current effort is focused on enhancing the mixing process of different passive species at the micro-scale, where a laminar flow regime damps turbulence effects. Chaotic advection is often used to improve mixing effects also at very low Reynolds numbers. In particular, we focus on passive micromixers, where chaotic advection is mainly achieved by properly selecting the geometry of microchannels. In such a context, reduced order modeling can play a role, especially in the design of new geometries. In this chapter, we verify the reliability and the computational benefits lead by a Hierarchical Model (HiMod) reduction when modeling the transport of a passive scalar in an S-shaped microchannel. Such a geometric configuration provides an ideal setting where to apply a HiMod approximation, which exploits the presence of a leading dynamics to commute the original three-dimensional model into a system of one-dimensional coupled problems. It can be proved that HiMod reduction guarantees a very good accuracy when compared with a high-fidelity model, despite a drastic reduction in terms of number of unknowns.}
    }
  6. P. Siena, M. Girfoglio, F. Ballarin, and G. Rozza, “Data-driven reduced order modelling for patient-specific hemodynamics of coronary artery bypass grafts with physical and geometrical parameters”, 2022.
    [BibTeX] [Abstract] [Download preprint]
    In this work the development of a machine learning-based Reduced Order Model (ROM) for the investigation of hemodynamics in a patient-specific configuration of Coronary Artery Bypass Graft (CABG) is proposed. The computational domain is referred to left branches of coronary arteries when a stenosis of the Left Main Coronary Artery (LMCA) occurs. The method extracts a reduced basis space from a collection of high-fidelity solutions via a Proper Orthogonal Decomposition (POD) algorithm and employs Artificial Neural Networks (ANNs) for the computation of the modal coefficients. The Full Order Model (FOM) is represented by the incompressible Navier-Stokes equations discretized using a Finite Volume (FV) technique. Both physical and geometrical parametrization are taken into account, the former one related to the inlet flow rate and the latter one related to the stenosis severity. With respect to the previous works focused on the development of a ROM framework for the evaluation of coronary artery disease, the novelties of our study include the use of the FV method in a patient-specific configuration, the use of a data-driven ROM technique and the mesh deformation strategy based on a Free Form Deformation (FFD) technique. The performance of our ROM approach is analyzed in terms of the error between full order and reduced order solutions as well as the speedup achieved at the online stage.
    @unpublished{SienaGirfoglioBallarinRozza2022,
    author = {P. Siena and M. Girfoglio and F. Ballarin and G. Rozza},
    title = {Data-driven reduced order modelling for patient-specific hemodynamics of coronary artery bypass grafts with physical and geometrical parameters},
    year = {2022},
    preprint = {https://arxiv.org/abs/2203.13682},
    abstract = {In this work the development of a machine learning-based Reduced Order Model (ROM) for the investigation of hemodynamics in a patient-specific configuration of Coronary Artery Bypass Graft (CABG) is proposed. The computational domain is referred to left branches of coronary arteries when a stenosis of the Left Main Coronary Artery (LMCA) occurs. The method extracts a reduced basis space from a collection of high-fidelity solutions via a Proper Orthogonal Decomposition (POD) algorithm and employs Artificial Neural Networks (ANNs) for the computation of the modal coefficients. The Full Order Model (FOM) is represented by the incompressible Navier-Stokes equations discretized using a Finite Volume (FV) technique. Both physical and geometrical parametrization are taken into account, the former one related to the inlet flow rate and the latter one related to the stenosis severity. With respect to the previous works focused on the development of a ROM framework for the evaluation of coronary artery disease, the novelties of our study include the use of the FV method in a patient-specific configuration, the use of a data-driven ROM technique and the mesh deformation strategy based on a Free Form Deformation (FFD) technique. The performance of our ROM approach is analyzed in terms of the error between full order and reduced order solutions as well as the speedup achieved at the online stage.}
    }
  7. M. Strazzullo, F. Ballarin, and G. Rozza, “POD-Galerkin model order reduction for parametrized nonlinear time-dependent optimal flow control: an application to shallow water equations”, Journal of Numerical Mathematics, 3(1), pp. 63-84, 2022.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]
    In this work we propose reduced order methods as a reliable strategy to efficiently solve parametrized optimal control problems governed by shallow waters equations in a solution tracking setting. The physical parametrized model we deal with is nonlinear and time dependent: this leads to very time consuming simulations which can be unbearable e.g. in a marine environmental monitoring plan application. Our aim is to show how reduced order modelling could help in studying different configurations and phenomena in a fast way. After building the optimality system, we rely on a POD-Galerkin reduction in order to solve the optimal control problem in a low dimensional reduced space. The presented theoretical framework is actually suited to general nonlinear time dependent optimal control problems. The proposed methodology is finally tested with a numerical experiment: the reduced optimal control problem governed by shallow waters equations reproduces the desired velocity and height profiles faster than the standard model, still remaining accurate.
    @article{StrazzulloBallarinRozza2020,
    author = {Maria Strazzullo and Francesco Ballarin and Gianluigi Rozza},
    title = {POD-Galerkin model order reduction for parametrized nonlinear time-dependent optimal flow control: an application to shallow water equations},
    year = {2022},
    doi = {10.1515/jnma-2020-0098},
    volume = {3},
    number = {1},
    pages = {63-84},
    journal = {Journal of Numerical Mathematics},
    preprint = {https://arxiv.org/abs/2003.09695},
    abstract = {In this work we propose reduced order methods as a reliable strategy to efficiently solve parametrized optimal control problems governed by shallow waters equations in a solution tracking setting. The physical parametrized model we deal with is nonlinear and time dependent: this leads to very time consuming simulations which can be unbearable e.g. in a marine environmental monitoring plan application. Our aim is to show how reduced order modelling could help in studying different configurations and phenomena in a fast way. After building the optimality system, we rely on a POD-Galerkin reduction in order to solve the optimal control problem in a low dimensional reduced space. The presented theoretical framework is actually suited to general nonlinear time dependent optimal control problems. The proposed methodology is finally tested with a numerical experiment: the reduced optimal control problem governed by shallow waters equations reproduces the desired velocity and height profiles faster than the standard model, still remaining accurate.}
    }

2021

  1. S. Ali, F. Ballarin, and G. Rozza, “A Reduced basis stabilization for the unsteady Stokes and Navier-Stokes equations”, 2021.
    [BibTeX] [Abstract] [Download preprint]
    In the Reduced Basis approximation of Stokes and Navier-Stokes problems, the Galerkin projection on the reduced spaces does not necessarily preserved the inf-sup stability even if the snapshots were generated through a stable full order method. Therefore, in this work we aim at building a stabilized Reduced Basis (RB) method for the approximation of unsteady Stokes and Navier-Stokes problems in parametric reduced order settings. This work extends the results presented for parametrized steady Stokes and Navier-Stokes problems in a work of ours \cite{Ali2018}. We apply classical residual-based stabilization techniques for finite element methods in full order, and then the RB method is introduced as Galerkin projection onto RB space. We compare this approach with supremizer enrichment options through several numerical experiments. We are interested to (numerically) guarantee the parametrized reduced inf-sup condition and to reduce the online computational costs.
    @unpublished{AliBallarinRozza2021,
    author = {S. Ali and F. Ballarin and G. Rozza},
    title = {A Reduced basis stabilization for the unsteady Stokes and Navier-Stokes equations},
    year = {2021},
    preprint = {https://arxiv.org/abs/2103.03553},
    abstract = {In the Reduced Basis approximation of Stokes and Navier-Stokes problems, the Galerkin projection on the reduced spaces does not necessarily preserved the inf-sup stability even if the snapshots were generated through a stable full order method. Therefore, in this work we aim at building a stabilized Reduced Basis (RB) method for the approximation of unsteady Stokes and Navier-Stokes problems in parametric reduced order settings. This work extends the results presented for parametrized steady Stokes and Navier-Stokes problems in a work of ours \cite{Ali2018}. We apply classical residual-based stabilization techniques for finite element methods in full order, and then the RB method is introduced as Galerkin projection onto RB space. We compare this approach with supremizer enrichment options through several numerical experiments. We are interested to (numerically) guarantee the parametrized reduced inf-sup condition and to reduce the online computational costs.}
    }
  2. G. Carere, M. Strazzullo, F. Ballarin, G. Rozza, and R. Stevenson, “A weighted POD-reduction approach for parametrized PDE-constrained Optimal Control Problems with random inputs and applications to environmental sciences”, Computers & Mathematics with Applications, 102, pp. 261-276, 2021.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]
    Reduced basis approximations of Optimal Control Problems (OCPs) governed by steady partial differential equations (PDEs) with random parametric inputs are analyzed and constructed. Such approximations are based on a Reduced Order Model, which in this work is constructed using the method of weighted Proper Orthogonal Decomposition. This Reduced Order Model then is used to efficiently compute the reduced basis approximation for any outcome of the random parameter. We demonstrate that such OCPs are well-posed by applying the adjoint approach, which also works in the presence of admissibility constraints and in the case of non linear-quadratic OCPs, and thus is more general than the conventional Lagrangian approach. We also show that a step in the construction of these Reduced Order Models, known as the aggregation step, is not fundamental and can in principle be skipped for noncoercive problems, leading to a cheaper online phase. Numerical applications in three scenarios from environmental science are considered, in which the governing PDE is steady and the control is distributed. Various parameter distributions are taken, and several implementations of the weighted Proper Orthogonal Decomposition are compared by choosing different quadrature rules.
