In this work we present a Reduced Order Model which is specifically designed to deal with turbulent flows in a finite volume setting. The method used to build the reduced order model is based on the idea of merging/combining projection-based techniques with data-driven reduction strategies. In particular, the work presents a mixed strategy that exploits a data-driven reduction method to approximate the eddy viscosity solution manifold and a classical POD-Galerkin projection approach for the velocity and the pressure fields, respectively. The newly proposed reduced order model has been validated on benchmark test cases in both steady and unsteady settings with Reynolds up to $Re=O(10^5)$.

VL - 416 UR - https://arxiv.org/abs/1907.09909 ER - TY - JOUR T1 - Efficient computation of bifurcation diagrams with a deflated approach to reduced basis spectral element method JF - Advances in Computational Mathematics Y1 - 2020 A1 - Pintore, Moreno A1 - Pichi, Federico A1 - Hess, Martin A1 - Rozza, Gianluigi A1 - Canuto, Claudio AB -The majority of the most common physical phenomena can be described using partial differential equations (PDEs). However, they are very often characterized by strong nonlinearities. Such features lead to the coexistence of multiple solutions studied by the bifurcation theory. Unfortunately, in practical scenarios, one has to exploit numerical methods to compute the solutions of systems of PDEs, even if the classical techniques are usually able to compute only a single solution for any value of a parameter when more branches exist. In this work we implemented an elaborated deflated continuation method, that relies on the spectral element method (SEM) and on the reduced basis (RB) one, to efficiently compute bifurcation diagrams with more parameters and more bifurcation points. The deflated continuation method can be obtained combining the classical continuation method and the deflation one: the former is used to entirely track each known branch of the diagram, while the latter is exploited to discover the new ones. Finally, when more than one parameter is considered, the efficiency of the computation is ensured by the fact that the diagrams can be computed during the online phase while, during the offline one, one only has to compute one-dimensional diagrams. In this work, after a more detailed description of the method, we will show the results that can be obtained using it to compute a bifurcation diagram associated with a problem governed by the Navier-Stokes equations.

UR - https://arxiv.org/abs/1912.06089 ER - TY - JOUR T1 - Efficient Geometrical parametrization for finite-volume based reduced order methods JF - International Journal for Numerical Methods in Engineering Y1 - 2020 A1 - Giovanni Stabile A1 - Matteo Zancanaro A1 - Gianluigi Rozza AB -In this work, we present an approach for the efficient treatment of parametrized geometries in the context of POD-Galerkin reduced order methods based on Finite Volume full order approximations. On the contrary to what is normally done in the framework of finite element reduced order methods, different geometries are not mapped to a common reference domain: the method relies on basis functions defined on an average deformed configuration and makes use of the Discrete Empirical Interpolation Method (D-EIM) to handle together non-affinity of the parametrization and non-linearities. In the first numerical example, different mesh motion strategies, based on a Laplacian smoothing technique and on a Radial Basis Function approach, are analyzed and compared on a heat transfer problem. Particular attention is devoted to the role of the non-orthogonal correction. In the second numerical example the methodology is tested on a geometrically parametrized incompressible Navier–Stokes problem. In this case, the reduced order model is constructed following the same segregated approach used at the full order level

VL - 121 UR - https://arxiv.org/abs/1901.06373 ER - TY - UNPB T1 - Enhancing CFD predictions in shape design problems by model and parameter space reduction Y1 - 2020 A1 - Marco Tezzele A1 - Nicola Demo A1 - Giovanni Stabile A1 - Andrea Mola A1 - Gianluigi Rozza AB -In this work we present an advanced computational pipeline for the approximation and prediction of the lift coefficient of a parametrized airfoil profile. The non-intrusive reduced order method is based on dynamic mode decomposition (DMD) and it is coupled with dynamic active subspaces (DyAS) to enhance the future state prediction of the target function and reduce the parameter space dimensionality. The pipeline is based on high-fidelity simulations carried out by the application of finite volume method for turbulent flows, and automatic mesh morphing through radial basis functions interpolation technique. The proposed pipeline is able to save 1/3 of the overall computational resources thanks to the application of DMD. Moreover exploiting DyAS and performing the regression on a lower dimensional space results in the reduction of the relative error in the approximation of the time-varying lift coefficient by a factor 2 with respect to using only the DMD.

UR - https://arxiv.org/abs/2001.05237 ER - TY - JOUR T1 - A hybrid reduced order method for modelling turbulent heat transfer problems JF - Computers & Fluids Y1 - 2020 A1 - Sokratia Georgaka A1 - Giovanni Stabile A1 - Kelbij Star A1 - Gianluigi Rozza A1 - Michael J. Bluck AB -A parametric, hybrid reduced order model approach based on the Proper Orthogonal Decomposition with both Galerkin projection and interpolation based on Radial Basis Functions method is presented. This method is tested against a case of turbulent non-isothermal mixing in a T-junction pipe, a common ow arrangement found in nuclear reactor cooling systems. The reduced order model is derived from the 3D unsteady, incompressible Navier-Stokes equations weakly coupled with the energy equation. For high Reynolds numbers, the eddy viscosity and eddy diffusivity are incorporated into the reduced order model with a Proper Orthogonal Decomposition (nested and standard) with Interpolation (PODI), where the interpolation is performed using Radial Basis Functions. The reduced order solver, obtained using a k-ω SST URANS full order model, is tested against the full order solver in a 3D T-junction pipe with parametric velocity inlet boundary conditions.

VL - 208 UR - https://arxiv.org/abs/1906.08725 ER - TY - CONF T1 - Non-Intrusive Polynomial Chaos Method Applied to Problems in Computational Fluid Dynamics with a Comparison to Proper Orthogonal Decomposition T2 - QUIET Selected Contributions Y1 - 2020 A1 - Saddam Hijazi A1 - Giovanni Stabile A1 - Andrea Mola A1 - Gianluigi Rozza ED - van Brummelen, Harald ED - Corsini, Alessandro ED - Perotto, Simona ED - Rozza, Gianluigi AB -In this work, Uncertainty Quantification (UQ) based on non-intrusive Polynomial Chaos Expansion (PCE) is applied to the CFD problem of the flow past an airfoil with parameterized angle of attack and inflow velocity. To limit the computational cost associated with each of the simulations required by the non-intrusive UQ algorithm used, we resort to a Reduced Order Model (ROM) based on Proper Orthogonal Decomposition (POD)-Galerkin approach. A first set of results is presented to characterize the accuracy of the POD-Galerkin ROM developed approach with respect to the Full Order Model (FOM) solver (OpenFOAM). A further analysis is then presented to assess how the UQ results are affected by substituting the FOM predictions with the surrogate ROM ones.

JF - QUIET Selected Contributions PB - Springer International Publishing UR - https://arxiv.org/abs/1901.02285 ER - TY - UNPB T1 - POD-Galerkin Model Order Reduction for Parametrized Nonlinear Time Dependent Optimal Flow Control: an Application to Shallow Water Equations Y1 - 2020 A1 - Maria Strazzullo A1 - Francesco Ballarin A1 - Gianluigi Rozza AB -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.

ER - TY - UNPB T1 - A POD-Galerkin reduced order model of a turbulent convective buoyant flow of sodium over a backward-facing step Y1 - 2020 A1 - Kelbij Star A1 - Giovanni Stabile A1 - Gianluigi Rozza A1 - Joris Degroote AB -A Finite-Volume based POD-Galerkin reduced order modeling strategy for steady-state Reynolds averaged Navier–Stokes (RANS) simulation is extended for low-Prandtl number flow. The reduced order model is based on a full order model for which the effects of buoyancy on the flow and heat transfer are characterized by varying the Richardson number. The Reynolds stresses are computed with a linear eddy viscosity model. A single gradient diffusion hypothesis, together with a local correlation for the evaluation of the turbulent Prandtl number, is used to model the turbulent heat fluxes. The contribution of the eddy viscosity and turbulent thermal diffusivity fields are considered in the reduced order model with an interpolation based data-driven method. The reduced order model is tested for buoyancy-aided turbulent liquid sodium flow over a vertical backward-facing step with a uniform heat flux applied on the wall downstream of the step. The wall heat flux is incorporated with a Neumann boundary condition in both the full order model and the reduced order model. The velocity and temperature profiles predicted with the reduced order model for the same and new Richardson numbers inside the range of parameter values are in good agreement with the RANS simulations. Also, the local Stanton number and skin friction distribution at the heated wall are qualitatively well captured. Finally, the reduced order simulations, performed on a single core, are about $10^5$ times faster than the RANS simulations that are performed on eight cores.

UR - https://arxiv.org/abs/2003.01114 ER - TY - JOUR T1 - Reduced Basis Model Order Reduction for Navier-Stokes equations in domains with walls of varying curvature JF - International Journal of Computational Fluid Dynamics Y1 - 2020 A1 - Hess, Martin A1 - Quaini, Annalisa A1 - Rozza, Gianluigi AB -We consider the Navier-Stokes equations in a channel with a narrowing and walls of varying curvature. By applying the empirical interpolation method to generate an affine parameter dependency, the offline-online procedure can be used to compute reduced order solutions for parameter variations. The reduced order space is computed from the steady-state snapshot solutions by a standard POD procedure. The model is discretised with high-order spectral element ansatz functions, resulting in 4752 degrees of freedom. The proposed reduced order model produces accurate approximations of steady-state solutions for a wide range of geometries and kinematic viscosity values. The application that motivated the present study is the onset of asymmetries (i.e., symmetry breaking bifurcation) in blood flow through a regurgitant mitral valve, depending on the Reynolds number and the valve shape. Through our computational study, we found that the critical Reynolds number for the symmetry breaking increases as the wall curvature increases.

VL - 34 UR - https://arxiv.org/abs/1901.03708 ER - TY - CONF T1 - A Reduced Order Approach for the Embedded Shifted Boundary FEM and a Heat Exchange System on Parametrized Geometries T2 - IUTAM Symposium on Model Order Reduction of Coupled Systems, Stuttgart, Germany, May 22–25, 2018 Y1 - 2020 A1 - Efthymios N. Karatzas A1 - Giovanni Stabile A1 - Nabib Atallah A1 - Guglielmo Scovazzi A1 - Gianluigi Rozza ED - Fehr, Jörg ED - Haasdonk, Bernard AB -A model order reduction technique is combined with an embedded boundary finite element method with a POD-Galerkin strategy. The proposed methodology is applied to parametrized heat transfer problems and we rely on a sufficiently refined shape-regular background mesh to account for parametrized geometries. In particular, the employed embedded boundary element method is the Shifted Boundary Method (SBM) recently proposed. This approach is based on the idea of shifting the location of true boundary conditions to a surrogate boundary, with the goal of avoiding cut cells near the boundary of the computational domain. This combination of methodologies has multiple advantages. In the first place, since the Shifted Boundary Method always relies on the same background mesh, there is no need to update the discretized parametric domain. Secondly, we avoid the treatment of cut cell elements, which usually need particular attention. Thirdly, since the whole background mesh is considered in the reduced basis construction, the SBM allows for a smooth transition of the reduced modes across the immersed domain boundary. The performances of the method are verified in two dimensional heat transfer numerical examples.

JF - IUTAM Symposium on Model Order Reduction of Coupled Systems, Stuttgart, Germany, May 22–25, 2018 PB - Springer International Publishing UR - https://arxiv.org/abs/1807.07753 ER - TY - CONF T1 - Reduced order methods for parametrized non-linear and time dependent optimal flow control problems, towards applications in biomedical and environmental sciences T2 - ENUMATH2019 proceedings Y1 - 2020 A1 - Maria Strazzullo A1 - Zakia Zainib A1 - Francesco Ballarin A1 - Gianluigi Rozza AB -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.

JF - ENUMATH2019 proceedings PB - Springer UR - https://arxiv.org/abs/1912.07886 ER - TY - JOUR T1 - A Reduced Order technique to study bifurcating phenomena: application to the Gross-Pitaevskii equation JF - SIAM Journal on Scientific Computing Y1 - 2020 A1 - Pichi, Federico A1 - Quaini, Annalisa A1 - Rozza, Gianluigi AB -We propose a computationally efficient framework to treat nonlinear partial differential equations having bifurcating solutions as one or more physical control parameters are varied. Our focus is on steady bifurcations. Plotting a bifurcation diagram entails computing multiple solutions of a parametrized, nonlinear problem, which can be extremely expensive in terms of computational time. In order to reduce these demanding computational costs, our approach combines a continuation technique and Newton's method with a Reduced Order Modeling (ROM) technique, suitably supplemented with a hyper-reduction method. To demonstrate the effectiveness of our ROM approach, we trace the steady solution branches of a nonlinear Schrödinger equation, called Gross-Pitaevskii equation, as one or two physical parameters are varied. In the two parameter study, we show that our approach is 60 times faster in constructing a bifurcation diagram than a standard Full Order Method.

UR - https://arxiv.org/abs/1907.07082 ER - TY - UNPB T1 - A supervised learning approach involving active subspaces for an efficient genetic algorithm in high-dimensional optimization problems Y1 - 2020 A1 - Nicola Demo A1 - Marco Tezzele A1 - Gianluigi Rozza AB -In this work, we present an extension of the genetic algorithm (GA) which exploits the active subspaces (AS) property to evolve the individuals on a lower dimensional space. In many cases, GA requires in fact more function evaluations than others optimization method to converge to the optimum. Thus, complex and high-dimensional functions may result intractable with the standard algorithm. To address this issue, we propose to linearly map the input parameter space of the original function onto its AS before the evolution, performing the mutation and mate processes in a lower dimensional space. In this contribution, we describe the novel method called ASGA, presenting differences and similarities with the standard GA method. We test the proposed method over n-dimensional benchmark functions – Rosenbrock, Ackley, Bohachevsky, Rastrigin, Schaffer N. 7, and Zakharov – and finally we apply it to an aeronautical shape optimization problem.

UR - https://arxiv.org/abs/2006.07282 ER - TY - JOUR T1 - Surface tension controls the onset of gyrification in brain organoids JF - Journal of the Mechanics and Physics of Solids Y1 - 2020 A1 - Davide Riccobelli A1 - Giulia Bevilacqua KW - Buckling KW - Embryogenesis KW - Morpho-elasticity KW - Post-buckling analysis KW - Surface tension AB -Understanding the mechanics of brain embryogenesis can provide insights on pathologies related to brain development, such as lissencephaly, a genetic disease which causes a reduction of the number of cerebral sulci. Recent experiments on brain organoids have confirmed that gyrification, i.e. the formation of the folded structures of the brain, is triggered by the inhomogeneous growth of the peripheral region. However, the rheology of these cellular aggregates and the mechanics of lissencephaly are still matter of debate. In this work, we develop a mathematical model of brain organoids based on the theory of morpho-elasticity. We describe them as non-linear elastic bodies, composed of a disk surrounded by a growing layer called cortex. The external boundary is subjected to a tissue surface tension due the intercellular adhesion forces. We show that the resulting surface energy is relevant at the small length scales of brain organoids and affects the mechanics of cellular aggregates. We perform a linear stability analysis of the radially symmetric configuration and we study the post-buckling behaviour through finite element simulations. We find that the process of gyrification is triggered by the cortex growth and modulated by the competition between two length scales: the radius of the organoid and the capillary length generated by surface tension. We show that a solid model can reproduce the results of the in-vitro experiments. Furthermore, we prove that the lack of brain sulci in lissencephaly is caused by a reduction of the cell stiffness: the softening of the organoid strengthens the role of surface tension, delaying or even inhibiting the onset of a mechanical instability at the free boundary.

VL - 134 UR - http://www.sciencedirect.com/science/article/pii/S0022509619304065 ER - TY - JOUR T1 - Activation of a muscle as a mapping of stress–strain curves JF - Extreme Mech. Lett. Y1 - 2019 A1 - Davide Riccobelli A1 - D. Ambrosi PB - Elsevier BV VL - 28 ER - TY - CONF T1 - A complete data-driven framework for the efficient solution of parametric shape design and optimisation in naval engineering problems T2 - VIII International Conference on Computational Methods in Marine Engineering Y1 - 2019 A1 - Demo, Nicola A1 - Tezzele, Marco A1 - Mola, Andrea A1 - Rozza, Gianluigi AB -In the reduced order modeling (ROM) framework, the solution of a parametric partial differential equation is approximated by combining the high-fidelity solutions of the problem at hand for several properly chosen configurations. Examples of the ROM application, in the naval field, can be found in [31, 24]. Mandatory ingredient for the ROM methods is the relation between the high-fidelity solutions and the parameters. Dealing with geometrical parameters, especially in the industrial context, this relation may be unknown and not trivial (simulations over hand morphed geometries) or very complex (high number of parameters or many nested morphing techniques). To overcome these scenarios, we propose in this contribution an efficient and complete data-driven framework involving ROM techniques for shape design and optimization, extending the pipeline presented in [7]. By applying the singular value decomposition (SVD) to the points coordinates defining the hull geometry –- assuming the topology is inaltered by the deformation –-, we are able to compute the optimal space which the deformed geometries belong to, hence using the modal coefficients as the new parameters we can reconstruct the parametric formulation of the domain. Finally the output of interest is approximated using the proper orthogonal decomposition with interpolation technique. To conclude, we apply this framework to a naval shape design problem where the bulbous bow is morphed to reduce the total resistance of the ship advancing in calm water.