    @article{CarereStrazzulloBallarinRozzaStevenson2021,
    author = {G. Carere and M. Strazzullo and F. Ballarin and G. Rozza and R. Stevenson},
    title = {A weighted POD-reduction approach for parametrized PDE-constrained Optimal Control Problems with random inputs and applications to environmental sciences},
    year = {2021},
    preprint = {https://arxiv.org/abs/2103.00632},
    journal = {Computers & Mathematics with Applications},
    volume = {102},
    pages = {261-276},
    doi = {10.1016/j.camwa.2021.10.020},
    abstract = {Reduced basis approximations of Optimal Control Problems (OCPs) governed by steady partial differential equations (PDEs) with random parametric inputs are analyzed and constructed. Such approximations are based on a Reduced Order Model, which in this work is constructed using the method of weighted Proper Orthogonal Decomposition. This Reduced Order Model then is used to efficiently compute the reduced basis approximation for any outcome of the random parameter. We demonstrate that such OCPs are well-posed by applying the adjoint approach, which also works in the presence of admissibility constraints and in the case of non linear-quadratic OCPs, and thus is more general than the conventional Lagrangian approach. We also show that a step in the construction of these Reduced Order Models, known as the aggregation step, is not fundamental and can in principle be skipped for noncoercive problems, leading to a cheaper online phase. Numerical applications in three scenarios from environmental science are considered, in which the governing PDE is steady and the control is distributed. Various parameter distributions are taken, and several implementations of the weighted Proper Orthogonal Decomposition are compared by choosing different quadrature rules.}
    }
  3. E. Fevola, F. Ballarin, L. Jiménez-Juan, S. Fremes, S. Grivet-Talocia, G. Rozza, and P. Triverio, “An optimal control approach to determine resistance-type boundary conditions from in-vivo data for cardiovascular simulations”, International Journal for Numerical Methods in Biomedical Engineering, 37(10), pp. e3516, 2021.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]
    The choice of appropriate boundary conditions is a fundamental step in computational fluid dynamics (CFD) simulations of the cardiovascular system. Boundary conditions, in fact, highly affect the computed pressure and flow rates, and consequently haemodynamic indicators such as wall shear stress, which are of clinical interest. Devising automated procedures for the selection of boundary conditions is vital to achieve repeatable simulations. However, the most common techniques do not automatically assimilate patient-specific data, relying instead on expensive and time-consuming manual tuning procedures. In this work, we propose a technique for the automated estimation of outlet boundary conditions based on optimal control. The values of resistive boundary conditions are set as control variables and optimized to match available patient-specific data. Experimental results on four aortic arches demonstrate that the proposed framework can assimilate 4D-Flow MRI data more accurately than two other common techniques based on Murray's law and Ohm's law.
    @article{FevolaBallarinJimenezJuanFremesGrivetTalociaRozzaTriverio2021,
    author = {E. Fevola and F. Ballarin and L. Jiménez-Juan and S. Fremes and S. Grivet-Talocia and G. Rozza and P. Triverio},
    title = {An optimal control approach to determine resistance-type boundary conditions from in-vivo data for cardiovascular simulations},
    year = {2021},
    preprint = {https://arxiv.org/abs/2104.13284},
    doi = {10.1002/cnm.3516},
    volume = {37},
    number = {10},
    pages = {e3516},
    journal = {International Journal for Numerical Methods in Biomedical Engineering},
    abstract = {The choice of appropriate boundary conditions is a fundamental step in computational fluid dynamics (CFD) simulations of the cardiovascular system. Boundary conditions, in fact, highly affect the computed pressure and flow rates, and consequently haemodynamic indicators such as wall shear stress, which are of clinical interest. Devising automated procedures for the selection of boundary conditions is vital to achieve repeatable simulations. However, the most common techniques do not automatically assimilate patient-specific data, relying instead on expensive and time-consuming manual tuning procedures. In this work, we propose a technique for the automated estimation of outlet boundary conditions based on optimal control. The values of resistive boundary conditions are set as control variables and optimized to match available patient-specific data. Experimental results on four aortic arches demonstrate that the proposed framework can assimilate 4D-Flow MRI data more accurately than two other common techniques based on Murray's law and Ohm's law.}
    }
  4. M. Girfoglio, L. Scandurra, F. Ballarin, G. Infantino, F. Nicolò, A. Montalto, G. Rozza, R. Scrofani, M. Comisso, and F. Musumeci, “Non-intrusive data-driven ROM framework for hemodynamics problems”, Acta Mechanica Sinica, 37, pp. 1183-1191, 2021.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]
    Reduced order modeling (ROM) techniques are numerical methods that approximate the solution to parametric partial differential equation (PDE) is approximated by properly combining the high-fidelity solutions of the problem obtained for several configurations, i.e. for several properly chosen values of the physical/geometrical parameters characterizing the problem. In this contribution, we propose an efficient non-intrusive data-driven framework involving ROM techniques in computational fluid dynamics (CFD) for hemodynamics applications. By starting from a database of high-fidelity solutions related to a certain values of the parameters, we apply the proper orthogonal decomposition with interpolation (PODI) and then reconstruct the variables of interest for new values of the parameters, i.e. different values from the ones included in the database. Furthermore, we present a preliminary web application through which one can run the ROM with a very user-friendly approach, without the need of having expertise in the numerical analysis and scientific computing field. The case study we have chosen to test the efficiency of our algorithm is represented by the aortic blood flow pattern in presence of a Left Ventricular Assist Device (LVAD) when varying the pump flow rate.
    @article{GirfoglioScandurraBallarinInfantinoNicoloMontaltoRozzaScrofaniComissoMusumeci2020,
    author = {Michele Girfoglio and Leonardo Scandurra and Francesco Ballarin and Giuseppe Infantino and Francesca Nicolò and Andrea Montalto and Gianluigi Rozza and Roberto Scrofani and Marina Comisso and Francesco Musumeci},
    title = {Non-intrusive data-driven ROM framework for hemodynamics problems},
    year = {2021},
    preprint = {https://arxiv.org/abs/2010.08139},
    journal = {Acta Mechanica Sinica},
    volume = {37},
    pages = {1183-1191},
    year = {2021},
    doi = {10.1007/s10409-021-01090-2},
    abstract = {Reduced order modeling (ROM) techniques are numerical methods that approximate the solution to parametric partial differential equation (PDE) is approximated by properly combining the high-fidelity solutions of the problem obtained for several configurations, i.e. for several properly chosen values of the physical/geometrical parameters characterizing the problem. In this contribution, we propose an efficient non-intrusive data-driven framework involving ROM techniques in computational fluid dynamics (CFD) for hemodynamics applications. By starting from a database of high-fidelity solutions related to a certain values of the parameters, we apply the proper orthogonal decomposition with interpolation (PODI) and then reconstruct the variables of interest for new values of the parameters, i.e. different values from the ones included in the database. Furthermore, we present a preliminary web application through which one can run the ROM with a very user-friendly approach, without the need of having expertise in the numerical analysis and scientific computing field. The case study we have chosen to test the efficiency of our algorithm is represented by the aortic blood flow pattern in presence of a Left Ventricular Assist Device (LVAD) when varying the pump flow rate.}
    }
  5. T. Kadeethum, F. Ballarin, and N. Bouklas, “Data-driven reduced order modeling of poroelasticity of heterogeneous media based on a discontinuous Galerkin approximation”, GEM – International Journal on Geomathematics, 12, pp. 12, 2021.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]
    We present a non-intrusive model reduction framework for linear poroelasticity problems in heterogeneous porous media using proper orthogonal decomposition (POD) and neural networks, based on the usual offline-online paradigm. As the conductivity of porous media can be highly heterogeneous and span several orders of magnitude, we utilize the interior penalty discontinuous Galerkin (DG) method as a full order solver to handle discontinuity and ensure local mass conservation during the offline stage. We then use POD as a data compression tool and compare the nested POD technique, in which time and uncertain parameter domains are compressed consecutively, to the classical POD method in which all domains are compressed simultaneously. The neural networks are finally trained to map the set of uncertain parameters, which could correspond to material properties, boundary conditions, or geometric characteristics, to the collection of coefficients calculated from an L2 projection over the reduced basis. We then perform a non-intrusive evaluation of the neural networks to obtain coefficients corresponding to new values of the uncertain parameters during the online stage. We show that our framework provides reasonable approximations of the DG solution, but it is significantly faster. Moreover, the reduced order framework can capture sharp discontinuities of both displacement and pressure fields resulting from the heterogeneity in the media conductivity, which is generally challenging for intrusive reduced order methods. The sources of error are presented, showing that the nested POD technique is computationally advantageous and still provides comparable accuracy to the classical POD method. We also explore the effect of different choices of the hyperparameters of the neural network on the framework performance.