JF - VIII International Conference on Computational Methods in Marine Engineering UR - https://arxiv.org/abs/1905.05982 ER - TY - RPRT T1 - A continuous dependence result for a dynamic debonding model in dimension one Y1 - 2019 A1 - Filippo Riva AB -In this paper we address the problem of continuous dependence on initial and boundary data for a one-dimensional debonding model describing a thin ﬁlm peeled away from a substrate. The system underlying the process couples the weakly damped wave equation with a Griﬃth’s criterion which rules the evolution of the debonded region. We show that under general convergence assumptions on the data the corresponding solutions converge to the limit one with respect to diﬀerent natural topologies.

PB - SISSA UR - http://preprints.sissa.it/xmlui/handle/1963/35329 U1 - 35640 U2 - Mathematics U4 - 1 ER - TY - UNPB T1 - Discontinuous Galerkin Model Order Reduction of Geometrically Parametrized Stokes Equation Y1 - 2019 A1 - Nirav Vasant Shah A1 - Martin Hess A1 - Gianluigi Rozza AB -The present work focuses on the geometric parametrization and the reduced order modeling of the Stokes equation. We discuss the concept of a parametrized geometry and its application within a reduced order modeling technique. The full order model is based on the discontinuous Galerkin method with an interior penalty formulation. We introduce the broken Sobolev spaces as well as the weak formulation required for an affine parameter dependency. The operators are transformed from a fixed domain to a parameter dependent domain using the affine parameter dependency. The proper orthogonal decomposition is used to obtain the basis of functions of the reduced order model. By using the Galerkin projection the linear system is projected onto the reduced space. During this process, the offline-online decomposition is used to separate parameter dependent operations from parameter independent operations. Finally this technique is applied to an obstacle test problem.The numerical outcomes presented include experimental error analysis, eigenvalue decay and measurement of online simulation time. Keywords: Discontinuous Galerkin method, Stokes flow, Geometric parametrization, Proper orthogonal decomposition.

UR - https://arxiv.org/abs/1912.09787 ER - TY - CONF T1 - Efficient Reduction in Shape Parameter Space Dimension for Ship Propeller Blade Design T2 - VIII International Conference on Computational Methods in Marine Engineering Y1 - 2019 A1 - Mola, Andrea A1 - Tezzele, Marco A1 - Gadalla, Mahmoud A1 - Valdenazzi, Federica A1 - Grassi, Davide A1 - Padovan, Roberta A1 - Rozza, Gianluigi AB -In this work, we present the results of a ship propeller design optimization campaign carried out in the framework of the research project PRELICA, funded by the Friuli Venezia Giulia regional government. The main idea of this work is to operate on a multidisciplinary level to identify propeller shapes that lead to reduced tip vortex-induced pressure and increased efficiency without altering the thrust. First, a specific tool for the bottom-up construction of parameterized propeller blade geometries has been developed. The algorithm proposed operates with a user defined number of arbitrary shaped or NACA airfoil sections, and employs arbitrary degree NURBS to represent the chord, pitch, skew and rake distribution as a function of the blade radial coordinate. The control points of such curves have been modified to generate, in a fully automated way, a family of blade geometries depending on as many as 20 shape parameters. Such geometries have then been used to carry out potential flow simulations with the Boundary Element Method based software PROCAL. Given the high number of parameters considered, such a preliminary stage allowed for a fast evaluation of the performance of several hundreds of shapes. In addition, the data obtained from the potential flow simulation allowed for the application of a parameter space reduction methodology based on active subspaces (AS) property, which suggested that the main propeller performance indices are, at a first but rather accurate approximation, only depending on a single parameter which is a linear combination of all the original geometric ones. AS analysis has also been used to carry out a constrained optimization exploiting response surface method in the reduced parameter space, and a sensitivity analysis based on such surrogate model. The few selected shapes were finally used to set up high fidelity RANS simulations and select an optimal shape.

JF - VIII International Conference on Computational Methods in Marine Engineering UR - https://arxiv.org/abs/1905.09815 ER - TY - JOUR T1 - An entropic interpolation proof of the HWI inequality JF - Stochastic Processes and their Applications Y1 - 2019 A1 - Ivan Gentil A1 - Christian Léonard A1 - Luigia Ripani A1 - Luca Tamanini KW - Entropic interpolations KW - Fisher information KW - Relative entropy KW - Schrödinger problem KW - Wasserstein distance AB -The HWI inequality is an “interpolation”inequality between the Entropy H, the Fisher information I and the Wasserstein distance W. We present a pathwise proof of the HWI inequality which is obtained through a zero noise limit of the Schrödinger problem. Our approach consists in making rigorous the Otto–Villani heuristics in Otto and Villani (2000) taking advantage of the entropic interpolations, which are regular both in space and time, rather than the displacement ones.

UR - http://www.sciencedirect.com/science/article/pii/S0304414918303454 ER - TY - JOUR T1 - Error estimates in weighted Sobolev norms for finite element immersed interface methods JF - Computers & Mathematics with Applications Y1 - 2019 A1 - Luca Heltai A1 - Nella Rotundo PB - Elsevier BV VL - 78 UR - https://doi.org/10.1016/j.camwa.2019.05.029 ER - TY - JOUR T1 - On the existence of elastic minimizers for initially stressed materials JF - Phil. Trans. R. Soc. A Y1 - 2019 A1 - Davide Riccobelli A1 - A. Agosti A1 - Pasquale Ciarletta PB - The Royal Society VL - 377 ER - TY - JOUR T1 - A Finite Volume approximation of the Navier-Stokes equations with nonlinear filtering stabilization JF - Computers & Fluids Y1 - 2019 A1 - Girfoglio, Michele A1 - Quaini, Annalisa A1 - Rozza, Gianluigi AB -We consider a Leray model with a nonlinear differential low-pass filter for the simulation of incompressible fluid flow at moderately large Reynolds number (in the range of a few thousands) with under-refined meshes. For the implementation of the model, we adopt the three-step algorithm Evolve-Filter-Relax (EFR). The Leray model has been extensively applied within a Finite Element (FE) framework. Here, we propose to combine the EFR algorithm with a computationally efficient Finite Volume (FV) method. Our approach is validated against numerical data available in the literature for the 2D flow past a cylinder and against experimental measurements for the 3D fluid flow in an idealized medical device, as recommended by the U.S. Food and Drug Administration. We will show that for similar levels of mesh refinement FV and FE methods provide significantly different results. Through our numerical experiments, we are able to provide practical directions to tune the parameters involved in the model. Furthermore, we are able to investigate the impact of mesh features (element type, non-orthogonality, local refinement, and element aspect ratio) and the discretization method for the convective term on the agreement between numerical solutions and experimental data.

VL - 187 UR - https://arxiv.org/abs/1901.05251 ER - TY - JOUR T1 - A Localized Reduced-Order Modeling Approach for PDEs with Bifurcating Solutions JF - Computer Methods in Applied Mechanics and Engineering Y1 - 2019 A1 - Hess, Martin A1 - Alla, Alessandro A1 - Quaini, Annalisa A1 - Rozza, Gianluigi A1 - Gunzburger, Max AB -Reduced-order modeling (ROM) commonly refers to the construction, based on a few solutions (referred to as snapshots) of an expensive discretized partial differential equation (PDE), and the subsequent application of low-dimensional discretizations of partial differential equations (PDEs) that can be used to more efficiently treat problems in control and optimization, uncertainty quantification, and other settings that require multiple approximate PDE solutions. In this work, a ROM is developed and tested for the treatment of nonlinear PDEs whose solutions bifurcate as input parameter values change. In such cases, the parameter domain can be subdivided into subregions, each of which corresponds to a different branch of solutions. Popular ROM approaches such as proper orthogonal decomposition (POD), results in a global low-dimensional basis that does no respect not take advantage of the often large differences in the PDE solutions corresponding to different subregions. Instead, in the new method, the k-means algorithm is used to cluster snapshots so that within cluster snapshots are similar to each other and are dissimilar to those in other clusters. This is followed by the construction of local POD bases, one for each cluster. The method also can detect which cluster a new parameter point belongs to, after which the local basis corresponding to that cluster is used to determine a ROM approximation. Numerical experiments show the effectiveness of the method both for problems for which bifurcation cause continuous and discontinuous changes in the solution of the PDE.

VL - 351 UR - https://arxiv.org/abs/1807.08851 ER - TY - JOUR T1 - A Note About the Strong Maximum Principle on RCD Spaces JF - Canadian Mathematical Bulletin Y1 - 2019 A1 - Nicola Gigli A1 - Chiara Rigoni AB -We give a direct proof of the strong maximum principle on finite dimensional RCD spaces based on the Laplacian comparison of the squared distance.

PB - Canadian Mathematical Society VL - 62 ER - TY - JOUR T1 - Parametric POD-Galerkin Model Order Reduction for Unsteady-State Heat Transfer Problems JF - Communications in Computational Physics Y1 - 2019 A1 - Sokratia Georgaka A1 - Giovanni Stabile A1 - Gianluigi Rozza A1 - Michael J. Bluck AB -A parametric reduced order model based on proper orthogonal decom- position with Galerkin projection has been developed and applied for the modeling of heat transport in T-junction pipes which are widely found in nuclear power plants. Thermal mixing of different temperature coolants in T-junction pipes leads to tem- perature fluctuations and this could potentially cause thermal fatigue in the pipe walls. The novelty of this paper is the development of a parametric ROM considering the three dimensional, incompressible, unsteady Navier-Stokes equations coupled with the heat transport equation in a finite volume approximation. Two different paramet- ric cases are presented in this paper: parametrization of the inlet temperatures and parametrization of the kinematic viscosity. Different training spaces are considered and the results are compared against the full order model.

VL - 27 UR - https://arxiv.org/abs/1808.05175 ER - TY - JOUR T1 - POD-Galerkin reduced order methods for combined Navier-Stokes transport equations based on a hybrid FV-FE solver JF - Computers & Mathematics with Applications Y1 - 2019 A1 - S. Busto A1 - G. Stabile A1 - G. Rozza A1 - M.E. Vázquez-Cendón AB -The purpose of this work is to introduce a novel POD-Galerkin strategy for the hybrid finite volume/finite element solver introduced in Bermúdez et al. 2014 and Busto et al. 2018. The interest is into the incompressible Navier-Stokes equations coupled with an additional transport equation. The full order model employed in this article makes use of staggered meshes. This feature will be conveyed to the reduced order model leading to the definition of reduced basis spaces in both meshes. The reduced order model presented herein accounts for velocity, pressure, and a transport-related variable. The pressure term at both the full order and the reduced order level is reconstructed making use of a projection method. More precisely, a Poisson equation for pressure is considered within the reduced order model. Results are verified against three-dimensional manufactured test cases. Moreover a modified version of the classical cavity test benchmark including the transport of a species is analysed.

UR - https://arxiv.org/abs/1810.07999 ER - TY - JOUR T1 - A reduced basis approach for PDEs on parametrized geometries based on the shifted boundary finite element method and application to a Stokes flow JF - Computer Methods in Applied Mechanics and Engineering Y1 - 2019 A1 - Karatzas, Efthymios N A1 - Stabile, Giovanni A1 - Nouveau, Leo A1 - Scovazzi, Guglielmo A1 - Rozza, Gianluigi AB -We propose a model order reduction technique integrating the Shifted Boundary Method (SBM) with a POD-Galerkin strategy. This approach allows to treat more complex parametrized domains in an efficient and straightforward way. The impact of the proposed approach is threefold. First, problems involving parametrizations of complex geometrical shapes and/or large domain deformations can be efficiently solved at full-order by means of the SBM, an unfitted boundary method that avoids remeshing and the tedious handling of cut cells by introducing an approximate surrogate boundary. Second, the computational effort is further reduced by the development of a reduced order model (ROM) technique based on a POD-Galerkin approach. Third, the SBM provides a smooth mapping from the true to the surrogate domain, and for this reason, the stability and performance of the reduced order basis are enhanced. This feature is the net result of the combination of the proposed ROM approach and the SBM. Similarly, the combination of the SBM with a projection-based ROM gives the great advantage of an easy and fast to implement algorithm considering geometrical parametrization with large deformations. The transformation of each geometry to a reference geometry (morphing) is in fact not required. These combined advantages will allow the solution of PDE problems more efficiently. We illustrate the performance of this approach on a number of two-dimensional Stokes flow problems.

VL - 347 UR - https://arxiv.org/abs/1807.07790 ER - TY - JOUR T1 - Reduced basis approaches for parametrized bifurcation problems held by non-linear Von Kármán equations Y1 - 2019 A1 - Pichi, Federico A1 - Rozza, Gianluigi AB -This work focuses on the computationally efficient detection of the buckling phenomena and bifurcation analysis of the parametric Von Kármán plate equations based on reduced order methods and spectral analysis. The computational complexity - due to the fourth order derivative terms, the non-linearity and the parameter dependence - provides an interesting benchmark to test the importance of the reduction strategies, during the construction of the bifurcation diagram by varying the parameter(s). To this end, together the state equations, we carry out also an analysis of the linearized eigenvalue problem, that allows us to better understand the physical behaviour near the bifurcation points, where we lose the uniqueness of solution. We test this automatic methodology also in the two parameter case, understanding the evolution of the first buckling mode. journal = Journal of Scientific Computing

VL - 81 UR - https://arxiv.org/abs/1804.02014 ER - TY - CONF T1 - Shape optimization through proper orthogonal decomposition with interpolation and dynamic mode decomposition enhanced by active subspaces T2 - VIII International Conference on Computational Methods in Marine Engineering Y1 - 2019 A1 - Tezzele, Marco A1 - Demo, Nicola A1 - Rozza, Gianluigi AB -We propose a numerical pipeline for shape optimization in naval engineering involving two different non-intrusive reduced order method (ROM) techniques. Such methods are proper orthogonal decomposition with interpolation (PODI) and dynamic mode decomposition (DMD). The ROM proposed will be enhanced by active subspaces (AS) as a pre-processing tool that reduce the parameter space dimension and suggest better sampling of the input space. We will focus on geometrical parameters describing the perturbation of a reference bulbous bow through the free form deformation (FFD) technique. The ROM are based on a finite volume method (FV) to simulate the multi-phase incompressible flow around the deformed hulls. In previous works we studied the reduction of the parameter space in naval engineering through AS [38, 10] focusing on different parts of the hull. PODI and DMD have been employed for the study of fast and reliable shape optimization cycles on a bulbous bow in [9]. The novelty of this work is the simultaneous reduction of both the input parameter space and the output fields of interest. In particular AS will be trained computing the total drag resistance of a hull advancing in calm water and its gradients with respect to the input parameters. DMD will improve the performance of each simulation of the campaign using only few snapshots of the solution fields in order to predict the regime state of the system. Finally PODI will interpolate the coefficients of the POD decomposition of the output fields for a fast approximation of all the fields at new untried parameters given by the optimization algorithm. This will result in a non-intrusive data-driven numerical optimization pipeline completely independent with respect to the full order solver used and it can be easily incorporated into existing numerical pipelines, from the reference CAD to the optimal shape.

JF - VIII International Conference on Computational Methods in Marine Engineering UR - https://arxiv.org/abs/1905.05483 ER - TY - CHAP T1 - A Spectral Element Reduced Basis Method in Parametric CFD T2 - Numerical Mathematics and Advanced Applications - ENUMATH 2017 Y1 - 2019 A1 - Hess, Martin W. A1 - Rozza, Gianluigi ED - Radu, Florin Adrian ED - Kumar, Kundan ED - Berre, Inga ED - Nordbotten, Jan Martin ED - Pop, Iuliu Sorin AB -We consider the Navier-Stokes equations in a channel with varying Reynolds numbers. The model is discretized with high-order spectral element ansatz functions, resulting in 14 259 degrees of freedom. The steady-state snapshot solu- tions define a reduced order space, which allows to accurately evaluate the steady- state solutions for varying Reynolds number with a reduced order model within a fixed-point iteration. In particular, we compare different aspects of implementing the reduced order model with respect to the use of a spectral element discretization. It is shown, how a multilevel static condensation in the pressure and velocity boundary degrees of freedom can be combined with a reduced order modelling approach to enhance computational times in parametric many-query scenarios.

JF - Numerical Mathematics and Advanced Applications - ENUMATH 2017 PB - Springer International Publishing VL - 126 UR - https://arxiv.org/abs/1712.06432 ER - TY - JOUR T1 - An authenticated theoretical modeling of electrified fluid jet in core–shell nanofibers production JF - JOURNAL OF INDUSTRIAL TEXTILES Y1 - 2018 A1 - Rafiei, S. A1 - Noroozi, B. A1 - Luca Heltai A1 - Haghi, A. K. VL - 47 ER - TY - JOUR T1 - Certified Reduced Basis Approximation for the Coupling of Viscous and Inviscid Parametrized Flow Models JF - Journal of Scientific Computing Y1 - 2018 A1 - Immanuel Martini A1 - Bernard Haasdonk A1 - Gianluigi Rozza VL - 74 UR - https://www.scopus.com/inward/record.uri?eid=2-s2.0-85017156114&doi=10.1007%2fs10915-017-0430-y&partnerID=40&md5=023ef0bb95713f4442d1fa374c92a964 ER - TY - CHAP T1 - Combined parameter and model reduction of cardiovascular problems by means of active subspaces and POD-Galerkin methods T2 - Mathematical and Numerical Modeling of the Cardiovascular System and Applications Y1 - 2018 A1 - Marco Tezzele A1 - Francesco Ballarin A1 - Gianluigi Rozza JF - Mathematical and Numerical Modeling of the Cardiovascular System and Applications PB - Springer ER - TY - JOUR T1 - A Comparison Between Active Strain and Active Stress in Transversely Isotropic Hyperelastic Materials JF - J. Elast. Y1 - 2018 A1 - Giulia Giantesio A1 - Alessandro Musesti A1 - Davide Riccobelli PB - Springer Nature ER - TY - CHAP T1 - Computational methods in cardiovascular mechanics T2 - Cardiovascular Mechanics Y1 - 2018 A1 - Auricchio, Ferdinando A1 - Conti, Michele A1 - Lefieux, Adrian A1 - Morganti, Simone A1 - Alessandro Reali A1 - Gianluigi Rozza A1 - Veneziani, Alessandro ED - Michel F. Labrosse AB -The introduction of computational models in cardiovascular sciences has been progressively bringing new and unique tools for the investigation of the physiopathology. Together with the dramatic improvement of imaging and measuring devices on one side, and of computational architectures on the other one, mathematical and numerical models have provided a new, clearly noninvasive, approach for understanding not only basic mechanisms but also patient-specific conditions, and for supporting the design and the development of new therapeutic options. The terminology in silico is, nowadays, commonly accepted for indicating this new source of knowledge added to traditional in vitro and in vivo investigations. The advantages of in silico methodologies are basically the low cost in terms of infrastructures and facilities, the reduced invasiveness and, in general, the intrinsic predictive capabilities based on the use of mathematical models. The disadvantages are generally identified in the distance between the real cases and their virtual counterpart required by the conceptual modeling that can be detrimental for the reliability of numerical simulations.

JF - Cardiovascular Mechanics PB - CRC Press UR - https://www.taylorfrancis.com/books/e/9781315280288/chapters/10.1201%2Fb21917-5 ER - TY - JOUR T1 - Dimension reduction in heterogeneous parametric spaces with application to naval engineering shape design problems JF - Advanced Modeling and Simulation in Engineering Sciences Y1 - 2018 A1 - Marco Tezzele A1 - Filippo Salmoiraghi A1 - Andrea Mola A1 - Gianluigi Rozza AB -We present the results of the first application in the naval architecture field of a methodology based on active subspaces properties for parameters space reduction. The physical problem considered is the one of the simulation of the hydrodynamic flow past the hull of a ship advancing in calm water. Such problem is extremely relevant at the preliminary stages of the ship design, when several flow simulations are typically carried out by the engineers to assess the dependence of the hull total resistance on the geometrical parameters of the hull, and others related with flows and hull properties. Given the high number of geometric and physical parameters which might affect the total ship drag, the main idea of this work is to employ the active subspaces properties to identify possible lower dimensional structures in the parameter space. Thus, a fully automated procedure has been implemented to produce several small shape perturbations of an original hull CAD geometry, in order to exploit the resulting shapes to run high fidelity flow simulations with different structural and physical parameters as well, and then collect data for the active subspaces analysis. The free form deformation procedure used to morph the hull shapes, the high fidelity solver based on potential flow theory with fully nonlinear free surface treatment, and the active subspaces analysis tool employed in this work have all been developed and integrated within SISSA mathLab as open source tools. The contribution will also discuss several details of the implementation of such tools, as well as the results of their application to the selected target engineering problem.

VL - 5 ER - TY - Generic T1 - An efficient shape parametrisation by free-form deformation enhanced by active subspace for hull hydrodynamic ship design problems in open source environment T2 - The 28th International Ocean and Polar Engineering Conference Y1 - 2018 A1 - Nicola Demo A1 - Marco Tezzele A1 - Andrea Mola A1 - Gianluigi Rozza KW - Active subspaces KW - Boundary element method KW - Dynamic mode decomposition KW - Fluid structure interaction KW - Free form deformation KW - Fully nonlinear potential KW - Numerical towing tank AB - In this contribution, we present the results of the application of a parameter space reduction methodology based on active subspaces to the hull hydrodynamic design problem. Several parametric deformations of an initial hull shape are considered to assess the influence of the shape parameters considered on the hull total drag. The hull resistance is typically computed by means of numerical simulations of the hydrodynamic flow past the ship. Given the high number of parameters involved - which might result in a high number of time consuming hydrodynamic simulations - assessing whether the parameters space can be reduced would lead to considerable computational cost reduction. Thus, the main idea of this work is to employ the active subspaces to identify possible lower dimensional structures in the parameter space, or to verify the parameter distribution in the position of the control points. To this end, a fully automated procedure has been implemented to produce several small shape perturbations of an original hull CAD geometry which are then used to carry out high-fidelity flow simulations and collect data for the active subspaces analysis. To achieve full automation of the open source pipeline described, both the free form deformation methodology employed for the hull perturbations and the solver based on unsteady potential flow theory, with fully nonlinear free surface treatment, are directly interfaced with CAD data structures and operate using IGES vendor-neutral file formats as input files. The computational cost of the fluid dynamic simulations is further reduced through the application of dynamic mode decomposition to reconstruct the steady state total drag value given only few initial snapshots of the simulation. The active subspaces analysis is here applied to the geometry of the DTMB-5415 naval combatant hull, which is which is a common benchmark in ship hydrodynamics simulations. JF - The 28th International Ocean and Polar Engineering Conference PB - International Society of Offshore and Polar Engineers CY - Sapporo, Japan UR - https://www.onepetro.org/conference-paper/ISOPE-I-18-481 ER - TY - ABST T1 - The Effort of Increasing Reynolds Number in Projection-Based Reduced Order Methods: from Laminar to Turbulent Flows Y1 - 2018 A1 - Saddam Hijazi A1 - Shafqat Ali A1 - Giovanni Stabile A1 - Francesco Ballarin A1 - Gianluigi Rozza ER - TY - RPRT T1 - Existence and uniqueness of dynamic evolutions for a one dimensional debonding model with damping Y1 - 2018 A1 - Lorenzo Nardini A1 - Filippo Riva AB -In this paper we analyse a one-dimensional debonding model for a thin film peeled from a substrate when friction is taken into account. It is described by the weakly damped wave equation whose domain, the debonded region, grows according to a Griffth's criterion. Firstly we prove that the equation admits a unique solution when the evolution of the debonding front is assigned. Finally we provide an existence and uniqueness result for the coupled problem given by the wave equation together with Griffth's criterion.

UR - http://preprints.sissa.it/xmlui/handle/1963/35319 U1 - 35629 U2 - Mathematics U4 - 1 ER - TY - JOUR T1 - EZyRB: Easy Reduced Basis method JF - The Journal of Open Source Software Y1 - 2018 A1 - Nicola Demo A1 - Marco Tezzele A1 - Gianluigi Rozza VL - 3 UR - https://joss.theoj.org/papers/10.21105/joss.00661 ER - TY - CHAP T1 - Failure of the Chain Rule in the Non Steady Two-Dimensional Setting T2 - Current Research in Nonlinear Analysis: In Honor of Haim Brezis and Louis Nirenberg Y1 - 2018 A1 - Stefano Bianchini A1 - Paolo Bonicatto ED - Rassias, Themistocles M. JF - Current Research in Nonlinear Analysis: In Honor of Haim Brezis and Louis Nirenberg PB - Springer International Publishing CY - Cham SN - 978-3-319-89800-1 UR - https://doi.org/10.1007/978-3-319-89800-1_2 ER - TY - JOUR T1 - Finite volume POD-Galerkin stabilised reduced order methods for the parametrised incompressible Navier-Stokes equations JF - Computers & Fluids Y1 - 2018 A1 - Giovanni Stabile A1 - Gianluigi Rozza PB - Elsevier {BV} UR - https://doi.org/10.1016/j.compfluid.2018.01.035 ER - TY - JOUR T1 - Free-form deformation, mesh morphing and reduced-order methods: enablers for efficient aerodynamic shape optimisation JF - International Journal of Computational Fluid Dynamics Y1 - 2018 A1 - Filippo Salmoiraghi A1 - Scardigli, Angela A1 - Telib, Haysam A1 - Gianluigi Rozza AB -In this work, we provide an integrated pipeline for the model-order reduction of turbulent flows around parametrised geometries in aerodynamics. In particular, free-form deformation is applied for geometry parametrisation, whereas two different reduced-order models based on proper orthogonal decomposition (POD) are employed in order to speed-up the full-order simulations: the first method exploits POD with interpolation, while the second one is based on domain decomposition. For the sampling of the parameter space, we adopt a Greedy strategy coupled with Constrained Centroidal Voronoi Tessellations, in order to guarantee a good compromise between space exploration and exploitation. The proposed framework is tested on an industrially relevant application, i.e. the front-bumper morphing of the DrivAer car model, using the finite-volume method for the full-order resolution of the Reynolds-Averaged Navier–Stokes equations.

PB - Taylor & Francis VL - 32 ER - TY - RPRT T1 - Long time existence for fully nonlinear NLS with small Cauchy data on the circle Y1 - 2018 A1 - Feola Roberto A1 - Felice Iandoli ER - TY - CONF T1 - Model Order Reduction by means of Active Subspaces and Dynamic Mode Decomposition for Parametric Hull Shape Design Hydrodynamics T2 - Technology and Science for the Ships of the Future: Proceedings of NAV 2018: 19th International Conference on Ship & Maritime Research Y1 - 2018 A1 - Marco Tezzele A1 - Nicola Demo A1 - Mahmoud Gadalla A1 - Andrea Mola A1 - Gianluigi Rozza AB - We present the results of the application of a parameter space reduction methodology based on active subspaces (AS) to the hull hydrodynamic design problem. Several parametric deformations of an initial hull shape are considered to assess the influence of the shape parameters on the hull wave resistance. Such problem is relevant at the preliminary stages of the ship design, when several flow simulations are carried out by the engineers to establish a certain sensibility with respect to the parameters, which might result in a high number of time consuming hydrodynamic simulations. The main idea of this work is to employ the AS to identify possible lower dimensional structures in the parameter space. The complete pipeline involves the use of free form deformation to parametrize and deform the hull shape, the full order solver based on unsteady potential flow theory with fully nonlinear free surface treatment directly interfaced with CAD, the use of dynamic mode decomposition to reconstruct the final steady state given only few snapshots of the simulation, and the reduction of the parameter space by AS, and shared subspace. Response surface method is used to minimize the total drag. JF - Technology and Science for the Ships of the Future: Proceedings of NAV 2018: 19th International Conference on Ship & Maritime Research PB - IOS Press CY - Trieste, Italy UR - http://ebooks.iospress.nl/publication/49270 ER - TY - JOUR T1 - Model Reduction for Parametrized Optimal Control Problems in Environmental Marine Sciences and Engineering JF - SIAM Journal on Scientific Computing Y1 - 2018 A1 - Maria Strazzullo A1 - Francesco Ballarin A1 - Mosetti, R. A1 - Gianluigi Rozza VL - 40 UR - https://doi.org/10.1137/17M1150591 ER - TY - JOUR T1 - Morpho-elastic model of the tortuous tumour vessels JF - Int. J. Non-Linear Mech. Y1 - 2018 A1 - Davide Riccobelli A1 - Pasquale Ciarletta PB - Elsevier BV VL - 107 ER - TY - JOUR T1 - Noncommutative Painlevé Equations and Systems of Calogero Type JF - Comm. Math. Phys Y1 - 2018 A1 - Marco Bertola A1 - Mattia Cafasso A1 - V. Rubtsov ER - TY - JOUR T1 - PyDMD: Python Dynamic Mode Decomposition JF - The Journal of Open Source Software Y1 - 2018 A1 - Nicola Demo A1 - Marco Tezzele A1 - Gianluigi Rozza VL - 3 UR - https://joss.theoj.org/papers/734e4326edd5062c6e8ee98d03df9e1d ER - TY - JOUR T1 - Recognizing the flat torus among RCD*(0,N) spaces via the study of the first cohomology group JF - Calculus of Variations and Partial Differential Equations Y1 - 2018 A1 - Nicola Gigli A1 - Chiara Rigoni AB -We prove that if the dimension of the first cohomology group of a $\mathsf{RCD}^\star (0,N)$ space is $N$, then the space is a flat torus. This generalizes a classical result due to Bochner to the non-smooth setting and also provides a first example where the study of the cohomology groups in such synthetic framework leads to geometric consequences.

VL - 57 UR - https://doi.org/10.1007/s00526-018-1377-z ER - TY - ABST T1 - A Reduced Basis approach for PDEs on parametrized geometries based on the Shifted Boundary Finite Element Method and application to fluid dynamics Y1 - 2018 A1 - Efthymios N. Karatzas A1 - Giovanni Stabile A1 - Leo Nouveau A1 - Guglielmo Scovazzi A1 - Gianluigi Rozza ER - TY - CHAP T1 - Reduced Basis Approximation and A Posteriori Error Estimation: Applications to Elasticity Problems in Several Parametric Settings T2 - Numerical Methods for PDEs Y1 - 2018 A1 - Huynh, D. B. P. A1 - Pichi, Federico A1 - Rozza, Gianluigi JF - Numerical Methods for PDEs VL - 15 UR - https://link.springer.com/chapter/10.1007/978-3-319-94676-4_8 ER - TY - ABST T1 - A Reduced Order Approach for the Embedded Shifted Boundary FEM and a Heat Exchange System on Parametrized Geometries Y1 - 2018 A1 - Efthymios N. Karatzas A1 - Giovanni Stabile A1 - N. Atallah A1 - Guglielmo Scovazzi A1 - Gianluigi Rozza ER - TY - CONF T1 - Shape Optimization by means of Proper Orthogonal Decomposition and Dynamic Mode Decomposition T2 - Technology and Science for the Ships of the Future: Proceedings of NAV 2018: 19th International Conference on Ship & Maritime Research Y1 - 2018 A1 - Nicola Demo A1 - Marco Tezzele A1 - Gianluca Gustin A1 - Gianpiero Lavini A1 - Gianluigi Rozza AB - Shape optimization is a challenging task in many engineering fields, since the numerical solutions of parametric system may be computationally expensive. This work presents a novel optimization procedure based on reduced order modeling, applied to a naval hull design problem. The advantage introduced by this method is that the solution for a specific parameter can be expressed as the combination of few numerical solutions computed at properly chosen parametric points. The reduced model is built using the proper orthogonal decomposition with interpolation (PODI) method. We use the free form deformation (FFD) for an automated perturbation of the shape, and the finite volume method to simulate the multiphase incompressible flow around the deformed hulls. Further computational reduction is done by the dynamic mode decomposition (DMD) technique: from few high dimensional snapshots, the system evolution is reconstructed and the final state of the simulation is faithfully approximated. Finally the global optimization algorithm iterates over the reduced space: the approximated drag and lift coefficients are projected to the hull surface, hence the resistance is evaluated for the new hulls until the convergence to the optimal shape is achieved. We will present the results obtained applying the described procedure to a typical Fincantieri cruise ship. JF - Technology and Science for the Ships of the Future: Proceedings of NAV 2018: 19th International Conference on Ship & Maritime Research PB - IOS Press CY - Trieste, Italy UR - http://ebooks.iospress.nl/publication/49229 ER - TY - JOUR T1 - Shape transitions in a soft incompressible sphere with residual stresses JF - Math. Mech. Solids Y1 - 2018 A1 - Davide Riccobelli A1 - Pasquale Ciarletta PB - SAGE Publications Sage UK: London, England VL - 23 ER - TY - JOUR T1 - Advances in Reduced order modelling for CFD: vortex shedding around a circular cylinder using a POD-Galerkin method JF - Communication in Applied Industrial Mathematics Y1 - 2017 A1 - Giovanni Stabile A1 - Saddam Hijazi A1 - Stefano Lorenzi A1 - Andrea Mola A1 - Gianluigi Rozza KW - finite volume, CFD KW - Reduced order methods AB -Vortex shedding around circular cylinders is a well known and studied phenomenon that appears in many engineering fields. In this work a Reduced Order Model (ROM) of the incompressible flow around a circular cylinder, built performing a Galerkin projection of the governing equations onto a lower dimensional space is presented. The reduced basis space is generated using a Proper Orthogonal Decomposition (POD) approach. In particular the focus is into (i) the correct reproduction of the pressure field, that in case of the vortex shedding phenomenon, is of primary importance for the calculation of the drag and lift coefficients; (ii) for this purpose the projection of the Governing equations (momentum equation and Poisson equation for pressure) is performed onto different reduced basis space for velocity and pressure, respectively; (iii) all the relevant modifications necessary to adapt standard finite element POD-Galerkin methods to a finite volume framework are presented. The accuracy of the reduced order model is assessed against full order results.

UR - https://arxiv.org/abs/1701.03424 ER - TY - JOUR T1 - On the Application of Reduced Basis Methods to Bifurcation Problems in Incompressible Fluid Dynamics JF - Journal of Scientific Computing Y1 - 2017 A1 - Giuseppe Pitton A1 - Gianluigi Rozza AB -In this paper we apply a reduced basis framework for the computation of flow bifurcation (and stability) problems in fluid dynamics. The proposed method aims at reducing the complexity and the computational time required for the construction of bifurcation and stability diagrams. The method is quite general since it can in principle be specialized to a wide class of nonlinear problems, but in this work we focus on an application in incompressible fluid dynamics at low Reynolds numbers. The validation of the reduced order model with the full order computation for a benchmark cavity flow problem is promising.