    @article{KadeethumBallarinBouklas2021,
    author = {T. Kadeethum and F. Ballarin and N. Bouklas},
    title = {Data-driven reduced order modeling of poroelasticity of heterogeneous media based on a discontinuous Galerkin approximation},
    year = {2021},
    journal = {GEM - International Journal on Geomathematics},
    volume = {12},
    pages = {12},
    preprint = {https://arxiv.org/abs/2101.11810},
    doi = {10.1007/s13137-021-00180-4},
    abstract = {We present a non-intrusive model reduction framework for linear poroelasticity problems in heterogeneous porous media using proper orthogonal decomposition (POD) and neural networks, based on the usual offline-online paradigm. As the conductivity of porous media can be highly heterogeneous and span several orders of magnitude, we utilize the interior penalty discontinuous Galerkin (DG) method as a full order solver to handle discontinuity and ensure local mass conservation during the offline stage. We then use POD as a data compression tool and compare the nested POD technique, in which time and uncertain parameter domains are compressed consecutively, to the classical POD method in which all domains are compressed simultaneously. The neural networks are finally trained to map the set of uncertain parameters, which could correspond to material properties, boundary conditions, or geometric characteristics, to the collection of coefficients calculated from an L2 projection over the reduced basis. We then perform a non-intrusive evaluation of the neural networks to obtain coefficients corresponding to new values of the uncertain parameters during the online stage. We show that our framework provides reasonable approximations of the DG solution, but it is significantly faster. Moreover, the reduced order framework can capture sharp discontinuities of both displacement and pressure fields resulting from the heterogeneity in the media conductivity, which is generally challenging for intrusive reduced order methods. The sources of error are presented, showing that the nested POD technique is computationally advantageous and still provides comparable accuracy to the classical POD method. We also explore the effect of different choices of the hyperparameters of the neural network on the framework performance.}
    }
  6. T. Kadeethum, S. Lee, F. Ballarin, J. Choo, and H. M. Nick, “A locally conservative mixed finite element framework for coupled hydro-mechanical-chemical processes in heterogeneous porous media”, Computers & Geosciences, 152, pp. 104774, 2021.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]
    This paper presents a mixed finite element framework for coupled hydro-mechanical–chemical processes in heterogeneous porous media. The framework combines two types of locally conservative discretization schemes: (1) an enriched Galerkin method for reactive flow, and (2) a three-field mixed finite element method for coupled fluid flow and solid deformation. This combination ensures local mass conservation, which is critical to flow and transport in heterogeneous porous media, with a relatively affordable computational cost. A particular class of the framework is constructed for calcite precipitation/dissolution reactions, incorporating their nonlinear effects on the fluid viscosity and solid deformation. Linearization schemes and algorithms for solving the nonlinear algebraic system are also presented. Through numerical examples of various complexity, we demonstrate that the proposed framework is a robust and efficient computational method for simulation of reactive flow and transport in deformable porous media, even when the material properties are strongly heterogeneous and anisotropic.
    @article{KadeethumLeeBallarinChooNick2020,
    author = {T. Kadeethum and S. Lee and F. Ballarin and J. Choo and H.M. Nick},
    title = {A locally conservative mixed finite element framework for coupled hydro-mechanical-chemical processes in heterogeneous porous media},
    year = {2021},
    journal = {Computers & Geosciences},
    volume = {152},
    pages = {104774},
    preprint = {https://arxiv.org/abs/2010.04994},
    doi = {10.1016/j.cageo.2021.104774},
    abstract = {This paper presents a mixed finite element framework for coupled hydro-mechanical–chemical processes in heterogeneous porous media. The framework combines two types of locally conservative discretization schemes: (1) an enriched Galerkin method for reactive flow, and (2) a three-field mixed finite element method for coupled fluid flow and solid deformation. This combination ensures local mass conservation, which is critical to flow and transport in heterogeneous porous media, with a relatively affordable computational cost. A particular class of the framework is constructed for calcite precipitation/dissolution reactions, incorporating their nonlinear effects on the fluid viscosity and solid deformation. Linearization schemes and algorithms for solving the nonlinear algebraic system are also presented. Through numerical examples of various complexity, we demonstrate that the proposed framework is a robust and efficient computational method for simulation of reactive flow and transport in deformable porous media, even when the material properties are strongly heterogeneous and anisotropic.}
    }
  7. T. Kadeethum, H. M. Nick, S. Lee, and F. Ballarin, “Enriched Galerkin Discretization for Modeling Poroelasticity and Permeability Alteration in Heterogeneous Porous Media”, Journal of Computational Physics, 427, pp. 110030, 2021.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]
    Accurate simulation of the coupled fluid flow and solid deformation in porous media is challenging, especially when the media permeability and storativity are heterogeneous. We apply the enriched Galerkin (EG) finite element method for the Biot's system. Block structure used to compose the enriched space and linearization and iterative schemes employed to solve the coupled media permeability alteration are illustrated. The open-source platform used to build the block structure is presented and illustrate that it helps the enriched Galerkin method easily adaptable to any existing discontinuous Galerkin codes. Subsequently, we compare the EG method with the classic continuous Galerkin (CG) and discontinuous Galerkin (DG) finite element methods. While these methods provide similar approximations for the pressure solution of Terzaghi's one-dimensional consolidation, the CG method produces spurious oscillations in fluid pressure and volumetric strain solutions at material interfaces that have permeability contrast and does not conserve mass locally. As a result, the flux approximation of the CG method is significantly different from the one of EG and DG methods, especially for the soft materials. The difference of flux approximation between EG and DG methods is insignificant; still, the EG method demands approximately two and three times fewer degrees of freedom than the DG method for two- and three-dimensional geometries, respectively. Lastly, we illustrate that the EG method produces accurate results even for much coarser meshes.
    @article{KadeethumNickLeeBallarin2020b,
    author = {T. Kadeethum and H.M. Nick and S. Lee and F. Ballarin},
    title = {Enriched Galerkin Discretization for Modeling Poroelasticity and Permeability Alteration in Heterogeneous Porous Media},
    journal = {Journal of Computational Physics},
    volume = {427},
    pages = {110030},
    year = {2021},
    doi = {10.1016/j.jcp.2020.110030},
    preprint = {https://arxiv.org/abs/2010.06653},
    abstract = {Accurate simulation of the coupled fluid flow and solid deformation in porous media is challenging, especially when the media permeability and storativity are heterogeneous. We apply the enriched Galerkin (EG) finite element method for the Biot's system. Block structure used to compose the enriched space and linearization and iterative schemes employed to solve the coupled media permeability alteration are illustrated. The open-source platform used to build the block structure is presented and illustrate that it helps the enriched Galerkin method easily adaptable to any existing discontinuous Galerkin codes. Subsequently, we compare the EG method with the classic continuous Galerkin (CG) and discontinuous Galerkin (DG) finite element methods. While these methods provide similar approximations for the pressure solution of Terzaghi's one-dimensional consolidation, the CG method produces spurious oscillations in fluid pressure and volumetric strain solutions at material interfaces that have permeability contrast and does not conserve mass locally. As a result, the flux approximation of the CG method is significantly different from the one of EG and DG methods, especially for the soft materials. The difference of flux approximation between EG and DG methods is insignificant; still, the EG method demands approximately two and three times fewer degrees of freedom than the DG method for two- and three-dimensional geometries, respectively. Lastly, we illustrate that the EG method produces accurate results even for much coarser meshes.}
    }
  8. M. Nonino, F. Ballarin, and G. Rozza, “A Monolithic and a Partitioned, Reduced Basis Method for Fluid-Structure Interaction Problems”, Fluids, 6(6), pp. 229, 2021.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]
    The aim of this work is to present a brief report concerning the various aspects of the Reduced Basis Method within Fluid-Structure Interaction problems. The idea is to adopt two different procedures and apply them to the same test case. First we preform a reduction technique that is based on a monolithic procedure, where we therefore solve all at once the fluid and the solid problem. Then we present an alternative reduction technique, that is based on a partitioned (or segregated) procedure instead: here the fluid and the solid problem are solved separately and then coupled through a fixed point strategy. The toy problem that we consider is the Turek-Hron benchmark test case, with a fluid Reynolds number Re = 100, which is known to lead to the formation of Karman vortexes in the fluid, and a periodically oscillating behaviour in the structure.
    @article{NoninoBallarinRozza2021,
    author = {M. Nonino and F. Ballarin and G. Rozza},
    title = {A Monolithic and a Partitioned, Reduced Basis Method for Fluid-Structure Interaction Problems},
    year = {2021},
    journal = {Fluids},
    volume = {6},
    number = {6},
    pages = {229},
    preprint = {https://arxiv.org/abs/2104.09882},
    doi = {10.3390/fluids6060229},
    abstract = {The aim of this work is to present a brief report concerning the various aspects of the Reduced Basis Method within Fluid-Structure Interaction problems. The idea is to adopt two different procedures and apply them to the same test case. First we preform a reduction technique that is based on a monolithic procedure, where we therefore solve all at once the fluid and the solid problem. Then we present an alternative reduction technique, that is based on a partitioned (or segregated) procedure instead: here the fluid and the solid problem are solved separately and then coupled through a fixed point strategy. The toy problem that we consider is the Turek-Hron benchmark test case, with a fluid Reynolds number Re = 100, which is known to lead to the formation of Karman vortexes in the fluid, and a periodically oscillating behaviour in the structure.}
    }
  9. F. Pichi, F. Ballarin, G. Rozza, and J. S. Hesthaven, “An artificial neural network approach to bifurcating phenomena in computational fluid dynamics”, 2021.
    [BibTeX] [Abstract] [Download preprint]
    This work deals with the investigation of bifurcating fluid phenomena using a reduced order modelling setting aided by artificial neural networks. We discuss the POD-NN approach dealing with non-smooth solutions set of nonlinear parametrized PDEs. Thus, we study the Navier-Stokes equations describing: (i) the Coanda effect in a channel, and (ii) the lid driven triangular cavity flow, in a physical/geometrical multi-parametrized setting, considering the effects of the domain's configuration on the position of the bifurcation points. Finally, we propose a reduced manifold-based bifurcation diagram for a non-intrusive recovery of the critical points evolution. Exploiting such detection tool, we are able to efficiently obtain information about the pattern flow behaviour, from symmetry breaking profiles to attaching/spreading vortices, even at high Reynolds numbers.