ER - TY - CHAP T1 - Certified Reduced Basis Method for Affinely Parametric Isogeometric Analysis NURBS Approximation T2 - Spectral and High Order Methods for Partial Differential Equations Y1 - 2017 A1 - Denis Devaud A1 - Gianluigi Rozza AB -In this work we apply reduced basis methods for parametric PDEs to an isogeometric formulation based on

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

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

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

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

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

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

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

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

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

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

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

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

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

JF - Spectral and High Order Methods for Partial Differential Equations PB - Springer CY - Heildeberg VL - 119 SN - 978-3-319-65869-8 ER - TY - JOUR T1 - On a certified smagorinsky reduced basis turbulence model JF - SIAM Journal on Numerical Analysis Y1 - 2017 A1 - Rebollo, T.C. A1 - E.D. Ávila A1 - Marmol, M.G. A1 - Francesco Ballarin A1 - Gianluigi Rozza VL - 55 UR - https://www.scopus.com/inward/record.uri?eid=2-s2.0-85039928218&doi=10.1137%2f17M1118233&partnerID=40&md5=221d9cd2bcc74121fcef93efd9d3d76c ER - TY - JOUR T1 - Clifford Tori and the singularly perturbed Cahn–Hilliard equation JF - Journal of Differential Equations Y1 - 2017 A1 - Matteo Rizzi KW - Cahn–Hilliard equation KW - Clifford Torus KW - Lyapunov–Schmidt reduction KW - Willmore surface AB -In this paper we construct entire solutions uε to the Cahn–Hilliard equation −ε2Δ(−ε2Δu+W′(u))+W″(u)(−ε2Δu+W′(u))=ε4λε(1−uε), under the volume constraint ∫R3(1−uε)2dx=82π2cε, with cε→1 as ε→0, whose nodal set approaches the Clifford Torus, that is the Torus with radii of ratio 1/2 embedded in R3, as ε→0. It is crucial that the Clifford Torus is a Willmore hypersurface and it is non-degenerate, up to conformal transformations. The proof is based on the Lyapunov–Schmidt reduction and on careful geometric expansions of the Laplacian.

VL - 262 UR - http://www.sciencedirect.com/science/article/pii/S0022039617300530 ER - TY - JOUR T1 - Computational reduction strategies for the detection of steady bifurcations in incompressible fluid-dynamics: Applications to Coanda effect in cardiology JF - Journal of Computational Physics Y1 - 2017 A1 - Giuseppe Pitton A1 - Annalisa Quaini A1 - Gianluigi Rozza KW - Parametrized Navier–Stokes equations KW - Reduced basis method KW - Stability of flows KW - Symmetry breaking bifurcation AB -We focus on reducing the computational costs associated with the hydrodynamic stability of solutions of the incompressible Navier–Stokes equations for a Newtonian and viscous fluid in contraction–expansion channels. In particular, we are interested in studying steady bifurcations, occurring when non-unique stable solutions appear as physical and/or geometric control parameters are varied. The formulation of the stability problem requires solving an eigenvalue problem for a partial differential operator. An alternative to this approach is the direct simulation of the flow to characterize the asymptotic behavior of the solution. Both approaches can be extremely expensive in terms of computational time. We propose to apply Reduced Order Modeling (ROM) techniques to reduce the demanding computational costs associated with the detection of a type of steady bifurcations in fluid dynamics. The application that motivated the present study is the onset of asymmetries (i.e., symmetry breaking bifurcation) in blood flow through a regurgitant mitral valve, depending on the Reynolds number and the regurgitant mitral valve orifice shape.

We show that equivariant Donaldson polynomials of compact toric surfaces can be calculated as residues of suitable combinations of Virasoro conformal blocks, by building on AGT correspondence between $\mathcal{N}=2$ supersymmetric gauge theories and two-dimensional conformal field theory. Talk presented by A.T. at the conference Interactions between Geometry and Physics — in honor of Ugo Bruzzo’s 60th birthday 17–22 August 2015, Guarujá, São Paulo, Brazil, mostly based on Bawane et al. (0000) and Bershtein et al. (0000).

VL - 118 UR - http://www.sciencedirect.com/science/article/pii/S0393044017300165 N1 - Interactions between Geometry and Physics. A Special Issue in Honor of Ugo Bruzzo’s 60th Birthday ER - TY - CHAP T1 - Model Reduction Methods T2 - Encyclopedia of Computational Mechanics Second Edition Y1 - 2017 A1 - Francisco Chinesta A1 - Antonio Huerta A1 - Gianluigi Rozza A1 - Karen Willcox AB -This chapter presents an overview of model order reduction – a new paradigm in the field of simulation-based engineering sciences, and one that can tackle the challenges and leverage the opportunities of modern ICT technologies. Despite the impressive progress attained by simulation capabilities and techniques, a number of challenging problems remain intractable. These problems are of different nature, but are common to many branches of science and engineering. Among them are those related to high-dimensional problems, problems involving very different time scales, models defined in degenerate domains with at least one of the characteristic dimensions much smaller than the others, model requiring real-time simulation, and parametric models. All these problems represent a challenge for standard mesh-based discretization techniques; yet the ability to solve these problems efficiently would open unexplored routes for real-time simulation, inverse analysis, uncertainty quantification and propagation, real-time optimization, and simulation-based control – critical needs in many branches of science and engineering. Model order reduction offers new simulation alternatives by circumventing, or at least alleviating, otherwise intractable computational challenges. In the present chapter, we revisit three of these model reduction techniques: proper orthogonal decomposition, proper generalized decomposition, and reduced basis methodologies.} preprint = {http://preprints.sissa.it/xmlui/bitstream/handle/1963/35194/ECM_MOR.pdf?sequence=1&isAllowed=y

JF - Encyclopedia of Computational Mechanics Second Edition PB - John Wiley & Sons ER - TY - JOUR T1 - A natural framework for isogeometric fluid-structure interaction based on BEM-shell coupling JF - COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING Y1 - 2017 A1 - Luca Heltai A1 - Kiendl, J. A1 - Antonio DeSimone A1 - Alessandro Reali VL - 316 UR - http://cdsads.u-strasbg.fr/abs/2017CMAME.316..522H ER - TY - JOUR T1 - Numerical modeling of hemodynamics scenarios of patient-specific coronary artery bypass grafts JF - Biomechanics and Modeling in Mechanobiology Y1 - 2017 A1 - Francesco Ballarin A1 - Elena Faggiano A1 - Andrea Manzoni A1 - Alfio Quarteroni A1 - Gianluigi Rozza A1 - Sonia Ippolito A1 - Roberto Scrofani VL - 16 UR - https://www.scopus.com/inward/record.uri?eid=2-s2.0-85015065851&doi=10.1007%2fs10237-017-0893-7&partnerID=40&md5=c388f20bd5de14187bad9ed7d9affbd0 ER - TY - JOUR T1 - POD-Galerkin reduced order methods for CFD using Finite Volume Discretisation: vortex shedding around a circular cylinder JF - Communications in Applied and Industrial Mathematics Y1 - 2017 A1 - Giovanni Stabile A1 - Saddam Hijazi A1 - Andrea Mola A1 - Stefano Lorenzi A1 - Gianluigi Rozza PB - Walter de Gruyter {GmbH} VL - 8 UR - https://doi.org/10.1515/caim-2017-0011 ER - TY - JOUR T1 - Rayleigh–Taylor instability in soft elastic layers JF - Phil. Trans. R. Soc. A Y1 - 2017 A1 - Davide Riccobelli A1 - Pasquale Ciarletta PB - The Royal Society VL - 375 ER - TY - JOUR T1 - Reduced Basis Methods for Uncertainty Quantification JF - SIAM/ASA Journal on Uncertainty Quantification Y1 - 2017 A1 - Peng Chen A1 - Alfio Quarteroni A1 - Gianluigi Rozza AB -In this work we review a reduced basis method for the solution of uncertainty quantification problems. Based on the basic setting of an elliptic partial differential equation with random input, we introduce the key ingredients of the reduced basis method, including proper orthogonal decomposition and greedy algorithms for the construction of the reduced basis functions, a priori and a posteriori error estimates for the reduced basis approximations, as well as its computational advantages and weaknesses in comparison with a stochastic collocation method [I. Babuška, F. Nobile, and R. Tempone, *SIAM Rev.*, 52 (2010), pp. 317--355]. We demonstrate its computational efficiency and accuracy for a benchmark problem with parameters ranging from a few to a few hundred dimensions. Generalizations to more complex models and applications to uncertainty quantification problems in risk prediction, evaluation of statistical moments, Bayesian inversion, and optimal control under uncertainty are also presented to illustrate how to use the reduced basis method in practice. Further challenges, advancements, and research opportunities are outlined.

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

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

JF - Model Reduction of Parametrized Systems PB - Springer International Publishing ER - TY - JOUR T1 - Solid tumors are poroelastic solids with a chemo-mechanical feedback on growth JF - J. Elast. Y1 - 2017 A1 - D. Ambrosi A1 - Pezzuto, S. A1 - Davide Riccobelli A1 - Stylianopoulos, T. A1 - Pasquale Ciarletta PB - Springer Netherlands VL - 129 ER - TY - JOUR T1 - Universality of the Peregrine Soliton in the Focusing Dynamics of the Cubic Nonlinear Schrödinger Equation JF - Phys. Rev. Lett. Y1 - 2017 A1 - Tikan, Alexey A1 - Billet, Cyril A1 - Gennady El A1 - Alexander Tovbis A1 - Marco Bertola A1 - Sylvestre, Thibaut A1 - Gustave, Francois A1 - Randoux, Stephane A1 - Genty, Goëry A1 - Suret, Pierre A1 - Dudley, John M. PB - American Physical Society VL - 119 UR - https://link.aps.org/doi/10.1103/PhysRevLett.119.033901 ER - TY - CONF T1 - Advances in geometrical parametrization and reduced order models and methods for computational fluid dynamics problems in applied sciences and engineering: overview and perspectives T2 - Proceedings of the ECCOMAS Congress 2016, VII European Conference on Computational Methods in Applied Sciences and Engineering, Y1 - 2016 A1 - Filippo Salmoiraghi A1 - Francesco Ballarin A1 - Giovanni Corsi A1 - Andrea Mola A1 - Marco Tezzele A1 - Gianluigi Rozza ED - Papadrakakis, M. ED - Papadopoulos, V. ED - Stefanou, G. ED - Plevris, V. AB -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.

JF - Proceedings of the ECCOMAS Congress 2016, VII European Conference on Computational Methods in Applied Sciences and Engineering, PB - ECCOMAS CY - Crete, Greece U1 - 35466 U2 - Mathematics U4 - 1 U5 - MAT/08 ER - TY - JOUR T1 - Critical points of a perturbed Otha-Kawasaki functional JF - arXiv preprint arXiv:1601.07093 Y1 - 2016 A1 - Matteo Rizzi ER - TY - JOUR T1 - Error Estimates of B-spline based finite-element method for the wind-driven ocean circulation JF - JOURNAL OF SCIENTIFIC COMPUTING Y1 - 2016 A1 - Rotundo, N. A1 - Kim, T. -Y. A1 - Jiang, W. A1 - Luca Heltai A1 - Fried, E. VL - 69 ER - TY - JOUR T1 - Exact results for N=2 supersymmetric gauge theories on compact toric manifolds and equivariant Donaldson invariants JF - Journal of High Energy Physics Y1 - 2016 A1 - Mikhail Bershtein A1 - Giulio Bonelli A1 - Massimiliano Ronzani A1 - Alessandro Tanzini AB -We provide a contour integral formula for the exact partition function of $\mathcal{N}=2$ supersymmetric $U(N)$ gauge theories on compact toric four-manifolds by means of supersymmetric localisation. We perform the explicit evaluation of the contour integral for $U(2)\; \mathcal{N}=2^\star$ theory on $\mathbb{P}^2$ for all instanton numbers. In the zero mass case, corresponding to the $\mathcal{N}=4$ supersymmetric gauge theory, we obtain the generating function of the Euler characteristics of instanton moduli spaces in terms of mock-modular forms. In the decoupling limit of infinite mass we find that the generating function of local and surface observables computes equivariant Donaldson invariants, thus proving in this case a longstanding conjecture by N. Nekrasov. In the case of vanishing first Chern class the resulting equivariant Donaldson polynomials are new.

VL - 2016 UR - https://doi.org/10.1007/JHEP07(2016)023 ER - TY - RPRT T1 - A fast virtual surgery platform for many scenarios haemodynamics of patient-specific coronary artery bypass grafts Y1 - 2016 A1 - Francesco Ballarin A1 - Elena Faggiano A1 - Andrea Manzoni A1 - Gianluigi Rozza A1 - Alfio Quarteroni A1 - Sonia Ippolito A1 - Roberto Scrofani A1 - Carlo Antona AB - 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. PB - Submitted UR - http://urania.sissa.it/xmlui/handle/1963/35240 U1 - 35545 U2 - Mathematics U4 - 1 U5 - MAT/08 ER - TY - THES T1 - Instanton counting on compact manifolds Y1 - 2016 A1 - Massimiliano Ronzani KW - Supersymmetry AB - In this thesis we analyze supersymmetric gauge theories on compact manifolds and their relation with representation theory of infinite Lie algebras associated to conformal field theories, and with the computation of geometric invariants and superconformal indices. The thesis contains the work done by the candidate during the doctorate programme at SISSA under the supervision of A. Tanzini and G. Bonelli. • in Chapter 2, we consider N = 2 supersymmetric gauge theories on four manifolds admitting an isometry. Generalized Killing spinor equations are derived from the consistency of supersymmetry algebrae and solved in the case of four manifolds admitting a U(1) isometry. This is used to explicitly compute the supersymmetric path integral on S2 × S2 via equivariant localization. The building blocks of the resulting partition function are shown to contain the three point functions and the conformal blocks of Liouville Gravity. • in Chapter 3, we provide a contour integral formula for the exact partition function of N = 2 supersymmetric U(N) gauge theories on compact toric four-manifolds by means of supersymmetric localisation. We perform the explicit evaluation of the contour integral for U(2) N = 2∗ theory on P2 for all instanton numbers. In the zero mass case, corresponding to the N = 4 supersymmetric gauge theory, we obtain the generating function of the Euler characteristics of instanton moduli spaces in terms of mock-modular forms. In the decoupling limit of infinite mass we find that the generating function of local and surface observables computes equivariant Donaldson invariants, thus proving in this case a long-standing conjecture by N. Nekrasov. In the case of vanishing first Chern class the resulting equivariant Donaldson polynomials are new. • in Chapter 4, we explore N = (1, 0) superconformal six-dimensional theories arising from M5 branes probing a transverse Ak singularity. Upon circle compactification to five dimensions, we describe this system with a dual pq-web of five-branes and propose the spectrum of basic five-dimensional in- stanton operators driving global symmetry enhancement. For a single M5 brane, we find that the exact partition function of the 5d quiver gauge theory matches the 6d (1, 0) index, which we compute by letter counting. We finally show which relations among vertex correlators of qW algebrae are implied by the S-duality of the pq-web. PB - SISSA UR - http://urania.sissa.it/xmlui/handle/1963/35219 U1 - 35521 U2 - Mathematics U4 - 1 U5 - MAT/07 ER - TY - RPRT T1 - Isogeometric analysis-based reduced order modelling for incompressible linear viscous flows in parametrized shapes Y1 - 2016 A1 - Filippo Salmoiraghi A1 - Francesco Ballarin A1 - Luca Heltai A1 - Gianluigi Rozza AB - 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. PB - Springer, AMOS Advanced Modelling and Simulation in Engineering Sciences UR - http://urania.sissa.it/xmlui/handle/1963/35199 U1 - 35493 U2 - Mathematics U4 - 1 U5 - MAT/08 ER - TY - CHAP T1 - Model Order Reduction: a survey T2 - Wiley Encyclopedia of Computational Mechanics, 2016 Y1 - 2016 A1 - Francisco Chinesta A1 - Antonio Huerta A1 - Gianluigi Rozza A1 - Karen Willcox JF - Wiley Encyclopedia of Computational Mechanics, 2016 PB - Wiley UR - http://urania.sissa.it/xmlui/handle/1963/35194 U1 - 35470 U2 - Mathematics U4 - 1 U5 - MAT/08 ER - TY - JOUR T1 - A multi-physics reduced order model for the analysis of Lead Fast Reactor single channel JF - Annals of Nuclear Energy, 87, 2 (2016): pp. 198-208 Y1 - 2016 A1 - Alberto Sartori A1 - Antonio Cammi A1 - Lelio Luzzi A1 - Gianluigi Rozza AB - In this work, a Reduced Basis method, with basis functions sampled by a Proper Orthogonal Decomposition technique, has been employed to develop a reduced order model of a multi-physics parametrized Lead-cooled Fast Reactor single-channel. Being the first time that a reduced order model is developed in this context, the work focused on a methodological approach and the coupling between the neutronics and the heat transfer, where the thermal feedbacks on neutronics are explicitly taken into account, in time-invariant settings. In order to address the potential of such approach, two different kinds of varying parameters have been considered, namely one related to a geometric quantity (i.e., the inner radius of the fuel pellet) and one related to a physical quantity (i.e., the inlet lead velocity). The capabilities of the presented reduced order model (ROM) have been tested and compared with a high-fidelity finite element model (upon which the ROM has been constructed) on different aspects. In particular, the comparison focused on the system reactivity prediction (with and without thermal feedbacks on neutronics), the neutron flux and temperature field reconstruction, and on the computational time. The outcomes provided by the reduced order model are in good agreement with the high-fidelity finite element ones, and a computational speed-up of at least three orders of magnitude is achieved as well. PB - Elsevier VL - 87 UR - http://urania.sissa.it/xmlui/handle/1963/35191 U1 - 35471 U2 - Mathematics U4 - 1 U5 - MAT/08 ER - TY - JOUR T1 - POD-Galerkin Method for Finite Volume Approximation of Navier-Stokes and RANS Equations Y1 - 2016 A1 - Stefano Lorenzi A1 - Antonio Cammi A1 - Lelio Luzzi A1 - Gianluigi Rozza AB - Numerical simulation of fluid flows requires important computational efforts but it is essential in engineering applications. Reduced Order Model (ROM) can be employed whenever fast simulations are required, or in general, whenever a trade-off between computational cost and solution accuracy is a preeminent issue as in process optimization and control. In this work, the efforts have been put to develop a ROM for Computational Fluid Dynamics (CFD) application based on Finite Volume approximation, starting from the results available in turbulent Reynold-Averaged Navier Stokes simulations in order to enlarge the application field of Proper Orthogonal Decomposition – Reduced Order Model (POD – ROM) technique to more industrial fields. The approach is tested in the classic benchmark of the numerical simulation of the 2D lid-driven cavity. In particular, two simulations at Re = 103 and Re = 105 have been considered in order to assess both a laminar and turbulent case. Some quantities have been compared with the Full Order Model in order to assess the performance of the proposed ROM procedure i.e., the kinetic energy of the system and the reconstructed quantities of interest (velocity, pressure and turbulent viscosity). In addition, for the laminar case, the comparison between the ROM steady-state solution and the data available in literature has been presented. The results have turned out to be very satisfactory both for the accuracy and the computational times. As a major outcome, the approach turns out not to be affected by the energy blow up issue characterizing the results obtained by classic turbulent POD-Galerkin methods. PB - Computer Methods in Applied Mechanics and Engineering, Elsevier U1 - 35502 U2 - Mathematics U4 - 1 U5 - MAT/08 ER - TY - JOUR T1 - POD–Galerkin monolithic reduced order models for parametrized fluid-structure interaction problems JF - International Journal Numerical Methods for Fluids Y1 - 2016 A1 - Francesco Ballarin A1 - Gianluigi Rozza AB - 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 PB - Wiley U1 - 35465 U2 - Mathematics U4 - 1 U5 - MAT/08 ER - TY - THES T1 - Qualitative properties and construction of solutions to some semilinear elliptic PDEs Y1 - 2016 A1 - Matteo Rizzi KW - moving planes method, maximum principle, Lyapunov-Schmidt reduction, Willmore surfaces, Otha-Kawasaki functional AB - This thesis is devoted to the study of elliptic equations. On the one hand, we study some qualitative properties, such as symmetry of solutions, on the other hand we explicitly construct some solutions vanishing near some fixed manifold. The main techniques are the moving planes method, in order to investigate the qualitative properties and the Lyapunov-Schmidt reduction. PB - SISSA U1 - 35500 U5 - MAT/05 ER - TY - JOUR T1 - A Reduced Basis Approach for Modeling the Movement of Nuclear Reactor Control Rods JF - NERS-14-1062; ASME J of Nuclear Rad Sci, 2, 2 (2016) 021019 Y1 - 2016 A1 - Alberto Sartori A1 - Antonio Cammi A1 - Lelio Luzzi A1 - Gianluigi Rozza AB - This work presents a reduced order model (ROM) aimed at simulating nuclear reactor control rods movement and featuring fast-running prediction of reactivity and neutron flux distribution as well. In particular, the reduced basis (RB) method (built upon a high-fidelity finite element (FE) approximation) has been employed. The neutronics has been modeled according to a parametrized stationary version of the multigroup neutron diffusion equation, which can be formulated as a generalized eigenvalue problem. Within the RB framework, the centroidal Voronoi tessellation is employed as a sampling technique due to the possibility of a hierarchical parameter space exploration, without relying on a “classical” a posteriori error estimation, and saving an important amount of computational time in the offline phase. Here, the proposed ROM is capable of correctly predicting, with respect to the high-fidelity FE approximation, both the reactivity and neutron flux shape. In this way, a computational speedup of at least three orders of magnitude is achieved. If a higher precision is required, the number of employed basis functions (BFs) must be increased. PB - ASME VL - 2 UR - http://urania.sissa.it/xmlui/handle/1963/35192 IS - 2 N1 - 8 pages U1 - 35473 U2 - Mathematics U4 - 1 ER - TY - JOUR T1 - Reduced basis approaches in time-dependent noncoercive settings for modelling the movement of nuclear reactor control rods JF - Communications in Computational Physics Y1 - 2016 A1 - Alberto Sartori A1 - Antonio Cammi A1 - Lelio Luzzi A1 - Gianluigi Rozza AB -In this work, two approaches, based on the certified Reduced Basis method, have been developed for simulating the movement of nuclear reactor control rods, in time-dependent non-coercive settings featuring a 3D geometrical framework. In particular, in a first approach, a piece-wise affine transformation based on subdomains division has been implemented for modelling the movement of one control rod. In the second approach, a “staircase” strategy has been adopted for simulating the movement of all the three rods featured by the nuclear reactor chosen as case study. The neutron kinetics has been modelled according to the so-called multi-group neutron diffusion, which, in the present case, is a set of ten coupled parametrized parabolic equations (two energy groups for the neutron flux, and eight for the precursors). Both the reduced order models, developed according to the two approaches, provided a very good accuracy compared with high-fidelity results, assumed as “truth” solutions. At the same time, the computational speed-up in the Online phase, with respect to the fine “truth” finite element discretization, achievable by both the proposed approaches is at least of three orders of magnitude, allowing a real-time simulation of the rod movement and control.

PB - SISSA UR - http://urania.sissa.it/xmlui/handle/1963/34963 IS - in press U1 - 35188 U2 - Mathematics ER - TY - JOUR T1 - Reduced basis method and domain decomposition for elliptic problems in networks and complex parametrized geometries JF - Computers and Mathematics with Applications Y1 - 2016 A1 - Laura Iapichino A1 - Alfio Quarteroni A1 - Gianluigi Rozza AB - The aim of this work is to solve parametrized partial differential equations in computational domains represented by networks of repetitive geometries by combining reduced basis and domain decomposition techniques. The main idea behind this approach is to compute once, locally and for few reference shapes, some representative finite element solutions for different values of the parameters and with a set of different suitable boundary conditions on the boundaries: these functions will represent the basis of a reduced space where the global solution is sought for. The continuity of the latter is assured by a classical domain decomposition approach. Test results on Poisson problem show the flexibility of the proposed method in which accuracy and computational time may be tuned by varying the number of reduced basis functions employed, or the set of boundary conditions used for defining locally the basis functions. The proposed approach simplifies the pre-computation of the reduced basis space by splitting the global problem into smaller local subproblems. Thanks to this feature, it allows dealing with arbitrarily complex network and features more flexibility than a classical global reduced basis approximation where the topology of the geometry is fixed. PB - Elsevier VL - 71 IS - 1 U1 - 35187 U2 - Mathematics U4 - 1 U5 - MAT/08 ER - TY - JOUR T1 - Symmetry enhancements via 5d instantons, qW-algebrae and (1,0) superconformal index JF - Journal of High Energy Physics Y1 - 2016 A1 - Benvenuti, Sergio A1 - Giulio Bonelli A1 - Massimiliano Ronzani A1 - Alessandro Tanzini AB -We explore $\mathcal{N}=(1,0)$ superconformal six-dimensional theories arising from M5 branes probing a transverse $A_k$ singularity. Upon circle compactification to 5 dimensions, we describe this system with a dual pq-web of five-branes and propose the spectrum of basic five-dimensional instanton operators driving global symmetry enhancement. For a single M5 brane, we find that the exact partition function of the 5d quiver gauge theory matches the 6d (1, 0) index, which we compute by letter counting. We finally show that S-duality of the pq-web implies new relations among vertex correlators of $q\mathcal{W}$ algebrae.

VL - 2016 UR - https://doi.org/10.1007/JHEP09(2016)053 ER - TY - JOUR T1 - Symmetry properties of some solutions to some semilinear elliptic equations JF - Annali della Scuola Normale Superiore di Pisa. Classe di scienze Y1 - 2016 A1 - Farina, Alberto A1 - Andrea Malchiodi A1 - Matteo Rizzi PB - Classe di Scienze VL - 16 ER - TY - JOUR T1 - Young towers for product systems JF - Discrete & Continuous Dynamical Systems - A Y1 - 2016 A1 - Stefano Luzzatto A1 - Marks Ruziboev AB -We show that the direct product of maps with Young towers admits a Young tower whose return times decay at a rate which is bounded above by the slowest of the rates of decay of the return times of the component maps. An application of this result, together with other results in the literature, yields various statistical properties for the direct product of various classes of systems, including Lorenz-like maps, multimodal maps, piecewise $C^2$ interval maps with critical points and singularities, Hénon maps and partially hyperbolic systems.

VL - 36 UR - http://aimsciences.org//article/id/18d4526e-470d-467e-967a-a0345ad4c642 ER - TY - BOOK T1 - Certified Reduced Basis Methods for Parametrized Partial Differential Equations T2 - Springer Briefs in Mathematics Y1 - 2015 A1 - Jan S Hesthaven A1 - Gianluigi Rozza A1 - Benjamin Stamm KW - a posteriori error bounds KW - empirical interpolation KW - parametrized partial differential equations KW - reduced basis methods, greedy algorithms AB -This book provides a thorough introduction to the mathematical and algorithmic aspects of certified reduced basis methods for parametrized partial differential equations. Central aspects ranging from model construction, error estimation and computational efficiency to empirical interpolation methods are discussed in detail for coercive problems. More advanced aspects associated with time-dependent problems, non-compliant and non-coercive problems and applications with geometric variation are also discussed as examples.

JF - Springer Briefs in Mathematics PB - Springer CY - Switzerland SN - 978-3-319-22469-5 ER - TY - JOUR T1 - Decay of correlations for invertible maps with non-Hölder observables JF - Dynamical Systems Y1 - 2015 A1 - Marks Ruziboev AB -An invertible dynamical system with some hyperbolic structure is considered. Upper estimates for the correlations of continuous observables are given in terms of modulus of continuity. The result is applied to certain Hénon maps and Solenoid maps with intermittency.

PB - Taylor & Francis VL - 30 UR - https://doi.org/10.1080/14689367.2015.1046816 ER - TY - RPRT T1 - Fast simulations of patient-specific haemodynamics of coronary artery bypass grafts based on a POD-Galerkin method and a vascular shape parametrization Y1 - 2015 A1 - Francesco Ballarin A1 - Elena Faggiano A1 - Sonia Ippolito A1 - Andrea Manzoni A1 - Alfio Quarteroni A1 - Gianluigi Rozza A1 - Roberto Scrofani AB - In this work a reduced-order computational framework for the study of haemodynamics in three-dimensional patient-specific configurations of coronary artery bypass grafts dealing with a wide range of scenarios is proposed. We combine several efficient algorithms to face at the same time both the geometrical complexity involved in the description of the vascular network and the huge computational cost entailed by time dependent patient-specific flow simulations. Medical imaging procedures allow to reconstruct patient-specific configurations from clinical data. A centerlines-based parametrization is proposed to efficiently handle geometrical variations. POD–Galerkin reduced-order models are employed to cut down large computational costs. This computational framework allows to characterize blood flows for different physical and geometrical variations relevant in the clinical practice, such as stenosis factors and anastomosis variations, in a rapid and reliable way. Several numerical results are discussed, highlighting the computational performance of the proposed framework, as well as its capability to perform sensitivity analysis studies, so far out of reach. UR - http://urania.sissa.it/xmlui/handle/1963/34623 U1 - 34824 U2 - Mathematics U4 - 1 U5 - MAT/08 ER - TY - JOUR T1 - A general existence result for the Toda system on compact surfaces JF - Advances in Mathematics Y1 - 2015 A1 - Luca Battaglia A1 - Aleks Jevnikar A1 - Andrea Malchiodi A1 - David Ruiz KW - Geometric PDEs KW - Min–max schemes KW - Variational methods AB -In this paper we consider the following Toda system of equations on a compact surface:−Δu1=2ρ1(h1eu1∫Σh1eu1dVg−1)−ρ2(h2eu2∫Σh2eu2dVg−1)−Δu1=−4π∑j=1mα1,j(δpj−1),−Δu2=2ρ2(h2eu2∫Σh2eu2dVg−1)−ρ1(h1eu1∫Σh1eu1dVg−1)−Δu2=−4π∑j=1mα2,j(δpj−1), which is motivated by the study of models in non-abelian Chern–Simons theory. Here h1,h2 are smooth positive functions, ρ1,ρ2 two positive parameters, pi points of the surface and α1,i,α2,j non-negative numbers. We prove a general existence result using variational methods. The same analysis applies to the following mean field equation−Δu=ρ1(heu∫ΣheudVg−1)−ρ2(he−u∫Σhe−udVg−1), which arises in fluid dynamics."

VL - 285 UR - http://www.sciencedirect.com/science/article/pii/S0001870815003072 ER - TY - THES T1 - Gibbs-Markov-Young Structures and Decay of Correlations Y1 - 2015 A1 - Marks Ruziboev KW - Decay of Correlations, GMY-towers AB - In this work we study mixing properties of discrete dynamical systems and related to them geometric structure. In the first chapter we show that the direct product of maps with Young towers admits a Young tower whose return times decay at a rate which is bounded above by the slowest of the rates of decay of the return times of the component maps. An application of this result, together with other results in the literature, yields various statistical properties for the direct product of various classes of systems, including Lorenz-like maps, multimodal maps, piecewise $C^2$ interval maps with critical points and singularities, H\'enon maps and partially hyperbolic systems. The second chapter is dedicated to the problem of decay of correlations for continuous observables. First we show that if the underlying system admits Young tower then the rate of decay of correlations for continuous observables can be estimated in terms of modulus of continuity and the decay rate of tail of Young tower. In the rest of the second chapter we study the relations between the rates of decay of correlations for smooth observables and continuous observables. We show that if the rates of decay of correlations is known for $C^r,$ observables ($r\ge 1$) then it is possible to obtain decay of correlations for continuous observables in terms of modulus of continuity. PB - SISSA U1 - 34677 U2 - Mathematics U4 - 1 U5 - MAT/05 ER - TY - JOUR T1 - Model order reduction of parameterized systems (MoRePaS): Preface to the special issue of advances in computational mathematics JF - Advances in Computational Mathematics Y1 - 2015 A1 - Peter Benner A1 - Mario Ohlberger A1 - Anthony Patera A1 - Gianluigi Rozza A1 - Sorensen, D.C. A1 - Karsten Urban VL - 41 ER - TY - JOUR T1 - Multilevel and weighted reduced basis method for stochastic optimal control problems constrained by Stokes equations JF - Numerische Mathematik, (2015), 36 p. Article in Press Y1 - 2015 A1 - Gianluigi Rozza A1 - Peng Chen A1 - Alfio Quarteroni AB - In this paper we develop and analyze a multilevel weighted reduced basis method for solving stochastic optimal control problems constrained by Stokes equations. We prove the analytic regularity of the optimal solution in the probability space under certain assumptions on the random input data. The finite element method and the stochastic collocation method are employed for the numerical approximation of the problem in the deterministic space and the probability space, respectively, resulting in many large-scale optimality systems to solve. In order to reduce the unaffordable computational effort, we propose a reduced basis method using a multilevel greedy algorithm in combination with isotropic and anisotropic sparse-grid techniques. A weighted a posteriori error bound highlights the contribution stemming from each method. Numerical tests on stochastic dimensions ranging from 10 to 100 demonstrate that our method is very efficient, especially for solving high-dimensional and large-scale optimization problems. PB - Springer UR - http://urania.sissa.it/xmlui/handle/1963/34491 U1 - 34680 U2 - Mathematics U4 - 1 U5 - MAT/08 ER - TY - JOUR T1 - N=2 supersymmetric gauge theories on S^2xS^2 and Liouville Gravity JF - Journal of High Energy Physics Y1 - 2015 A1 - Aditya Bawane A1 - Giulio Bonelli A1 - Massimiliano Ronzani A1 - Alessandro Tanzini AB -We consider $\mathcal{N}=2$ supersymmetric gauge theories on four manifolds admitting an isometry. Generalized Killing spinor equations are derived from the consistency of supersymmetry algebrae and solved in the case of four manifolds admitting a $U(1)$ isometry. This is used to explicitly compute the supersymmetric path integral on $S^2 \times S^2$ via equivariant localization. The building blocks of the resulting partition function are shown to contain the three point functions and the conformal blocks of Liouville Gravity.

VL - 2015 UR - https://doi.org/10.1007/JHEP07(2015)054 ER - TY - JOUR T1 - The phototransduction machinery in the rod outer segment has a strong efficacy gradient Y1 - 2015 A1 - Monica Mazzolini A1 - Giuseppe Facchetti A1 - L. Andolfi A1 - R. Proietti Zaccaria A1 - S. Tuccio A1 - J. Treud A1 - Claudio Altafini A1 - Enzo M. Di Fabrizio A1 - Marco Lazzarino A1 - G. Rapp A1 - Vincent Torre PB - National Academy of Sciences UR - http://urania.sissa.it/xmlui/handle/1963/35157 N1 - Open Access article U1 - 35382 U2 - Neuroscience ER - TY - JOUR T1 - Reduced basis approximation and a-posteriori error estimation for the coupled Stokes-Darcy system JF - Advances in Computational Mathematics Y1 - 2015 A1 - Immanuel Martini A1 - Gianluigi Rozza A1 - Bernard Haasdonk KW - Domain decomposition KW - Error estimation KW - Non-coercive problem KW - Porous medium equation KW - Reduced basis method KW - Stokes flow AB -The coupling of a free flow with a flow through porous media has many potential applications in several fields related with computational science and engineering, such as blood flows, environmental problems or food technologies. We present a reduced basis method for such coupled problems. The reduced basis method is a model order reduction method applied in the context of parametrized systems. Our approach is based on a heterogeneous domain decomposition formulation, namely the Stokes-Darcy problem. Thanks to an offline/online-decomposition, computational times can be drastically reduced. At the same time the induced error can be bounded by fast evaluable a-posteriori error bounds. In the offline-phase the proposed algorithms make use of the decomposed problem structure. Rigorous a-posteriori error bounds are developed, indicating the accuracy of certain lifting operators used in the offline-phase as well as the accuracy of the reduced coupled system. Also, a strategy separately bounding pressure and velocity errors is extended. Numerical experiments dealing with groundwater flow scenarios demonstrate the efficiency of the approach as well as the limitations regarding a-posteriori error estimation.