    @unpublished{PichiBallarinRozzaHesthaven2021,
    author = {F. Pichi and F. Ballarin and G. Rozza and J. S. Hesthaven},
    title = {An artificial neural network approach to bifurcating phenomena in computational fluid dynamics},
    year = {2021},
    preprint = {https://arxiv.org/abs/2109.10765},
    abstract = {This work deals with the investigation of bifurcating fluid phenomena using a reduced order modelling setting aided by artificial neural networks. We discuss the POD-NN approach dealing with non-smooth solutions set of nonlinear parametrized PDEs. Thus, we study the Navier-Stokes equations describing: (i) the Coanda effect in a channel, and (ii) the lid driven triangular cavity flow, in a physical/geometrical multi-parametrized setting, considering the effects of the domain's configuration on the position of the bifurcation points. Finally, we propose a reduced manifold-based bifurcation diagram for a non-intrusive recovery of the critical points evolution. Exploiting such detection tool, we are able to efficiently obtain information about the pattern flow behaviour, from symmetry breaking profiles to attaching/spreading vortices, even at high Reynolds numbers.}
    }
  10. N. V. Shah, M. Girfoglio, P. Quintela, G. Rozza, A. Lengomin, F. Ballarin, and P. Barral, “Finite element based model order reduction for parametrized one-way coupled steady state linear thermomechanical problems”, 2021.
    [BibTeX] [Abstract] [Download preprint]
    This contribution focuses on the development of Model Order Reduction (MOR) for one-way coupled steady state linear thermomechanical problems in a finite element setting. We apply Proper Orthogonal Decomposition (POD) for the computation of reduced basis space. On the other hand, for the evaluation of the modal coefficients, we use two different methodologies: the one based on the Galerkin projection (G) and the other one based on Artificial Neural Network (ANN). We aim at comparing POD-G and POD-ANN in terms of relevant features including errors and computational efficiency. In this context, both physical and geometrical parametrization are considered. We also carry out a validation of the Full Order Model (FOM) based on customized benchmarks in order to provide a complete computational pipeline. The framework proposed is applied to a relevant industrial problem related to the investigation of thermomechanical phenomena arising in blast furnace hearth walls.
    @unpublished{ShahGirfoglioQuintelaRozzaLengominBallarinBarral2021,
    author = {N. V. Shah and M. Girfoglio and P. Quintela and G. Rozza and A. Lengomin and F. Ballarin and P. Barral},
    title = {Finite element based model order reduction for parametrized one-way coupled steady state linear thermomechanical problems},
    year = {2021},
    preprint = {https://arxiv.org/abs/2111.08534},
    abstract = {This contribution focuses on the development of Model Order Reduction (MOR) for one-way coupled steady state linear thermomechanical problems in a finite element setting. We apply Proper Orthogonal Decomposition (POD) for the computation of reduced basis space. On the other hand, for the evaluation of the modal coefficients, we use two different methodologies: the one based on the Galerkin projection (G) and the other one based on Artificial Neural Network (ANN). We aim at comparing POD-G and POD-ANN in terms of relevant features including errors and computational efficiency. In this context, both physical and geometrical parametrization are considered. We also carry out a validation of the Full Order Model (FOM) based on customized benchmarks in order to provide a complete computational pipeline. The framework proposed is applied to a relevant industrial problem related to the investigation of thermomechanical phenomena arising in blast furnace hearth walls.}
    }
  11. M. Strazzullo, M. Girfoglio, F. Ballarin, T. Iliescu, and G. Rozza, “Consistency of the Full and Reduced Order Models for Evolve-Filter-Relax Regularization of Convection-Dominated, Marginally-Resolved Flows”, 2021.
    [BibTeX] [Abstract] [Download preprint]
    Numerical stabilization is often used to eliminate (alleviate) the spurious oscillations generally produced by full order models (FOMs) in under-resolved or marginally-resolved simulations of convection-dominated flows. In this paper, we investigate the role of numerical stabilization in reduced order models (ROMs) of marginally-resolved convection-dominated flows. Specifically, we investigate the FOM-ROM consistency, i.e., whether the numerical stabilization is beneficial both at the FOM and the ROM level. As a numerical stabilization strategy, we focus on the evolve-filter-relax (EFR) regularization algorithm, which centers around spatial filtering. To investigate the FOM-ROM consistency, we consider two ROM strategies: (I) the EFR-ROM, in which the EFR stabilization is used at the FOM level, but not at the ROM level; and (ii) the EFR-EFRROM, in which the EFR stabilization is used both at the FOM and at the ROM level. We compare the EFR-ROM with the EFR-EFRROM in the numerical simulation of a 2D flow past a circular cylinder in the convection-dominated, marginally-resolved regime. We also perform model reduction with respect to both time and Reynolds number. Our numerical investigation shows that the EFR-EFRROM is more accurate than the EFR-ROM, which suggests that FOM-ROM consistency is beneficial in convection-dominated,marginally-resolved flows.
    @unpublished{StrazzulloGirfoglioBallarinIliescuRozza2021,
    author = {M. Strazzullo and M. Girfoglio and F. Ballarin and T. Iliescu and G. Rozza},
    title = {Consistency of the Full and Reduced Order Models for Evolve-Filter-Relax Regularization of Convection-Dominated, Marginally-Resolved Flows},
    year = {2021},
    preprint = {https://arxiv.org/abs/2110.05093},
    abstract = {Numerical stabilization is often used to eliminate (alleviate) the spurious oscillations generally produced by full order models (FOMs) in under-resolved or marginally-resolved simulations of convection-dominated flows. In this paper, we investigate the role of numerical stabilization in reduced order models (ROMs) of marginally-resolved convection-dominated flows. Specifically, we investigate the FOM-ROM consistency, i.e., whether the numerical stabilization is beneficial both at the FOM and the ROM level. As a numerical stabilization strategy, we focus on the evolve-filter-relax (EFR) regularization algorithm, which centers around spatial filtering. To investigate the FOM-ROM consistency, we consider two ROM strategies: (I) the EFR-ROM, in which the EFR stabilization is used at the FOM level, but not at the ROM level; and (ii) the EFR-EFRROM, in which the EFR stabilization is used both at the FOM and at the ROM level. We compare the EFR-ROM with the EFR-EFRROM in the numerical simulation of a 2D flow past a circular cylinder in the convection-dominated, marginally-resolved regime. We also perform model reduction with respect to both time and Reynolds number. Our numerical investigation shows that the EFR-EFRROM is more accurate than the EFR-ROM, which suggests that FOM-ROM consistency is beneficial in convection-dominated,marginally-resolved flows.}
    }
  12. M. Strazzullo, F. Ballarin, and G. Rozza, “A Certified Reduced Basis Method for Linear Parametrized Parabolic Optimal Control Problems in Space-Time Formulation”, 2021.
    [BibTeX] [Abstract] [Download preprint]
    In this work, we propose to efficently solve time dependent parametrized optimal control problems governed by parabolic partial differential equations through the certified reduced basis method. In particular, we will exploit an error estimator procedure, based on easy-to-compute quantities which guarantee a rigorous and efficient bound for the error of the involved variables. First of all, we propose the analysis of the problem at hand, proving its well-posedness thanks to Nečas - Babuška theory for distributed and boundary controls in a space-time formulation. Then, we derive error estimators to apply a Greedy method during the offline stage, in order to perform, during the online stage, a Galerkin projection onto a low-dimensional space spanned by properly chosen high-fidelity solutions. We tested the error estimators on two model problems governed by a Graetz flow: a physical parametrized distributed optimal control problem and a boundary optimal control problem with physical and geometrical parameters. The results have been compared to a previously proposed bound, based on the exact computation of the Babu\v ska inf-sup constant, in terms of reliability and computational costs. We remark that our findings still hold in the steady setting and we propose a brief insight also for this simpler formulation.
    @unpublished{StrazzulloBallarinRozza2021,
    author = {M. Strazzullo and F. Ballarin and G. Rozza},
    title = {A Certified Reduced Basis Method for Linear Parametrized Parabolic Optimal Control Problems in Space-Time Formulation},
    year = {2021},
    preprint = {https://arxiv.org/abs/2103.00460},
    abstract = {In this work, we propose to efficently solve time dependent parametrized optimal control problems governed by parabolic partial differential equations through the certified reduced basis method. In particular, we will exploit an error estimator procedure, based on easy-to-compute quantities which guarantee a rigorous and efficient bound for the error of the involved variables. First of all, we propose the analysis of the problem at hand, proving its well-posedness thanks to Nečas - Babuška theory for distributed and boundary controls in a space-time formulation. Then, we derive error estimators to apply a Greedy method during the offline stage, in order to perform, during the online stage, a Galerkin projection onto a low-dimensional space spanned by properly chosen high-fidelity solutions. We tested the error estimators on two model problems governed by a Graetz flow: a physical parametrized distributed optimal control problem and a boundary optimal control problem with physical and geometrical parameters. The results have been compared to a previously proposed bound, based on the exact computation of the Babu\v ska inf-sup constant, in terms of reliability and computational costs. We remark that our findings still hold in the steady setting and we propose a brief insight also for this simpler formulation.}
    }
  13. M. Strazzullo, Z. Zainib, F. Ballarin, and G. Rozza, “Reduced Order Methods for Parametrized Non-linear and Time Dependent Optimal Flow Control Problems, Towards Applications in Biomedical and Environmental Sciences”, in Numerical Mathematics and Advanced Applications ENUMATH 2019, 2021, pp. 841–850.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]
    We introduce reduced order methods as an efficient strategy to solve parametrized non-linear and time dependent optimal flow control problems governed by partial differential equations. Indeed, optimal control problems require a huge computational effort in order to be solved, most of all in a physical and/or geometrical parametrized setting. Reduced order methods are a reliably suitable approach, increasingly gaining popularity, to achieve rapid and accurate optimal solutions in several fields, such as in biomedical and environmental sciences. In this work, we exploit POD-Galerkin reduction over a parametrized optimality system, derived from Karush-Kuhn-Tucker conditions. The methodology presented is tested on two boundary control problems, governed respectively by (i) time dependent Stokes equations and (ii) steady non-linear Navier-Stokes equations.