VL - special issue for MoRePaS 2012 IS - in press ER - TY - JOUR T1 - Reduced basis approximation of parametrized advection-diffusion PDEs with high Péclet number JF - Lecture Notes in Computational Science and Engineering Y1 - 2015 A1 - Pacciarini, P. A1 - Gianluigi Rozza AB -In this work we show some results about the reduced basis approximation of advection dominated parametrized problems, i.e. advection-diffusion problems with high Péclet number. These problems are of great importance in several engineering applications and it is well known that their numerical approximation can be affected by instability phenomena. In this work we compare two possible stabilization strategies in the framework of the reduced basis method, by showing numerical results obtained for a steady advection-diffusion problem.

VL - 103 ER - TY - JOUR T1 - Reduced basis approximation of parametrized optimal flow control problems for the Stokes equations JF - Computers and Mathematics with Applications Y1 - 2015 A1 - Federico Negri A1 - Andrea Manzoni A1 - Gianluigi Rozza AB -This paper extends the reduced basis method for the solution of parametrized optimal control problems presented in Negri et al. (2013) to the case of noncoercive (elliptic) equations, such as the Stokes equations. We discuss both the theoretical properties-with particular emphasis on the stability of the resulting double nested saddle-point problems and on aggregated error estimates-and the computational aspects of the method. Then, we apply it to solve a benchmark vorticity minimization problem for a parametrized bluff body immersed in a two or a three-dimensional flow through boundary control, demonstrating the effectivity of the methodology.

VL - 69 ER - TY - JOUR T1 - Supremizer stabilization of POD-Galerkin approximation of parametrized Navier-Stokes equations Y1 - 2015 A1 - Francesco Ballarin A1 - Andrea Manzoni A1 - Alfio Quarteroni A1 - Gianluigi Rozza AB - In this work, we present a stable proper orthogonal decomposition–Galerkin approximation for parametrized steady incompressible Navier–Stokes equations with low Reynolds number. PB - Wiley UR - http://urania.sissa.it/xmlui/handle/1963/34701 U1 - 34915 U2 - Mathematics U4 - 1 U5 - MAT/08 ER - TY - JOUR T1 - Buckling dynamics of a solvent-stimulated stretched elastomeric sheet Y1 - 2014 A1 - Alessandro Lucantonio A1 - Matthieu Roché A1 - Paola Nardinocchi A1 - Howard A. Stone AB - When stretched uniaxially, a thin elastic sheet may exhibit buckling. The occurrence of buckling depends on the geometrical properties of the sheet and the magnitude of the applied strain. Here we show that an elastomeric sheet initially stable under uniaxial stretching can destabilize when exposed to a solvent that swells the elastomer. We demonstrate experimentally and computationally that the features of the buckling pattern depend on the magnitude of stretching, and this observation offers a new way for controlling the shape of a swollen homogeneous thin sheet. PB - Royal Society of Chemistry UR - http://urania.sissa.it/xmlui/handle/1963/34967 U1 - 35197 U2 - Physics U4 - 1 ER - TY - JOUR T1 - Comparison between reduced basis and stochastic collocation methods for elliptic problems Y1 - 2014 A1 - Peng Chen A1 - Alfio Quarteroni A1 - Gianluigi Rozza AB - The stochastic collocation method (Babuška et al. in SIAM J Numer Anal 45(3):1005-1034, 2007; Nobile et al. in SIAM J Numer Anal 46(5):2411-2442, 2008a; SIAM J Numer Anal 46(5):2309-2345, 2008b; Xiu and Hesthaven in SIAM J Sci Comput 27(3):1118-1139, 2005) has recently been applied to stochastic problems that can be transformed into parametric systems. Meanwhile, the reduced basis method (Maday et al. in Comptes Rendus Mathematique 335(3):289-294, 2002; Patera and Rozza in Reduced basis approximation and a posteriori error estimation for parametrized partial differential equations Version 1.0. Copyright MIT, http://augustine.mit.edu, 2007; Rozza et al. in Arch Comput Methods Eng 15(3):229-275, 2008), primarily developed for solving parametric systems, has been recently used to deal with stochastic problems (Boyaval et al. in Comput Methods Appl Mech Eng 198(41-44):3187-3206, 2009; Arch Comput Methods Eng 17:435-454, 2010). In this work, we aim at comparing the performance of the two methods when applied to the solution of linear stochastic elliptic problems. Two important comparison criteria are considered: (1), convergence results of the approximation error; (2), computational costs for both offline construction and online evaluation. Numerical experiments are performed for problems from low dimensions O (1) to moderate dimensions O (10) and to high dimensions O (100). The main result stemming from our comparison is that the reduced basis method converges better in theory and faster in practice than the stochastic collocation method for smooth problems, and is more suitable for large scale and high dimensional stochastic problems when considering computational costs. PB - Springer UR - http://urania.sissa.it/xmlui/handle/1963/34727 U1 - 34916 U2 - Mathematics U4 - 1 ER - TY - JOUR T1 - Comparison of a Modal Method and a Proper Orthogonal Decomposition approach for multi-group time-dependent reactor spatial kinetics JF - Annals of Nuclear Energy Y1 - 2014 A1 - Alberto Sartori A1 - Davide Baroli A1 - Antonio Cammi A1 - Davide Chiesa A1 - Lelio Luzzi A1 - Roberto R. Ponciroli A1 - Ezio Previtali A1 - Marco E. Ricotti A1 - Gianluigi Rozza A1 - Monica Sisti AB -In this paper, two modelling approaches based on a Modal Method (MM) and on the Proper Orthogonal Decomposition (POD) technique, for developing a control-oriented model of nuclear reactor spatial kinetics, are presented and compared. Both these methods allow developing neutronics description by means of a set of ordinary differential equations. The comparison of the outcomes provided by the two approaches focuses on the capability of evaluating the reactivity and the neutron flux shape in different reactor configurations, with reference to a TRIGA Mark II reactor. The results given by the POD-based approach are higher-fidelity with respect to the reference solution than those computed according to the MM-based approach, in particular when the perturbation concerns a reduced region of the core. If the perturbation is homogeneous throughout the core, the two approaches allow obtaining comparable accuracy results on the quantities of interest. As far as the computational burden is concerned, the POD approach ensures a better efficiency rather than direct Modal Method, thanks to the ability of performing a longer computation in the preprocessing that leads to a faster evaluation during the on-line phase.

PB - Elsevier VL - 71 UR - http://urania.sissa.it/xmlui/handle/1963/35039 U1 - 35270 U2 - Physics U4 - 1 ER - TY - JOUR T1 - On conjugate times of LQ optimal control problems Y1 - 2014 A1 - Andrei A. Agrachev A1 - Luca Rizzi A1 - Pavel Silveira KW - Optimal control, Lagrange Grassmannian, Conjugate point AB - Motivated by the study of linear quadratic optimal control problems, we consider a dynamical system with a constant, quadratic Hamiltonian, and we characterize the number of conjugate times in terms of the spectrum of the Hamiltonian vector field $\vec{H}$. We prove the following dichotomy: the number of conjugate times is identically zero or grows to infinity. The latter case occurs if and only if $\vec{H}$ has at least one Jordan block of odd dimension corresponding to a purely imaginary eigenvalue. As a byproduct, we obtain bounds from below on the number of conjugate times contained in an interval in terms of the spectrum of $\vec{H}$. PB - Springer UR - http://hdl.handle.net/1963/7227 N1 - 14 pages, 1 figure U1 - 7261 U2 - Mathematics U4 - 1 U5 - MAT/05 ANALISI MATEMATICA ER - TY - THES T1 - The curvature of optimal control problems with applications to sub-Riemannian geometry Y1 - 2014 A1 - Luca Rizzi KW - Sub-Riemannian geometry AB - Optimal control theory is an extension of the calculus of variations, and deals with the optimal behaviour of a system under a very general class of constraints. This field has been pioneered by the group of mathematicians led by Lev Pontryagin in the second half of the 50s and nowadays has countless applications to the real worlds (robotics, trains, aerospace, models for human behaviour, human vision, image reconstruction, quantum control, motion of self-propulsed micro-organism). In this thesis we introduce a novel definition of curvature for an optimal control problem. In particular it works for any sub-Riemannian and sub-Finsler structure. Related problems, such as comparison theorems for sub-Riemannian manifolds, LQ optimal control problem and Popp's volume and are also investigated. PB - SISSA UR - http://hdl.handle.net/1963/7321 N1 - The PhD thesis is composed of 211 pages and is recorded in PDF format U1 - 7367 U2 - Mathematics U4 - 1 U5 - MAT/03 GEOMETRIA ER - TY - JOUR T1 - Efficient geometrical parametrisation techniques of interfaces for reduced-order modelling: application to fluid–structure interaction coupling problems JF - International Journal of Computational Fluid Dynamics Y1 - 2014 A1 - Forti, D. A1 - Gianluigi Rozza AB - We present some recent advances and improvements in shape parametrisation techniques of interfaces for reduced-order modelling with special attention to fluid–structure interaction problems and the management of structural deformations, namely, to represent them into a low-dimensional space (by control points). This allows to reduce the computational effort, and to significantly simplify the (geometrical) deformation procedure, leading to more efficient and fast reduced-order modelling applications in this kind of problems. We propose an efficient methodology to select the geometrical control points for the radial basis functions based on a modal greedy algorithm to improve the computational efficiency in view of more complex fluid–structure applications in several fields. The examples provided deal with aeronautics and wind engineering. VL - 28 ER - TY - CHAP T1 - Fundamentals of Reduced Basis Method for problems governed by parametrized PDEs and applications T2 - Separated representations and PGD-based model reduction : fundamentals and applications Y1 - 2014 A1 - Gianluigi Rozza KW - reduced basis method, linear elasticity, heat transfer, error bounds, parametrized PDEs AB -In this chapter we consider Reduced Basis (RB) approximations of parametrized Partial Differential Equations (PDEs). The the idea behind RB is to decouple the generation and projection stages (Offline/Online computational procedures) of the approximation process in order to solve parametrized PDEs in a fast, inexpensive and reliable way. The RB method, especially applied to 3D problems, allows great computational savings with respect to the classical Galerkin Finite Element (FE) Method. The standard FE method is typically ill suited to (i) iterative contexts like optimization, sensitivity analysis and many-queries in general, and (ii) real time evaluation. We consider for simplicity coercive PDEs. We discuss all the steps to set up a RB approximation, either from an analytical and a numerical point of view. Then we present an application of the RB method to a steady thermal conductivity problem in heat transfer with emphasis on geometrical and physical parameters.

JF - Separated representations and PGD-based model reduction : fundamentals and applications T3 - CISM International Centre for Mechanical Sciences PB - Springer CY - Wien VL - 554 ER - TY - JOUR T1 - An improvement on geometrical parameterizations by transfinite maps JF - Comptes Rendus Mathematique Y1 - 2014 A1 - Jäggli, C. A1 - Laura Iapichino A1 - Gianluigi Rozza AB - We present a method to generate a non-affine transfinite map from a given reference domain to a family of deformed domains. The map is a generalization of the Gordon-Hall transfinite interpolation approach. It is defined globally over the reference domain. Once we have computed some functions over the reference domain, the map can be generated by knowing the parametric expressions of the boundaries of the deformed domain. Being able to define a suitable map from a reference domain to a desired deformation is useful for the management of parameterized geometries. VL - 352 ER - TY - JOUR T1 - A model for crack growth with branching and kinking JF - Asymptotic Analysis Y1 - 2014 A1 - Simone Racca KW - quasistatic crack evolution, branching, kinking, Griffith\\\'s criterion AB -We study an evolution model for fractured elastic materials in the 2-dimensional case, for which the crack path is not assumed to be known a priori. We introduce some general assumptions on the structure of the fracture sets suitable to remove the restrictions on the regularity of the crack sets and to allow for kinking and branching to develop. In addition we define the front of the fracture and its velocity. By means of a time-discretization approach, we prove the existence of a continuous-time evolution that satisfies an energy inequality and a stability criterion. The energy balance also takes into account the energy dissipated at the front of the fracture. The stability criterion is stated in the framework of Griffith's theory, in terms of the energy release rate, when the crack grows at least at one point of its front.

PB - SISSA VL - 89 UR - https://content.iospress.com/articles/asymptotic-analysis/asy1233 IS - 1-2 U1 - 6293 U2 - Mathematics U4 - 1 U5 - MAT/05 ANALISI MATEMATICA ER - TY - JOUR T1 - Model Order Reduction in Fluid Dynamics: Challenges and Perspectives Y1 - 2014 A1 - Toni Lassila A1 - Andrea Manzoni A1 - Alfio Quarteroni A1 - Gianluigi Rozza AB - This chapter reviews techniques of model reduction of fluid dynamics systems. Fluid systems are known to be difficult to reduce efficiently due to several reasons. First of all, they exhibit strong nonlinearities - which are mainly related either to nonlinear convection terms and/or some geometric variability - that often cannot be treated by simple linearization. Additional difficulties arise when attempting model reduction of unsteady flows, especially when long-term transient behavior needs to be accurately predicted using reduced order models and more complex features, such as turbulence or multiphysics phenomena, have to be taken into consideration. We first discuss some general principles that apply to many parametric model order reduction problems, then we apply them on steady and unsteady viscous flows modelled by the incompressible Navier-Stokes equations. We address questions of inf-sup stability, certification through error estimation, computational issues and-in the unsteady case - long-time stability of the reduced model. Moreover, we provide an extensive list of literature references. PB - Springer U1 - 34923 U2 - Mathematics U4 - 1 ER - TY - RPRT T1 - Rate-independent damage in thermo-viscoelastic materials with inertia Y1 - 2014 A1 - Giuliano Lazzaroni A1 - Riccarda Rossi A1 - Marita Thomas A1 - Rodica Toader AB - We present a model for rate-independent, unidirectional, partial damage in visco-elastic materials with inertia and thermal effects. The damage process is modeled by means of an internal variable, governed by a rate-independent flow rule. The heat equation and the momentum balance for the displacements are coupled in a highly nonlinear way. Our assumptions on the corresponding energy functional also comprise the case of the Ambrosio-Tortorelli phase-field model (without passage to the brittle limit). We discuss a suitable weak formulation and prove an existence theorem obtained with the aid of a (partially) decoupled time-discrete scheme and variational convergence methods. We also carry out the asymptotic analysis for vanishing viscosity and inertia and obtain a fully rate-independent limit model for displacements and damage, which is Independent of temperature. PB - SISSA UR - http://urania.sissa.it/xmlui/handle/1963/7444 U1 - 7542 ER - TY - CONF T1 - Reduced basis method for the Stokes equations in decomposable domains using greedy optimization T2 - ECMI 2014 proceedings Y1 - 2014 A1 - Laura Iapichino A1 - Alfio Quarteroni A1 - Gianluigi Rozza A1 - Volkwein, Stefan JF - ECMI 2014 proceedings ER - TY - BOOK T1 - Reduced Order Methods for Modeling and Computational Reduction T2 - MS&A Y1 - 2014 A1 - Alfio Quarteroni A1 - Gianluigi Rozza KW - reduced order methods, MOR, ROM, POD, RB, greedy, CFD, Numerical Analysis AB -This monograph addresses the state of the art of reduced order methods for modeling and computational reduction of complex parametrized systems, governed by ordinary and/or partial differential equations, with a special emphasis on real time computing techniques and applications in computational mechanics, bioengineering and computer graphics.

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

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

JF - MS&A PB - Springer CY - Milano VL - 9 ER - TY - Generic T1 - A reduced order model for multi-group time-dependent parametrized reactor spatial kinetics T2 - 22nd International Conference on Nuclear Engineering ICONE22 Y1 - 2014 A1 - Alberto Sartori A1 - Davide Baroli A1 - Antonio Cammi A1 - Lelio Luzzi A1 - Gianluigi Rozza AB -

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

JF - 22nd International Conference on Nuclear Engineering ICONE22 PB - American Society of Mechanical Engineers (ASME) CY - Prague, Czech Republic SN - 978-079184595-0 UR - http://urania.sissa.it/xmlui/handle/1963/35123 N1 - 2014 22nd International Conference on Nuclear Engineering, ICONE 2014; Prague; Czech Republic; 7 July 2014 through 11 July 2014; Code 109131; U1 - 35360 U2 - Mathematics U4 - 1 ER - TY - JOUR T1 - Shape Optimization by Free-Form Deformation: Existence Results and Numerical Solution for Stokes Flows Y1 - 2014 A1 - Francesco Ballarin A1 - Andrea Manzoni A1 - Gianluigi Rozza A1 - Sandro Salsa AB - 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. PB - Springer UR - http://urania.sissa.it/xmlui/handle/1963/34698 U1 - 34914 U2 - Mathematics U4 - 1 ER - TY - RPRT T1 - Some remarks on a model for rate-independent damage in thermo-visco-elastodynamics Y1 - 2014 A1 - Giuliano Lazzaroni A1 - Riccarda Rossi A1 - Marita Thomas A1 - Rodica Toader AB - This note deals with the analysis of a model for partial damage, where the rateindependent, unidirectional flow rule for the damage variable is coupled with the rate-dependent heat equation, and with the momentum balance featuring inertia and viscosity according to Kelvin-Voigt rheology. The results presented here combine the approach from Roubicek [1] with the methods from Lazzaroni/Rossi/Thomas/Toader [2] and extend the analysis to the setting of inhomogeneous time-dependent Dirichlet data. PB - SISSA UR - http://urania.sissa.it/xmlui/handle/1963/7463 U1 - 7566 ER - TY - JOUR T1 - Stabilized reduced basis method for parametrized advection-diffusion PDEs JF - Computer Methods in Applied Mechanics and Engineering Y1 - 2014 A1 - Pacciarini, P. A1 - Gianluigi Rozza AB -In this work, we propose viable and efficient strategies for the stabilization of the reduced basis approximation of an advection dominated problem. In particular, we investigate the combination of a classic stabilization method (SUPG) with the Offline-Online structure of the RB method. We explain why the stabilization is needed in both stages and we identify, analytically and numerically, which are the drawbacks of a stabilization performed only during the construction of the reduced basis (i.e. only in the Offline stage). We carry out numerical tests to assess the performances of the ``double'' stabilization both in steady and unsteady problems, also related to heat transfer phenomena.

VL - 274 ER - TY - CONF T1 - Stabilized reduced basis method for parametrized scalar advection-diffusion problems at higher Péclet number: Roles of the boundary layers and inner fronts T2 - 11th World Congress on Computational Mechanics, WCCM 2014, 5th European Conference on Computational Mechanics, ECCM 2014 and 6th European Conference on Computational Fluid Dynamics, ECFD 2014 Y1 - 2014 A1 - Pacciarini, P. A1 - Gianluigi Rozza AB -Advection-dominated problems, which arise in many engineering situations, often require a fast and reliable approximation of the solution given some parameters as inputs. In this work we want to investigate the coupling of the reduced basis method - which guarantees rapidity and reliability - with some classical stabilization techiques to deal with the advection-dominated condition. We provide a numerical extension of the results presented in [1], focusing in particular on problems with curved boundary layers and inner fronts whose direction depends on the parameter.

JF - 11th World Congress on Computational Mechanics, WCCM 2014, 5th European Conference on Computational Mechanics, ECCM 2014 and 6th European Conference on Computational Fluid Dynamics, ECFD 2014 UR - https://infoscience.epfl.ch/record/203327/files/ECCOMAS_PP_GR.pdf ER - TY - JOUR T1 - A variational model for the quasi-static growth of fractional dimensional brittle fractures Y1 - 2014 A1 - Simone Racca A1 - Rodica Toader KW - Variational models AB -We propose a variational model for the irreversible quasi-static evolution of brittle fractures having fractional Hausdorff dimension in the setting of two-dimensional antiplane and plane elasticity. The evolution along such irregular crack paths can be obtained as $\Gamma$-limit of evolutions along one-dimensional cracks when the fracture toughness tends to zero.

PB - European Mathematical Society UR - http://hdl.handle.net/1963/6983 U1 - 6973 U2 - Mathematics U4 - -1 ER - TY - JOUR T1 - A weighted empirical interpolation method: A priori convergence analysis and applications Y1 - 2014 A1 - Peng Chen A1 - Alfio Quarteroni A1 - Gianluigi Rozza AB - We extend the classical empirical interpolation method [M. Barrault, Y. Maday, N.C. Nguyen and A.T. Patera, An empirical interpolation method: application to efficient reduced-basis discretization of partial differential equations. Compt. Rend. Math. Anal. Num. 339 (2004) 667-672] to a weighted empirical interpolation method in order to approximate nonlinear parametric functions with weighted parameters, e.g. random variables obeying various probability distributions. A priori convergence analysis is provided for the proposed method and the error bound by Kolmogorov N-width is improved from the recent work [Y. Maday, N.C. Nguyen, A.T. Patera and G.S.H. Pau, A general, multipurpose interpolation procedure: the magic points. Commun. Pure Appl. Anal. 8 (2009) 383-404]. We apply our method to geometric Brownian motion, exponential Karhunen-Loève expansion and reduced basis approximation of non-affine stochastic elliptic equations. We demonstrate its improved accuracy and efficiency over the empirical interpolation method, as well as sparse grid stochastic collocation method. PB - EDP Sciences UR - http://urania.sissa.it/xmlui/handle/1963/35021 U1 - 35253 U2 - Mathematics U4 - 1 U5 - MAT/05 ER - TY - JOUR T1 - A combination between the reduced basis method and the ANOVA expansion: On the computation of sensitivity indices JF - Comptes Rendus Mathematique. Volume 351, Issue 15-16, August 2013, Pages 593-598 Y1 - 2013 A1 - Denis Devaud A1 - Andrea Manzoni A1 - Gianluigi Rozza KW - Partial differential equations AB -We consider a method to efficiently evaluate in a real-time context an output based on the numerical solution of a partial differential equation depending on a large number of parameters. We state a result allowing to improve the computational performance of a three-step RB-ANOVA-RB method. This is a combination of the reduced basis (RB) method and the analysis of variations (ANOVA) expansion, aiming at compressing the parameter space without affecting the accuracy of the output. The idea of this method is to compute a first (coarse) RB approximation of the output of interest involving all the parameter components, but with a large tolerance on the a posteriori error estimate; then, we evaluate the ANOVA expansion of the output and freeze the least important parameter components; finally, considering a restricted model involving just the retained parameter components, we compute a second (fine) RB approximation with a smaller tolerance on the a posteriori error estimate. The fine RB approximation entails lower computational costs than the coarse one, because of the reduction of parameter dimensionality. Our result provides a criterion to avoid the computation of those terms in the ANOVA expansion that are related to the interaction between parameters in the bilinear form, thus making the RB-ANOVA-RB procedure computationally more feasible.

PB - Elsevier UR - http://hdl.handle.net/1963/7389 U1 - 7434 U2 - Mathematics U4 - 1 U5 - MAT/05 ANALISI MATEMATICA ER - TY - RPRT T1 - The curvature: a variational approach Y1 - 2013 A1 - Andrei A. Agrachev A1 - Davide Barilari A1 - Luca Rizzi KW - Crurvature, subriemannian metric, optimal control problem AB - The curvature discussed in this paper is a rather far going generalization of the Riemannian sectional curvature. We define it for a wide class of optimal control problems: a unified framework including geometric structures such as Riemannian, sub-Riemannian, Finsler and sub-Finsler structures; a special attention is paid to the sub-Riemannian (or Carnot-Caratheodory) metric spaces. Our construction of the curvature is direct and naive, and it is similar to the original approach of Riemann. Surprisingly, it works in a very general setting and, in particular, for all sub-Riemannian spaces. PB - SISSA UR - http://hdl.handle.net/1963/7226 N1 - 88 pages, 10 figures, (v2) minor typos corrected, (v3) added sections on Finsler manifolds, slow growth distributions, Heisenberg group U1 - 7260 U2 - Mathematics U4 - 1 U5 - MAT/03 GEOMETRIA ER - TY - JOUR T1 - Free Form Deformation Techniques Applied to 3D Shape Optimization Problems JF - Communications in Applied and Industrial Mathematics Y1 - 2013 A1 - Anwar Koshakji A1 - Alfio Quarteroni A1 - Gianluigi Rozza AB - The purpose of this work is to analyse and study an efficient parametrization technique for a 3D shape optimization problem. After a brief review of the techniques and approaches already available in literature, we recall the Free Form Deformation parametrization, a technique which proved to be efficient and at the same time versatile, allowing to manage complex shapes even with few parameters. We tested and studied the FFD technique by establishing a path, from the geometry definition, to the method implementation, and finally to the simulation and to the optimization of the shape. In particular, we have studied a bulb and a rudder of a race sailing boat as model applications, where we have tested a complete procedure from Computer-Aided-Design to build the geometrical model to discretization and mesh generation. ER - TY - JOUR T1 - Monads for framed sheaves on Hirzebruch surfaces Y1 - 2013 A1 - Claudio Bartocci A1 - Ugo Bruzzo A1 - Claudio L.S. Rava KW - Monads, framed sheaves, Hirzebruch surfaces AB - We define monads for framed torsion-free sheaves on Hirzebruch surfaces and use them to construct moduli spaces for these objects. These moduli spaces are smooth algebraic varieties, and we show that they are fine by constructing a universal monad. U1 - 7292 U2 - Mathematics U4 - 1 U5 - MAT/03 GEOMETRIA ER - TY - JOUR T1 - Nonabelian Lie algebroid extensions Y1 - 2013 A1 - Ugo Bruzzo A1 - Igor Mencattini A1 - Pietro Tortella A1 - Vladimir Rubtsov KW - Lie algebroids, nonabelian extensions, spectral sequences AB -We classify nonabelian extensions of Lie algebroids in the holomorphic or algebraic category, and introduce and study a spectral sequence that one can attach to any such extension and generalizes the Hochschild-Serre spectral sequence associated to an ideal in a Lie algebra. We compute the differentials of the spectral sequence up to $d_2$

U1 - 7293 U2 - Mathematics U4 - 1 U5 - MAT/03 GEOMETRIA ER - TY - JOUR T1 - Reduced basis approximation and a posteriori error estimation for Stokes flows in parametrized geometries: roles of the inf-sup stability constants JF - Numerische Mathematik, 2013 Y1 - 2013 A1 - Gianluigi Rozza A1 - Phuong Huynh A1 - Andrea Manzoni KW - parametrized Stokes equations AB - In this paper we review and we extend the reduced basis approximation and a posteriori error estimation for steady Stokes flows in a ffinely parametrized geometries, focusing on the role played by the Brezzi\\\'s and Babu ska\\\'s stability constants. The crucial ingredients of the methodology are a Galerkin projection onto a low-dimensional space of basis functions properly selected, an a ne parametric dependence enabling to perform competitive Off ine-Online splitting in the computational\\r\\nprocedure and a rigorous a posteriori error estimation on eld variables.\\r\\nThe combination of these three factors yields substantial computational savings which are at the basis of an e fficient model order reduction, ideally suited for real-time simulation and many-query contexts (e.g. optimization, control or parameter identi cation). In particular, in this work we focus on i) the stability of the reduced basis approximation based on the Brezzi\\\'s saddle point theory and the introduction of a supremizer operator on the pressure terms, ii) a rigorous a posteriori error estimation procedure for velocity and pressure elds based on the Babu ska\\\'s inf-sup constant (including residuals calculations), iii) the computation of a lower bound of the stability constant, and iv) di erent options for the reduced basis spaces construction. We present some illustrative results for both\\r\\ninterior and external steady Stokes flows in parametrized geometries representing two parametrized classical Poiseuille and Couette \\r\\nflows, a channel contraction and a simple flow control problem around a curved obstacle. PB - Springer UR - http://hdl.handle.net/1963/6339 U1 - 6269 U2 - Mathematics U4 - 1 U5 - MAT/08 ANALISI NUMERICA ER - TY - CHAP T1 - Reduced Basis Approximation for the Structural-Acoustic Design based on Energy Finite Element Analysis (RB-EFEA) T2 - CEMRACS 2013 - Modelling and simulation of complex systems: stochastic and deterministic approaches Y1 - 2013 A1 - Denis Devaud A1 - Gianluigi Rozza JF - CEMRACS 2013 - Modelling and simulation of complex systems: stochastic and deterministic approaches VL - 48 ER - TY - JOUR T1 - Reduced basis method for parametrized elliptic optimal control problems JF - SIAM Journal on Scientific Computing Y1 - 2013 A1 - Federico Negri A1 - Gianluigi Rozza A1 - Andrea Manzoni A1 - Alfio Quarteroni AB - We propose a suitable model reduction paradigm-the certified reduced basis method (RB)-for the rapid and reliable solution of parametrized optimal control problems governed by partial differential equations. In particular, we develop the methodology for parametrized quadratic optimization problems with elliptic equations as a constraint and infinite-dimensional control variable. First, we recast the optimal control problem in the framework of saddle-point problems in order to take advantage of the already developed RB theory for Stokes-type problems. Then, the usual ingredients of the RB methodology are called into play: a Galerkin projection onto a low-dimensional space of basis functions properly selected by an adaptive procedure; an affine parametric dependence enabling one to perform competitive offline-online splitting in the computational procedure; and an efficient and rigorous a posteriori error estimate on the state, control, and adjoint variables as well as on the cost functional. Finally, we address some numerical tests that confirm our theoretical results and show the efficiency of the proposed technique. VL - 35 ER - TY - RPRT T1 - A Reduced Computational and Geometrical Framework for Inverse Problems in Haemodynamics Y1 - 2013 A1 - Toni Lassila A1 - Andrea Manzoni A1 - Alfio Quarteroni A1 - Gianluigi Rozza PB - SISSA U1 - 6571 U2 - Mathematics U4 - 1 U5 - MAT/08 ANALISI NUMERICA ER - TY - RPRT T1 - A reduced-order strategy for solving inverse Bayesian identification problems in physiological flows Y1 - 2013 A1 - Toni Lassila A1 - Andrea Manzoni A1 - Alfio Quarteroni A1 - Gianluigi Rozza PB - SISSA U1 - 6555 U2 - Mathematics U4 - 1 U5 - MAT/08 ANALISI NUMERICA ER - TY - RPRT T1 - Reduction Strategies for Shape Dependent Inverse Problems in Haemodynamics Y1 - 2013 A1 - Toni Lassila A1 - Andrea Manzoni A1 - Gianluigi Rozza PB - SISSA U1 - 6554 U2 - Mathematics U4 - 1 U5 - MAT/08 ANALISI NUMERICA ER - TY - THES T1 - Some models of crack growth in brittle materials Y1 - 2013 A1 - Simone Racca KW - Quasi-static crack evolution PB - SISSA U1 - 7205 U2 - Mathematics U4 - 1 U5 - MAT/05 ANALISI MATEMATICA ER - TY - JOUR T1 - Stochastic optimal robin boundary control problems of advection-dominated elliptic equations JF - SIAM Journal on Numerical Analysis Y1 - 2013 A1 - Peng Chen A1 - Alfio Quarteroni A1 - Gianluigi Rozza AB - In this work we deal with a stochastic optimal Robin boundary control problem constrained by an advection-diffusion-reaction elliptic equation with advection-dominated term. We assume that the uncertainty comes from the advection field and consider a stochastic Robin boundary condition as control function. A stochastic saddle point system is formulated and proved to be equivalent to the first order optimality system for the optimal control problem, based on which we provide the existence and uniqueness of the optimal solution as well as some results on stochastic regularity with respect to the random variables. Stabilized finite element approximations in physical space and collocation approximations in stochastic space are applied to discretize the optimality system. A global error estimate in the product of physical space and stochastic space for the numerical approximation is derived. Illustrative numerical experiments are provided. VL - 51 ER - TY - JOUR T1 - A variational Analysis of the Toda System on Compact Surfaces JF - Communications on Pure and Applied Mathematics, Volume 66, Issue 3, March 2013, Pages 332-371 Y1 - 2013 A1 - Andrea Malchiodi A1 - David Ruiz AB - In this paper we consider the Toda system of equations on a compact surface. We will give existence results by using variational methods in a non coercive case. A key tool in our analysis is a new Moser-Trudinger type inequality under suitable conditions on the center of mass and the scale of concentration of the two components u_1, u_2. PB - Wiley UR - http://hdl.handle.net/1963/6558 N1 - pre-peer version, to appear in Comm. Pure Applied Math U1 - 6489 U2 - Mathematics U4 - 1 U5 - MAT/05 ANALISI MATEMATICA ER - TY - JOUR T1 - A weighted reduced basis method for elliptic partial differential equations with random input data JF - SIAM Journal on Numerical Analysis Y1 - 2013 A1 - Peng Chen A1 - Alfio Quarteroni A1 - Gianluigi Rozza AB - In this work we propose and analyze a weighted reduced basis method to solve elliptic partial differential equations (PDEs) with random input data. The PDEs are first transformed into a weighted parametric elliptic problem depending on a finite number of parameters. Distinctive importance of the solution at different values of the parameters is taken into account by assigning different weights to the samples in the greedy sampling procedure. A priori convergence analysis is carried out by constructive approximation of the exact solution with respect to the weighted parameters. Numerical examples are provided for the assessment of the advantages of the proposed method over the reduced basis method and the stochastic collocation method in both univariate and multivariate stochastic problems. VL - 51 ER - TY - JOUR T1 - Boundary control and shape optimization for the robust design of bypass anastomoses under uncertainty JF - Mathematical Modelling and Numerical Analysis, in press, 2012-13 Y1 - 2012 A1 - Toni Lassila A1 - Andrea Manzoni A1 - Alfio Quarteroni A1 - Gianluigi Rozza KW - shape optimization AB - We review the optimal design of an arterial bypass graft following either a (i) boundary optimal control approach, or a (ii) shape optimization formulation. The main focus is quantifying and treating the uncertainty in the residual flow when the hosting artery is not completely occluded,\\r\\nfor which the worst-case in terms of recirculation e ffects is inferred to correspond to a strong ori fice flow through near-complete occlusion. A worst-case optimal control approach is applied to the steady\\r\\nNavier-Stokes equations in 2D to identify an anastomosis angle and a cu ed shape that are robust with respect to a possible range of residual \\r\\nflows. We also consider a reduced order modelling framework\\r\\nbased on reduced basis methods in order to make the robust design problem computationally feasible. The results obtained in 2D are compared with simulations in a 3D geometry but without model\\r\\nreduction or the robust framework. PB - Cambridge University Press UR - http://hdl.handle.net/1963/6337 U1 - 6267 U2 - Mathematics U4 - 1 U5 - MAT/08 ANALISI NUMERICA ER - TY - JOUR T1 - On a class of vector fields with discontinuity of divide-by-zero type and its applications JF - Journal of dynamical and control systems Y1 - 2012 A1 - Roberta Ghezzi A1 - Alexey O. Remizov AB -We study phase portraits and singular points of vector fields of a special type, that is, vector fields whose components are fractions with a common denominator vanishing on a smooth regular hypersurface in the phase space. We assume also some additional conditions, which are fulfilled, for instance, if the vector field is divergence-free. This problem is motivated by a large number of applications. In this paper, we consider three of them in the framework of differential geometry: singularities of geodesic flows in various singular metrics on surfaces.