    @InProceedings{StrazzulloZainibBallarinRozza2019,
    author = {Maria Strazzullo and Zakia Zainib and Francesco Ballarin and Gianluigi Rozza},
    editor = {Vermolen, Fred J. and Vuik, Cornelis},
    title = {Reduced Order Methods for Parametrized Non-linear and Time Dependent Optimal Flow Control Problems, Towards Applications in Biomedical and Environmental Sciences},
    booktitle = {Numerical Mathematics and Advanced Applications ENUMATH 2019},
    year = {2021},
    publisher = {Springer International Publishing},
    pages = {841--850},
    preprint = {https://arxiv.org/abs/1912.07886},
    doi = {10.1007/978-3-030-55874-1_83},
    abstract = {We introduce reduced order methods as an efficient strategy to solve parametrized non-linear and time dependent optimal flow control problems governed by partial differential equations. Indeed, optimal control problems require a huge computational effort in order to be solved, most of all in a physical and/or geometrical parametrized setting. Reduced order methods are a reliably suitable approach, increasingly gaining popularity, to achieve rapid and accurate optimal solutions in several fields, such as in biomedical and environmental sciences. In this work, we exploit POD-Galerkin reduction over a parametrized optimality system, derived from Karush-Kuhn-Tucker conditions. The methodology presented is tested on two boundary control problems, governed respectively by (i) time dependent Stokes equations and (ii) steady non-linear Navier-Stokes equations.}
    }
  14. Z. Zainib, F. Ballarin, S. Fremes, P. Triverio, L. Jiménez-Juan, and G. Rozza, “Reduced order methods for parametric optimal flow control in coronary bypass grafts, towards patient-specific data assimilation”, International Journal for Numerical Methods in Biomedical Engineering, 37(12), pp. e3367, 2021.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]
    Coronary artery bypass grafts (CABG) surgery is an invasive procedure performed to circumvent partial or complete blood flow blockage in coronary artery disease (CAD). In this work, we apply a numerical optimal flow control model to patient-specific geometries of CABG, reconstructed from clinical images of real-life surgical cases, in parameterized settings. The aim of these applications is to match known physiological data with numerical hemodynamics corresponding to different scenarios, arisen by tuning some parameters. Such applications are an initial step towards matching patient-specific physiological data in patient-specific vascular geometries as best as possible. Two critical challenges that reportedly arise in such problems are, (i). lack of robust quantification of meaningful boundary conditions required to match known data as best as possible and (ii). high computational cost. In this work, we utilize unknown control variables in the optimal flow control problems to take care of the first challenge. Moreover, to address the second challenge, we propose a time-efficient and reliable computational environment for such parameterized problems by projecting them onto a low-dimensional solution manifold through proper orthogonal decomposition (POD)–Galerkin.
    @article{ZainibBallarinFremesTriverioJimenezJuanRozza2020,
    author = {Zakia Zainib and Francesco Ballarin and Stephen Fremes and Piero Triverio and Laura Jiménez-Juan and Gianluigi Rozza},
    title = {Reduced order methods for parametric optimal flow control in coronary bypass grafts, towards patient-specific data assimilation},
    year = {2021},
    preprint = {https://arxiv.org/abs/1911.01409},
    doi = {10.1002/cnm.3367},
    volume = {37},
    number = {12},
    pages = {e3367},
    journal = {International Journal for Numerical Methods in Biomedical Engineering},
    abstract = {Coronary artery bypass grafts (CABG) surgery is an invasive procedure performed to circumvent partial or complete blood flow blockage in coronary artery disease (CAD). In this work, we apply a numerical optimal flow control model to patient-specific geometries of CABG, reconstructed from clinical images of real-life surgical cases, in parameterized settings. The aim of these applications is to match known physiological data with numerical hemodynamics corresponding to different scenarios, arisen by tuning some parameters. Such applications are an initial step towards matching patient-specific physiological data in patient-specific vascular geometries as best as possible.
    Two critical challenges that reportedly arise in such problems are, (i). lack of robust quantification of meaningful boundary conditions required to match known data as best as possible and (ii). high computational cost. In this work, we utilize unknown control variables in the optimal flow control problems to take care of the first challenge. Moreover, to address the second challenge, we propose a time-efficient and reliable computational environment for such parameterized problems by projecting them onto a low-dimensional solution manifold through proper orthogonal decomposition (POD)--Galerkin.}
    }
  15. M. Zancanaro, F. Ballarin, S. Perotto, and G. Rozza, “Hierarchical Model Reduction Techniques for Flow Modeling in a Parametrized Setting”, Multiscale Modeling & Simulation, 19(1), pp. 267-293, 2021.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]
    In this work we focus on two different methods to deal with parametrized partial differential equations in an efficient and accurate way. Starting from high fidelity approximations built via the hierarchical model reduction discretization, we consider two approaches, both based on a projection model reduction technique. The two methods differ for the algorithm employed during the construction of the reduced basis. In particular, the former employs the proper orthogonal decomposition, while the latter relies on a greedy algorithm according to the certified reduced basis technique. The two approaches are preliminarily compared on two-dimensional scalar and vector test cases.
    @article{ZancanaroBallarinPerottoRozza2019,
    author = {Matteo Zancanaro and Francesco Ballarin and Simona Perotto and Gianluigi Rozza},
    title = {Hierarchical Model Reduction Techniques for Flow Modeling in a Parametrized Setting},
    journal = {Multiscale Modeling \& Simulation},
    volume = {19},
    number = {1},
    pages = {267-293},
    year = {2021},
    doi = {10.1137/19M1285330},
    preprint = {https://arxiv.org/abs/1909.01668},
    abstract = {In this work we focus on two different methods to deal with parametrized partial differential equations in an efficient and accurate way. Starting from high fidelity approximations built via the hierarchical model reduction discretization, we consider two approaches, both based on a projection model reduction technique. The two methods differ for the algorithm employed during the construction of the reduced basis. In particular, the former employs the proper orthogonal decomposition, while the latter relies on a greedy algorithm according to the certified reduced basis technique. The two approaches are preliminarily compared on two-dimensional scalar and vector test cases.}
    }

2020

  1. S. Ali, F. Ballarin, and G. Rozza, “Stabilized reduced basis methods for parametrized steady Stokes and Navier-Stokes equations”, Computers & Mathematics with Applications, 80(11), pp. 2399-2416, 2020.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]
    It is well known in the Reduced Basis approximation of saddle point problems that the Galerkin projection on the reduced space does not guarantee the inf-sup approximation stability even if a stable high fidelity method was used to generate snapshots. For problems in computational fluid dynamics, the lack of inf-sup stability is reflected by the inability to accurately approximate the pressure field. In this context, inf-sup stability is usually recovered through the enrichment of the velocity space with suitable supremizer functions. The main goal of this work is to propose an alternative approach, which relies on the residual based stabilization techniques customarily employed in the Finite Element literature, such as Brezzi-Pitkaranta, Franca-Hughes, streamline upwind Petrov-Galerkin, Galerkin Least Square. In the spirit of offline-online reduced basis computational splitting, two such options are proposed, namely offline-only stabilization and offline-online stabilization. These approaches are then compared to (and combined with) the state of the art supremizer enrichment approach. Numerical results are discussed, highlighting that the proposed methodology allows to obtain smaller reduced basis spaces (i.e., neglecting supremizer enrichment) for which a modified inf-sup stability is still preserved at the reduced order level.
    @article{AliBallarinRozza2020,
    author = {Shafqat Ali and Francesco Ballarin and Gianluigi Rozza},
    title = {Stabilized reduced basis methods for parametrized steady Stokes and Navier-Stokes equations},
    year = {2020},
    preprint = {https://arxiv.org/abs/2001.00820},
    journal = {Computers & Mathematics with Applications},
    volume = {80},
    number = {11},
    pages = {2399-2416},
    doi = {10.1016/j.camwa.2020.03.019},
    abstract = {It is well known in the Reduced Basis approximation of saddle point problems that the Galerkin projection on the reduced space does not guarantee the inf-sup approximation stability even if a stable high fidelity method was used to generate snapshots. For problems in computational fluid dynamics, the lack of inf-sup stability is reflected by the inability to accurately approximate the pressure field. In this context, inf-sup stability is usually recovered through the enrichment of the velocity space with suitable supremizer functions. The main goal of this work is to propose an alternative approach, which relies on the residual based stabilization techniques customarily employed in the Finite Element literature, such as Brezzi-Pitkaranta, Franca-Hughes, streamline upwind Petrov-Galerkin, Galerkin Least Square. In the spirit of offline-online reduced basis computational splitting, two such options are proposed, namely offline-only stabilization and offline-online stabilization. These approaches are then compared to (and combined with) the state of the art supremizer enrichment approach. Numerical results are discussed, highlighting that the proposed methodology allows to obtain smaller reduced basis spaces (i.e., neglecting supremizer enrichment) for which a modified inf-sup stability is still preserved at the reduced order level.}
    }
  2. F. Ballarin, T. Chacón Rebollo, E. Delgado Ávila, M. Gómez Mármol, and G. Rozza, “Certified Reduced Basis VMS-Smagorinsky model for natural convection flow in a cavity with variable height”, Computers & Mathematics with Applications, 80(5), pp. 973-989, 2020.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]
    In this work we present a Reduced Basis VMS-Smagorinsky Boussinesq model, applied to natural convection problems in a variable height cavity, in which the buoyancy forces are involved. We take into account in this problem both physical and geometrical parametrizations, considering the Rayleigh number as a parameter, so as the height of the cavity. We perform an Empirical Interpolation Method to approximate the sub-grid eddy viscosity term that lets us obtain an affine decomposition with respect to the parameters. We construct an a posteriori error estimator, based upon the Brezzi–Rappaz–Raviart theory, used in the greedy algorithm for the selection of the basis functions. Finally we present several numerical tests for different parameter configuration.
    @article{BallarinChaconDelgadoGomezRozza2020,
    author = {Ballarin, Francesco and Chacón Rebollo, Tomás and Delgado Ávila, Enrique and Gómez Mármol, Macarena and Rozza, Gianluigi},
    title = {Certified Reduced Basis VMS-Smagorinsky model for natural convection flow in a cavity with variable height},
    journal = {Computers & Mathematics with Applications},
    volume = {80},
    number = {5},
    pages = {973-989},
    year = {2020},
    preprint = {https://arxiv.org/abs/1902.05729},
    doi = {10.1016/j.camwa.2020.05.013},
    abstract = {In this work we present a Reduced Basis VMS-Smagorinsky Boussinesq model, applied to natural convection problems in a variable height cavity, in which the buoyancy forces are involved. We take into account in this problem both physical and geometrical parametrizations, considering the Rayleigh number as a parameter, so as the height of the cavity. We perform an Empirical Interpolation Method to approximate the sub-grid eddy viscosity term that lets us obtain an affine decomposition with respect to the parameters. We construct an a posteriori error estimator, based upon the Brezzi–Rappaz–Raviart theory, used in the greedy algorithm for the selection of the basis functions. Finally we present several numerical tests for different parameter configuration.}
    }
  3. M. Girfoglio, F. Ballarin, G. Infantino, F. Nicolò, A. Montalto, G. Rozza, R. Scrofani, M. Comisso, and F. Musumeci, “Non-intrusive PODI-ROM for patient-specific aortic blood flow in presence of a LVAD device”, 2020.
    [BibTeX] [Abstract] [Download preprint]
    Left ventricular assist devices (LVADs) are used to provide haemodynamic support to patients with critical cardiac failure. Severe complications can occur because of the modifications of the blood flow in the aortic region. In this work, the effect of a continuous flow LVAD device on the aortic flow is investigated by means of a non-intrusive reduced order model (ROM) built using the proper orthogonal decomposition with interpolation (PODI) method. The full order model (FOM) is represented by the incompressible Navier-Stokes equations discretized by using a Finite Volume (FV) technique, coupled with three-element Windkessel models to enforce outlet boundary conditions in a multi-scale approach. A patient-specific framework is proposed: a personalized geometry reconstructed from Computed Tomography (CT) images is used and the individualization of the coefficients of the three-element Windkessel models is based on experimental data provided by the Right Heart Catheterization (RCH) and Echocardiography (ECHO) tests. Pre-surgery configuration is also considered at FOM level in order to further validate the model. A parametric study with respect to the LVAD flow rate is considered. The accuracy of the reduced order model is assessed against results obtained with the full order model.
    @unpublished{GirfoglioBallarinInfantinoNicoloMontaltoRozzaScrofaniComissoMusumeci2020,
    author = {Michele Girfoglio and Francesco Ballarin and Giuseppe Infantino and Francesca Nicolò and Andrea Montalto and Gianluigi Rozza and Roberto Scrofani and Marina Comisso and Francesco Musumeci},
    title = {Non-intrusive PODI-ROM for patient-specific aortic blood flow in presence of a LVAD device},
    year = {2020},
    preprint = {https://arxiv.org/abs/2007.03527},
    abstract = {Left ventricular assist devices (LVADs) are used to provide haemodynamic support to patients with critical cardiac failure. Severe complications can occur because of the modifications of the blood flow in the aortic region. In this work, the effect of a continuous flow LVAD device on the aortic flow is investigated by means of a non-intrusive reduced order model (ROM) built using the proper orthogonal decomposition with interpolation (PODI) method. The full order model (FOM) is represented by the incompressible Navier-Stokes equations discretized by using a Finite Volume (FV) technique, coupled with three-element Windkessel models to enforce outlet boundary conditions in a multi-scale approach. A patient-specific framework is proposed: a personalized geometry reconstructed from Computed Tomography (CT) images is used and the individualization of the coefficients of the three-element Windkessel models is based on experimental data provided by the Right Heart Catheterization (RCH) and Echocardiography (ECHO) tests. Pre-surgery configuration is also considered at FOM level in order to further validate the model. A parametric study with respect to the LVAD flow rate is considered. The accuracy of the reduced order model is assessed against results obtained with the full order model.}
    }
  4. 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”, in Numerical Methods for Flows: FEF 2017 Selected Contributions, H. van Brummelen, A. Corsini, S. Perotto, and G. Rozza (eds.), Springer International Publishing, pp. 245–264, 2020.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]
    We present in this double contribution 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.
    @inbook{HijaziAliStabileBallarinRozza2018,
    author = {Hijazi, Saddam and Ali, Shafqat and Stabile, Giovanni and Ballarin, Francesco and Rozza, Gianluigi},
    editor = {van Brummelen, Harald and Corsini, Alessandro and Perotto, Simona and Rozza, Gianluigi},
    chapter = {The Effort of Increasing Reynolds Number in Projection-Based Reduced Order Methods: From Laminar to Turbulent Flows},
    bookTitle = {Numerical Methods for Flows: FEF 2017 Selected Contributions},
    year = {2020},
    publisher = {Springer International Publishing},
    pages = {245--264},
    abstract = {We present in this double contribution 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.},
    doi = {10.1007/978-3-030-30705-9_22},
    preprint = {https://arxiv.org/abs/1807.11370},
    }
  5. T. Kadeethum, H. M. Nick, S. Lee, and F. Ballarin, “Flow in porous media with low dimensional fractures by employing enriched Galerkin method”, Advances in Water Resources, 142, pp. 103620, 2020.
    [BibTeX] [Abstract] [View on publisher website]
    This paper presents the enriched Galerkin discretization for modeling fluid flow in fractured porous media using the mixed-dimensional approach. The proposed method has been tested against published benchmarks. Since fracture and porous media discontinuities can significantly influence single- and multi-phase fluid flow, the heterogeneous and anisotropic matrix permeability setting is utilized to assess the enriched Galerkin performance in handling the discontinuity within the matrix domain and between the matrix and fracture domains. Our results illustrate that the enriched Galerkin method has the same advantages as the discontinuous Galerkin method; for example, it conserves local and global fluid mass, captures the pressure discontinuity, and provides the optimal error convergence rate. However, the enriched Galerkin method requires much fewer degrees of freedom than the discontinuous Galerkin method in its classical form. The pressure solutions produced by both methods are similar regardless of the conductive or non-conductive fractures or heterogeneity in matrix permeability. This analysis shows that the enriched Galerkin scheme reduces the computational costs while offering the same accuracy as the discontinuous Galerkin so that it can be applied for large-scale flow problems. Furthermore, the results of a time-dependent problem for a three-dimensional geometry reveal the value of correctly capturing the discontinuities as barriers or highly-conductive fractures.
    @article{KadeethumNickLeeBallarin2020,
    author = {T. Kadeethum and H.M. Nick and S. Lee and F. Ballarin},
    title = {Flow in porous media with low dimensional fractures by employing enriched Galerkin method},
    journal = {Advances in Water Resources},
    volume = {142},
    pages = {103620},
    year = {2020},
    doi = {10.1016/j.advwatres.2020.103620},
    abstract = {This paper presents the enriched Galerkin discretization for modeling fluid flow in fractured porous media using the mixed-dimensional approach. The proposed method has been tested against published benchmarks. Since fracture and porous media discontinuities can significantly influence single- and multi-phase fluid flow, the heterogeneous and anisotropic matrix permeability setting is utilized to assess the enriched Galerkin performance in handling the discontinuity within the matrix domain and between the matrix and fracture domains. Our results illustrate that the enriched Galerkin method has the same advantages as the discontinuous Galerkin method; for example, it conserves local and global fluid mass, captures the pressure discontinuity, and provides the optimal error convergence rate. However, the enriched Galerkin method requires much fewer degrees of freedom than the discontinuous Galerkin method in its classical form. The pressure solutions produced by both methods are similar regardless of the conductive or non-conductive fractures or heterogeneity in matrix permeability. This analysis shows that the enriched Galerkin scheme reduces the computational costs while offering the same accuracy as the discontinuous Galerkin so that it can be applied for large-scale flow problems. Furthermore, the results of a time-dependent problem for a three-dimensional geometry reveal the value of correctly capturing the discontinuities as barriers or highly-conductive fractures.}
    }
  6. E. N. Karatzas, M. Nonino, F. Ballarin, and G. Rozza, “A Reduced Order Cut Finite Element method for geometrically parametrized steady and unsteady Navier-Stokes problems”, 2020.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]
    This work focuses on steady and unsteady Navier-Stokes equations in a reduced order modeling framework. The methodology proposed is based on a Proper Orthogonal Decomposition within a levelset geometry description and the problems of interest are discretized with an unfitted mesh Finite Element Method. We construct and investigate a unified and geometry independent reduced basis which overcomes many barriers and complications of the past, that may occur whenever geometrical morphings are taking place. By employing a geometry independent reduced basis, we are able to avoid remeshing and transformation to reference configurations, and we are able to handle complex geometries. This combination of a fixed background mesh in a fixed extended background geometry with reduced order techniques appears beneficial and advantageous in many industrial and engineering applications, which could not be resolved efficiently in the past.
    @unpublished{KaratzasNoninoBallarinRozza2020,
    author = {Efthymios N. Karatzas and Monica Nonino and Francesco Ballarin and Gianluigi Rozza},
    title = {A Reduced Order Cut Finite Element method for geometrically parametrized steady and unsteady Navier-Stokes problems},
    year = {2020},
    preprint = {https://arxiv.org/abs/2010.04953},
    doi = {10.1016/j.camwa.2021.07.016},
    abstract = {This work focuses on steady and unsteady Navier-Stokes equations in a reduced order modeling framework. The methodology proposed is based on a Proper Orthogonal Decomposition within a levelset geometry description and the problems of interest are discretized with an unfitted mesh Finite Element Method. We construct and investigate a unified and geometry independent reduced basis which overcomes many barriers and complications of the past, that may occur whenever geometrical morphings are taking place. By employing a geometry independent reduced basis, we are able to avoid remeshing and transformation to reference configurations, and we are able to handle complex geometries. This combination of a fixed background mesh in a fixed extended background geometry with reduced order techniques appears beneficial and advantageous in many industrial and engineering applications, which could not be resolved efficiently in the past.}
    }
  7. E. N. Karatzas, F. Ballarin, and G. Rozza, “Projection-based reduced order models for a cut finite element method in parametrized domains”, Computers & Mathematics with Applications, 79(3), pp. 833–851, 2020.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]
    This work presents a reduced order modelling technique built on a high fidelity embedded mesh finite element method. Such methods, and in particular the CutFEM method, are attractive in the generation of projection-based reduced order models thanks to their capabilities to seamlessly handle large deformations of parametrized domains. The combination of embedded methods and reduced order models allows us to obtain fast evaluation of parametrized problems, avoiding remeshing as well as the reference domain formulation, often used in the reduced order modelling for boundary fitted finite element formulations. The resulting novel methodology is presented on linear elliptic and Stokes problems, together with several test cases to assess its capability. The role of a proper extension and transport of embedded solutions to a common background is analyzed in detail.
    @article{KaratzasBallarinRozza2020,
    author = {Karatzas, Efthymios N. and Ballarin, Francesco and Rozza, Gianluigi},
    title = {Projection-based reduced order models for a cut finite element method in parametrized domains},
    journal = {Computers & Mathematics with Applications},
    volume = {79},
    number = {3},
    pages = {833--851},
    year = {2020},
    doi = {10.1016/j.camwa.2019.08.003},
    preprint = {https://arxiv.org/abs/1901.03846},
    abstract = {This work presents a reduced order modelling technique built on a high fidelity embedded mesh finite element method. Such methods, and in particular the CutFEM method, are attractive in the generation of projection-based reduced order models thanks to their capabilities to seamlessly handle large deformations of parametrized domains. The combination of embedded methods and reduced order models allows us to obtain fast evaluation of parametrized problems, avoiding remeshing as well as the reference domain formulation, often used in the reduced order modelling for boundary fitted finite element formulations. The resulting novel methodology is presented on linear elliptic and Stokes problems, together with several test cases to assess its capability. The role of a proper extension and transport of embedded solutions to a common background is analyzed in detail.}
    }
  8. S. Perotto, M. G. Carlino, and F. Ballarin, “Model Reduction by Separation of Variables: A Comparison Between Hierarchical Model Reduction and Proper Generalized Decomposition”, in Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018, 2020, pp. 61–77.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]
    Hierarchical Model reduction and Proper Generalized Decomposition both exploit separation of variables to perform a model reduction. After setting the basics, we exemplify these techniques on some standard elliptic problems to highlight pros and cons of the two procedures, both from a methodological and a numerical viewpoint.
    @InProceedings{PerottoCarlinoBallarin2020,
    author = {Perotto, Simona and Carlino, Michele Giuliano and Ballarin, Francesco},
    editor = {Sherwin, Spencer J. and Moxey, David and Peir{\'o}, Joaquim and Vincent, Peter E. and Schwab, Christoph},
    title = {Model Reduction by Separation of Variables: A Comparison Between Hierarchical Model Reduction and Proper Generalized Decomposition},
    booktitle = {Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018},
    year = {2020},
    publisher = {Springer International Publishing},
    pages = {61--77},
    preprint = {https://arxiv.org/abs/1811.11486},
    doi = {10.1007/978-3-030-39647-3_4},
    abstract = {Hierarchical Model reduction and Proper Generalized Decomposition both exploit separation of variables to perform a model reduction. After setting the basics, we exemplify these techniques on some standard elliptic problems to highlight pros and cons of the two procedures, both from a methodological and a numerical viewpoint.}
    }
  9. F. Pichi, M. Strazzullo, F. Ballarin, and G. Rozza, “Driving bifurcating parametrized nonlinear PDEs by optimal control strategies: application to Navier-Stokes equations with model order reduction”, 2020.
    [BibTeX] [Abstract] [Download preprint]
    This work deals with optimal control problems as a strategy to drive bifurcating solutions of nonlinear parametrized partial differential equations towards a desired branch. Indeed, for these governing equations, multiple solution configurations can arise from the same parametric instance. We thus aim at describing how optimal control allows to change the solution profile and the stability of state solution branches. First of all, a general framework for nonlinear optimal control problems is presented in order to reconstruct each branch of optimal solutions, discussing in detail the stability properties of the obtained controlled solutions. Then, we apply the proposed framework to several optimal control problems governed by bifurcating Navier-Stokes equations in a sudden-expansion channel, describing the qualitative and quantitative effect of the control over a pitchfork bifurcation and commenting in detail the stability eigenvalue analysis of the controlled state. Finally, we propose reduced order modeling as a tool to efficiently and reliably solve parametric stability analysis of such optimal control systems, which can be unbearable to perform with standard discretization techniques such as the Finite Element Method.
    @unpublished{PichiStrazzulloBallarinRozza2020,
    author = {Federico Pichi and Maria Strazzullo and Francesco Ballarin and Gianluigi Rozza},
    title = {Driving bifurcating parametrized nonlinear PDEs by optimal control strategies: application to Navier-Stokes equations with model order reduction},
    year = {2020},
    preprint = {https://arxiv.org/abs/2010.13506},
    abstract = {This work deals with optimal control problems as a strategy to drive bifurcating solutions of nonlinear parametrized partial differential equations towards a desired branch. Indeed, for these governing equations, multiple solution configurations can arise from the same parametric instance. We thus aim at describing how optimal control allows to change the solution profile and the stability of state solution branches. First of all, a general framework for nonlinear optimal control problems is presented in order to reconstruct each branch of optimal solutions, discussing in detail the stability properties of the obtained controlled solutions. Then, we apply the proposed framework to several optimal control problems governed by bifurcating Navier-Stokes equations in a sudden-expansion channel, describing the qualitative and quantitative effect of the control over a pitchfork bifurcation and commenting in detail the stability eigenvalue analysis of the controlled state. Finally, we propose reduced order modeling as a tool to efficiently and reliably solve parametric stability analysis of such optimal control systems, which can be unbearable to perform with standard discretization techniques such as the Finite Element Method.}
    }
  10. G. Rozza, M. Hess, G. Stabile, M. Tezzele, and F. Ballarin, “Basic Ideas and Tools for Projection-Based Model Reduction of Parametric Partial Differential Equations”, in Model Order Reduction Volume 2: Snapshot-Based Methods and Algorithms, P. Benner, S. Grivet-Talocia, A. Quarteroni, G. Rozza, W. H. A. Schilders, and L. M. Silveira (eds.), , 2020.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]
    We provide first the functional analysis background required for reduced order modeling and present the underlying concepts of reduced basis model reduction. The projection-based model reduction framework under affinity assumptions, offline-online decomposition and error estimation is introduced. Several tools for geometry parametrizations, such as free form deformation, radial basis function interpolation and inverse distance weighting interpolation are explained. The empirical interpolation method is introduced as a general tool to deal with non-affine parameter dependency and non-linear problems. The discrete and matrix versions of the empirical interpolation are considered as well. Active subspaces properties are discussed to reduce high-dimensional parameter spaces as a pre-processing step. Several examples illustrate the methodologies.
    @inbook{RozzaHessStabileTezzeleBallarin2020,
    author = {Gianluigi Rozza and Martin Hess and Giovanni Stabile and Marco Tezzele and Francesco Ballarin},
    chapter = {Basic Ideas and Tools for Projection-Based Model Reduction of Parametric Partial Differential Equations},
    year = {2020},
    booktitle = {Model Order Reduction Volume 2: Snapshot-Based Methods and Algorithms},
    editor = {P. Benner and S. Grivet-Talocia and A. Quarteroni and G. Rozza and W. H. A. Schilders and L. M. Silveira},
    preprint = {https://arxiv.org/abs/1911.08954},
    url = {https://www.degruyter.com/view/title/565388},
    abstract = {We provide first the functional analysis background required for reduced order modeling and present the underlying concepts of reduced basis model reduction. The projection-based model reduction framework under affinity assumptions, offline-online decomposition and error estimation is introduced. Several tools for geometry parametrizations, such as free form deformation, radial basis function interpolation and inverse distance weighting interpolation are explained. The empirical interpolation method is introduced as a general tool to deal with non-affine parameter dependency and non-linear problems. The discrete and matrix versions of the empirical interpolation are considered as well. Active subspaces properties are discussed to reduce high-dimensional parameter spaces as a pre-processing step. Several examples illustrate the methodologies.}
    }
  11. M. Strazzullo, F. Ballarin, and G. Rozza, “POD–Galerkin Model Order Reduction for Parametrized Time Dependent Linear Quadratic Optimal Control Problems in Saddle Point Formulation”, Journal of Scientific Computing, 83(3), pp. 55, 2020.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]
    In this work we deal with parametrized time dependent optimal control problems governed by partial differential equations. We aim at extending the standard saddle point framework of steady constraints to time dependent cases. We provide an analysis of the well-posedness of this formulation both for parametrized scalar parabolic constraint and Stokes governing equations and we propose reduced order methods as an effective strategy to solve them. Indeed, on one hand, parametrized time dependent optimal control is a very powerful mathematical model which is able to describe several physical phenomena, on the other, it requires a huge computational effort. Reduced order methods are a suitable approach to have rapid and accurate simulations. We rely on POD–Galerkin reduction over the physical and geometrical parameters of the optimality system in a space-time formulation. Our theoretical results and our methodology are tested on two examples: a boundary time dependent optimal control for a Graetz flow and a distributed optimal control governed by time dependent Stokes equations. With these two test cases the convenience of the reduced order modelling is further extended to the field of time dependent optimal control.
    @article{StrazzulloBallarinRozza2020,
    author = {Maria Strazzullo and Francesco Ballarin and Gianluigi Rozza},
    title = {POD--Galerkin Model Order Reduction for Parametrized Time Dependent Linear Quadratic Optimal Control Problems in Saddle Point Formulation},
    journal = {Journal of Scientific Computing},
    volume = {83},
    number = {3},
    pages = {55},
    year = {2020},
    preprint = {https://arxiv.org/abs/1909.09631},
    doi = {10.1007/s10915-020-01232-x},
    abstract = {In this work we deal with parametrized time dependent optimal control problems governed by partial differential equations. We aim at extending the standard saddle point framework of steady constraints to time dependent cases. We provide an analysis of the well-posedness of this formulation both for parametrized scalar parabolic constraint and Stokes governing equations and we propose reduced order methods as an effective strategy to solve them. Indeed, on one hand, parametrized time dependent optimal control is a very powerful mathematical model which is able to describe several physical phenomena, on the other, it requires a huge computational effort. Reduced order methods are a suitable approach to have rapid and accurate simulations. We rely on POD–Galerkin reduction over the physical and geometrical parameters of the optimality system in a space-time formulation. Our theoretical results and our methodology are tested on two examples: a boundary time dependent optimal control for a Graetz flow and a distributed optimal control governed by time dependent Stokes equations. With these two test cases the convenience of the reduced order modelling is further extended to the field of time dependent optimal control.}
    }

2019

  1. F. Ballarin, A. D’Amario, S. Perotto, and G. Rozza, “A POD-selective inverse distance weighting method for fast parametrized shape morphing”, International Journal for Numerical Methods in Engineering, 117(8), pp. 860–884, 2019.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]
    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.
    @article{BallarinDAmarioPerottoRozza2019,
    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 = {2019},
    preprint = {https://arxiv.org/abs/1710.09243},
    doi = {10.1002/nme.5982},
    volume = {117},
    number = {8},
    pages = {860--884},
    journal = {International Journal for Numerical Methods in Engineering},
    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.}
    }
  2. M. Nonino, F. Ballarin, G. Rozza, and Y. Maday, “Overcoming slowly decaying Kolmogorov n-width by transport maps: application to model order reduction of fluid dynamics and fluid–structure interaction problems”, 2019.
    [BibTeX] [Abstract] [Download preprint]
    In this work we focus on reduced order modelling for problems for which the resulting reduced basis spaces show a slow decay of the Kolmogorov n-width, or, in practical calculations, its computational surrogate given by the magnitude of the eigenvalues returned by a proper orthogonal decomposition on the solution manifold. In particular, we employ an additional preprocessing during the offline phase of the reduced basis method, in order to obtain smaller reduced basis spaces. Such preprocessing is based on the composition of the snapshots with a transport map, that is a family of smooth and invertible mappings that map the physical domain of the problem into itself. Two test cases are considered: a fluid moving in a domain with deforming walls, and a fluid past a rotating cylinder. Comparison between the results of the novel offline stage and the standard one is presented.
    @unpublished{NoninoBallarinRozzaMaday2019,
    author = {Monica Nonino and Francesco Ballarin and Gianluigi Rozza and Yvon Maday},
    title = {Overcoming slowly decaying Kolmogorov n-width by transport maps: application to model order reduction of fluid dynamics and fluid--structure interaction problems},
    year = {2019},
    preprint = {https://arxiv.org/abs/1911.06598},
    abstract = {In this work we focus on reduced order modelling for problems for which the resulting reduced basis spaces show a slow decay of the Kolmogorov n-width, or, in practical calculations, its computational surrogate given by the magnitude of the eigenvalues returned by a proper orthogonal decomposition on the solution manifold. In particular, we employ an additional preprocessing during the offline phase of the reduced basis method, in order to obtain smaller reduced basis spaces. Such preprocessing is based on the composition of the snapshots with a transport map, that is a family of smooth and invertible mappings that map the physical domain of the problem into itself. Two test cases are considered: a fluid moving in a domain with deforming walls, and a fluid past a rotating cylinder. Comparison between the results of the novel offline stage and the standard one is presented.}
    }
  3. G. Stabile, F. Ballarin, G. Zuccarino, and G. Rozza, “A reduced order variational multiscale approach for turbulent flows”, Advances in Computational Mathematics, 45(5), pp. 2349-2368, 2019.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]
    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.
    @article{StabileBallarinZuccarinoRozza2019,
    author = {Stabile, Giovanni and Ballarin, Francesco and Zuccarino, Giacomo and Rozza, Gianluigi},
    title = {A reduced order variational multiscale approach for turbulent flows},
    year = {2019},
    journal = {Advances in Computational Mathematics},
    volume = {45},
    number = {5},
    pages = {2349-2368},
    doi = {10.1007/s10444-019-09712-x},
    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.}
    }
  4. L. Venturi, D. Torlo, F. Ballarin, and G. Rozza, “Weighted Reduced Order Methods for Parametrized Partial Differential Equations with Random Inputs”, in Uncertainty Modeling for Engineering Applications, F. Canavero (ed.), Springer International Publishing, pp. 27–40, 2019.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]
    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.
    @inbook{VenturiTorloBallarinRozza2019,
    author = {Venturi, Luca and Torlo, Davide and Ballarin, Francesco and Rozza, Gianluigi},
    chapter = {Weighted Reduced Order Methods for Parametrized Partial Differential Equations with Random Inputs},
    year = {2019},
    booktitle = {Uncertainty Modeling for Engineering Applications},
    editor = {Canavero, Flavio},
    publisher = {Springer International Publishing},
    pages = {27--40},
    preprint = {https://arxiv.org/abs/1805.00828},
    doi = {10.1007/978-3-030-04870-9_2},
    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”, Journal of Scientific Computing, 81(1), pp. 136–153, 2019.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]
    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.
    @article{VenturiBallarinRozza2018,
    author = {Venturi, Luca and Ballarin, Francesco and Rozza, Gianluigi},
    title = {A Weighted POD Method for Elliptic PDEs with Random Inputs},
    year = {2019},
    preprint = {https://arxiv.org/abs/1802.08724},
    journal={Journal of Scientific Computing},
    volume={81},
    number={1},
    pages={136--153},
    doi={10.1007/s10915-018-0830-7},
    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.}
    }

2018

  1. 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), 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{StrazzulloBallarinMosettiRozza2018,
    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.}
    }
  2. M. Tezzele, F. Ballarin, and G. Rozza, “Combined parameter and model reduction of cardiovascular problems by means of active subspaces and POD-Galerkin methods”, in Mathematical and Numerical Modeling of the Cardiovascular System and Applications, D. Boffi, L. F. Pavarino, G. Rozza, S. Scacchi, and C. Vergara (eds.), Cham: Springer International Publishing, pp. 185–207, 2018.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]
    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.
    @inbook{TezzeleBallarinRozza2018,
    author = {Tezzele, Marco and Ballarin, Francesco and Rozza, Gianluigi},
    chapter = {Combined parameter and model reduction of cardiovascular problems by means of active subspaces and POD-Galerkin methods},
    address = {Cham},
    booktitle = {Mathematical and Numerical Modeling of the Cardiovascular System and Applications},
    doi = {10.1007/978-3-319-96649-6_8},
    editor = {Boffi, Daniele and Pavarino, Luca F. and Rozza, Gianluigi and Scacchi, Simone and Vergara, Christian},
    isbn = {978-3-319-96649-6},
    pages = {185--207},
    publisher = {Springer International Publishing},
    year = {2018},
    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.}
    }
  3. D. Torlo, F. Ballarin, and G. Rozza, “Stabilized Weighted Reduced Basis Methods for Parametrized Advection Dominated Problems with Random Inputs”, SIAM/ASA Journal on Uncertainty Quantification, 6(4), pp. 1475-1502, 2018.
    [BibTeX] [Abstract] [Download preprint] [View on publisher website]
    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 the wRB (weighted reduced basis) method for stochastic parametrized problems with the stabilized RB (reduced basis) method, which is the integration of classical stabilization methods (streamline/upwind Petrov–Galerkin (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.
    @article{TorloBallarinRozza2018,
    author = {Torlo, Davide and Ballarin, Francesco and Rozza, Gianluigi},
    title = {Stabilized Weighted Reduced Basis Methods for Parametrized Advection Dominated Problems with Random Inputs},
    journal = {SIAM/ASA Journal on Uncertainty Quantification},
    volume = {6},
    number = {4},
    pages = {1475-1502},
    year = {2018},
    doi = {10.1137/17M1163517},
    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 the wRB (weighted reduced basis) method for stochastic parametrized problems with the stabilized RB (reduced basis) method, which is the integration of classical stabilization methods (streamline/upwind Petrov--Galerkin (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.}
    }

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), 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://preprints.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, vol. 17, 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},
    volume = {17},
    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. 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), 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},
    }

2016

  1. 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), 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}
    }
  2. 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, 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 = {http://preprints.sissa.it/xmlui/bitstream/handle/1963/34623/REPORT.pdf?sequence=1&isAllowed=y}
    }
  3. 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}
    }
  4. 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), 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},
    }

2015

  1. 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), 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}
    }
  2. 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}
    }

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), 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}
    }