PB - Springer VL - 18 IS - 1 U1 - 7038 U2 - Mathematics U4 - -1 ER - TY - JOUR T1 - Decompositions of large-scale biological systems based on dynamical properties JF - Bioinformatics (Oxford, England). 2012 Jan; 28(1):76-83 Y1 - 2012 A1 - Nicola Soranzo A1 - Fahimeh Ramezani A1 - Giovanni Iacono A1 - Claudio Altafini AB - MOTIVATION: Given a large-scale biological network represented as an influence graph, in this article we investigate possible decompositions of the network aimed at highlighting specific dynamical properties.\\r\\nRESULTS: The first decomposition we study consists in finding a maximal directed acyclic subgraph of the network, which dynamically corresponds to searching for a maximal open-loop subsystem of the given system. Another dynamical property investigated is strong monotonicity. We propose two methods to deal with this property, both aimed at decomposing the system into strongly monotone subsystems, but with different structural characteristics: one method tends to produce a single large strongly monotone component, while the other typically generates a set of smaller disjoint strongly monotone subsystems.\\r\\nAVAILABILITY: Original heuristics for the methods investigated are described in the article. PB - Oxford University Press UR - http://hdl.handle.net/1963/5226 U1 - 5049 U2 - Physics U3 - Functional Analysis and Applications U4 - -1 ER - TY - JOUR T1 - Deformed Lorentz symmetry and relative locality in a curved/expanding spacetime JF - Phys. Rev. D 86 (2012) 124035 Y1 - 2012 A1 - Giovanni Amelino-Camelia A1 - Antonino Marciano A1 - Marco Matassa A1 - Giacomo Rosati KW - Doubly special relativity AB - The interest of part of the quantum-gravity community in the possibility of\r\nPlanck-scale-deformed Lorentz symmetry is also fueled by the opportunities for testing the relevant scenarios with analyses, from a signal-propagation perspective, of observations of bursts of particles from cosmological distances. In this respect the fact that so far the implications of deformed Lorentz symmetry have been investigated only for flat (Minkowskian) spacetimes represents a very significant limitation, since for propagation over cosmological distances the curvature/expansion of spacetime is evidently tangible. We here provide a significant step toward filling this gap by exhibiting an explicit example of Planck-scale-deformed relativistic symmetries of a spacetime with constant rate of expansion (deSitterian). Technically we obtain the first ever example of a relativistic theory of worldlines of particles with 3 nontrivial relativistic invariants: a large speed scale (\"speed-of-light scale\"), a large distance scale (inverse of the \"expansion-rate scale\"), and a large momentum scale (\"Planck scale\"). We address some of the challenges that had obstructed success for previous attempts by exploiting the recent understanding of the connection between deformed Lorentz symmetry and relativity of spacetime locality. We also offer a preliminary analysis of the differences between the scenario we here propose and the most studied scenario for broken (rather than deformed) Lorentz symmetry in expanding spacetimes. PB - American Physical Society N1 - 12 pages, 5 figures U1 - 6496 U2 - Physics U4 - -1 ER - TY - JOUR T1 - A formula for Popp\'s volume in sub-Riemannian geometry JF - Analysis and Geometry in Metric Spaces, vol. 1 (2012), pages : 42-57 Y1 - 2012 A1 - Luca Rizzi A1 - Davide Barilari KW - subriemannian, volume, Popp, control AB - For an equiregular sub-Riemannian manifold M, Popp\'s volume is a smooth\r\nvolume which is canonically associated with the sub-Riemannian structure, and\r\nit is a natural generalization of the Riemannian one. In this paper we prove a\r\ngeneral formula for Popp\'s volume, written in terms of a frame adapted to the\r\nsub-Riemannian distribution. As a first application of this result, we prove an\r\nexplicit formula for the canonical sub-Laplacian, namely the one associated\r\nwith Popp\'s volume. Finally, we discuss sub-Riemannian isometries, and we prove\r\nthat they preserve Popp\'s volume. We also show that, under some hypotheses on\r\nthe action of the isometry group of M, Popp\'s volume is essentially the unique\r\nvolume with such a property. PB - SISSA UR - http://hdl.handle.net/1963/6501 N1 - 16 pages, minor revisions U1 - 6446 U2 - Mathematics U4 - 1 U5 - MAT/03 GEOMETRIA ER - TY - JOUR T1 - Frobenius manifold for the dispersionless Kadomtsev-Petviashvili equation JF - Communications in Mathematical Physics 311 (2012) 557-594 Y1 - 2012 A1 - Andrea Raimondo AB - We consider a Frobenius structure associated with the dispersionless\\r\\nKadomtsev-Petviashvili equation. This is done, essentially, by applying a\\r\\ncontinuous analogue of the finite dimensional theory in the space of Schwartz\\r\\nfunctions on the line. The potential of the Frobenius manifold is found to be a\\r\\nlogarithmic potential with quadratic external field. Following the construction\\r\\nof the principal hierarchy, we construct a set of infinitely many commuting\\r\\nflows, which extends the classical dKP hierarchy. PB - Springer UR - http://hdl.handle.net/1963/6040 U1 - 5931 U2 - Mathematics U3 - Mathematical Physics U4 - -1 ER - TY - RPRT T1 - A Fully Coupled Immersed Finite Element Method for Fluid Structure Interaction via the Deal.II Library Y1 - 2012 A1 - Luca Heltai A1 - Saswati Roy A1 - Francesco Costanzo KW - Finite Element Method KW - Immersed Boundary Method KW - Immersed Finite Element Method AB - We present the implementation of a solution scheme for fluid-structure\\r\\ninteraction problems via the finite element software library deal.II. The\\r\\nsolution scheme is an immersed finite element method in which two independent discretizations are used for the fluid and immersed deformable body. In this type of formulation the support of the equations of motion of the fluid is extended to cover the union of the solid and fluid domains. The equations of motion over the extended solution domain govern the flow of a fluid under the action of a body force field. This body force field informs the fluid of the presence of the immersed solid. The velocity field of the immersed solid is the restriction over the immersed domain of the velocity field in the extended equations of motion. The focus of this paper is to show how the determination of the motion of the immersed domain is carried out in practice. We show that our implementation is general, that is, it is not dependent on a specific choice of the finite element spaces over the immersed solid and the extended fluid domains. We present some preliminary results concerning the accuracy of the proposed method. PB - SISSA UR - http://hdl.handle.net/1963/6255 N1 - 28 pages, 9 figures U1 - 6172 U2 - Mathematics U3 - Functional Analysis and Applications U4 - 1 U5 - MAT/08 ANALISI NUMERICA ER - TY - CHAP T1 - Generalized reduced basis methods and n-width estimates for the approximation of the solution manifold of parametric PDEs T2 - Springer, Indam Series, Vol. 4, 2012 Y1 - 2012 A1 - Toni Lassila A1 - Andrea Manzoni A1 - Alfio Quarteroni A1 - Gianluigi Rozza KW - solution manifold AB - The set of solutions of a parameter-dependent linear partial di fferential equation with smooth coe fficients typically forms a compact manifold in a Hilbert space. In this paper we review the generalized reduced basis method as a fast computational tool for the uniform approximation of the solution manifold. We focus on operators showing an affi ne parametric dependence, expressed as a linear combination of parameter-independent operators through some smooth, parameter-dependent scalar functions. In the case that the parameter-dependent operator has a dominant term in its affi ne expansion, one can prove the existence of exponentially convergent uniform approximation spaces for the entire solution manifold. These spaces can be constructed without any assumptions on the parametric regularity of the manifold \\r\\nonly spatial regularity of the solutions is required. The exponential convergence rate is then inherited by the generalized reduced basis method. We provide a numerical example related to parametrized elliptic\\r\\nequations con rming the predicted convergence rates. JF - Springer, Indam Series, Vol. 4, 2012 PB - Springer UR - http://hdl.handle.net/1963/6340 U1 - 6270 U2 - Mathematics U4 - 1 U5 - MAT/08 ANALISI NUMERICA ER - TY - JOUR T1 - On localization in holomorphic equivariant cohomology JF - Central European Journal of Mathematics, Volume 10, Issue 4, August 2012, Pages 1442-1454 Y1 - 2012 A1 - Ugo Bruzzo A1 - Vladimir Rubtsov KW - Lie algebroids AB - We prove a localization formula for a "holomorphic equivariant cohomology" attached to the Atiyah algebroid of an equivariant holomorphic vector bundle. This generalizes Feng-Ma, Carrell-Liebermann, Baum-Bott and K. Liu's localization formulas. PB - Springer UR - http://hdl.handle.net/1963/6584 U1 - 6543 U2 - Mathematics U4 - 1 ER - TY - Generic T1 - Reduction strategies for PDE-constrained oprimization problems in Haemodynamics T2 - European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS 2012) J. Eberhardsteiner et.al. (eds.), Vienna, Austria, 10-14 sept. 2012 Y1 - 2012 A1 - Gianluigi Rozza A1 - Andrea Manzoni A1 - Federico Negri KW - inverse problems AB - Solving optimal control problems for many different scenarios obtained by varying a set of parameters in the state system is a computationally extensive task. In this paper we present a new reduced framework for the formulation, the analysis and the numerical solution of parametrized PDE-constrained optimization problems. This framework is based on a suitable saddle-point formulation of the optimal control problem and exploits the reduced basis method for the rapid and reliable solution of parametrized PDEs, leading to a relevant computational reduction with respect to traditional discretization techniques such as the finite element method. This allows a very efficient evaluation of state solutions and cost functionals, leading to an effective solution of repeated optimal control problems, even on domains of variable shape, for which a further (geometrical) reduction is pursued, relying on flexible shape parametrization techniques. This setting is applied to the solution of two problems arising from haemodynamics, dealing with both data reconstruction and data assimilation over domains of variable shape,\\r\\nwhich can be recast in a common PDE-constrained optimization formulation. JF - European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS 2012) J. Eberhardsteiner et.al. (eds.), Vienna, Austria, 10-14 sept. 2012 UR - http://hdl.handle.net/1963/6338 U1 - 6268 U2 - Mathematics U4 - 1 U5 - MAT/08 ANALISI NUMERICA ER - TY - JOUR T1 - Simulation-based uncertainty quantification of human arterial network hemodynamics JF - International Journal Numerical Methods Biomedical Engineering Y1 - 2012 A1 - Peng Chen A1 - Alfio Quarteroni A1 - Gianluigi Rozza KW - uncertainty quantification, mathematical modelling of the cardiovascular system, fluid-structure interaction AB - This work aims at identifying and quantifying uncertainties from various sources in human cardiovascular\r\nsystem based on stochastic simulation of a one dimensional arterial network. A general analysis of\r\ndifferent uncertainties and probability characterization with log-normal distribution of these uncertainties\r\nis introduced. Deriving from a deterministic one dimensional fluid structure interaction model, we establish\r\nthe stochastic model as a coupled hyperbolic system incorporated with parametric uncertainties to describe\r\nthe blood flow and pressure wave propagation in the arterial network. By applying a stochastic collocation\r\nmethod with sparse grid technique, we study systemically the statistics and sensitivity of the solution with\r\nrespect to many different uncertainties in a relatively complete arterial network with potential physiological\r\nand pathological implications for the first time. PB - Wiley U1 - 6467 U2 - Mathematics U4 - 1 U5 - MAT/08 ANALISI NUMERICA ER - TY - JOUR T1 - A Viscosity-driven crack evolution JF - Advances in Calculus of Variations 5 (2012) 433-483 Y1 - 2012 A1 - Simone Racca AB -We present a model of crack growth in brittle materials which couples dissipative effects on the crack tip and viscous effects. We consider the 2 -dimensional antiplane case with pre-assigned crack path, and firstly prove an existence result for a rate-dependent evolution problem by means of time-discretization. The next goal is to describe the rate-independent evolution as limit of the rate-dependent ones when the dissipative and viscous effects vanish. The rate-independent evolution satisfies a Griffith’s criterion for the crack growth, but, in general, it does not fulfil a global minimality condition; its fracture set may exhibit jump discontinuities with respect to time. Under suitable regularity assumptions, the quasi-static crack growth is described by solving a finite-dimensional problem.

PB - SISSA UR - http://hdl.handle.net/1963/5130 U1 - 4944 U2 - Mathematics U3 - Functional Analysis and Applications U4 - -1 ER - TY - JOUR T1 - Cluster solutions for the Schrödinger-Poisson-Slater problem around a local minimum of the potential JF - Rev. Mat. Iberoamericana Y1 - 2011 A1 - David Ruiz A1 - Giusi Vaira PB - Real Sociedad Matemática Española VL - 27 UR - https://projecteuclid.org:443/euclid.rmi/1296828834 ER - TY - JOUR T1 - New improved Moser-Trudinger inequalities and singular Liouville equations on compact surfaces JF - Geometric and Functional Analysis 21 (2011) 1196-1217 Y1 - 2011 A1 - Andrea Malchiodi A1 - David Ruiz AB - We consider a singular Liouville equation on a compact surface, arising from the study of Chern-Simons vortices in a self dual regime. Using new improved versions of the Moser-Trudinger inequalities (whose main feature is to be scaling invariant) and a variational scheme, we prove new existence results. PB - Springer UR - http://hdl.handle.net/1963/4099 U1 - 305 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - SBV regularity for Hamilton-Jacobi equations in R^n JF - Arch. Rational Mech. Anal. 200 (2011) 1003-1021 Y1 - 2011 A1 - Stefano Bianchini A1 - Camillo De Lellis A1 - Roger Robyr AB -In this paper we study the regularity of viscosity solutions to the following Hamilton-Jacobi equations $$ \partial_t u + H(D_{x} u)=0 \qquad \textrm{in}\quad \Omega\subset \mathbb{R}\times \mathbb{R}^{n} . $$ In particular, under the assumption that the Hamiltonian $H\in C^2(\mathbb{R}^n)$ is uniformly convex, we prove that $D_{x}u$ and $\partial_t u$ belong to the class $SBV_{loc}(\Omega)$.

PB - Springer UR - http://hdl.handle.net/1963/4911 U1 - 4695 U2 - Mathematics U3 - Functional Analysis and Applications U4 - -1 ER - TY - RPRT T1 - Cohomology of Skew-holomorphic lie algebroids Y1 - 2010 A1 - Ugo Bruzzo A1 - Vladimir Rubtsov AB - We introduce the notion of skew-holomorphic Lie algebroid on a complex manifold, and explore some cohomologies theories that one can associate to it. Examples are given in terms of holomorphic Poisson structures of various sorts. UR - http://hdl.handle.net/1963/3853 U1 - 856 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - Existence of planar curves minimizing length and curvature JF - Proc. Steklov Inst. Math. 270 (2010) 43-56 Y1 - 2010 A1 - Ugo Boscain A1 - Grégoire Charlot A1 - Francesco Rossi AB - In this paper we consider the problem of reconstructing a curve that is partially hidden or corrupted by minimizing the functional $\\\\int \\\\sqrt{1+K_\\\\gamma^2} ds$, depending both on length and curvature $K$. We fix starting and ending points as well as initial and final directions.\\nFor this functional we discuss the problem of existence of minimizers on various functional spaces. We find non-existence of minimizers in cases in which initial and final directions are considered with orientation. In this case, minimizing sequences of trajectories can converge to curves with angles.\\nWe instead prove existence of minimizers for the \\\"time-reparameterized\\\" functional $$\\\\int \\\\| \\\\dot\\\\gamma(t) \\\\|\\\\sqrt{1+K_\\\\ga^2} dt$$ for all boundary conditions if initial and final directions are considered regardless to orientation. In this case, minimizers can present cusps (at most two) but not angles. PB - Springer UR - http://hdl.handle.net/1963/4107 U1 - 297 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Homogeneous binary trees as ground states of quantum critical Hamiltonians JF - Phys. Rev. A 81 (2010) 062335 Y1 - 2010 A1 - Pietro Silvi A1 - Vittorio Giovannetti A1 - Simone Montangero A1 - Matteo Rizzi A1 - J. Ignacio Cirac A1 - Rosario Fazio AB -Many-body states whose wave-function admits a representation in terms of a uniform binary-tree tensor decomposition are shown to obey to power-law two-body correlations functions. Any such state can be associated with the ground state of a translational invariant Hamiltonian which, depending on the dimension of the systems sites, involve at most couplings between third-neighboring sites. A detailed analysis of their spectra shows that they admit an exponentially large ground space.

PB - American Physical Society UR - http://hdl.handle.net/1963/3909 U1 - 800 U2 - Physics U3 - Condensed Matter Theory ER - TY - JOUR T1 - Homogeneous multiscale entanglement renormalization ansatz tensor networks for quantum critical systems JF - New J. Phys. 12 (2010) 075018 Y1 - 2010 A1 - Matteo Rizzi A1 - Simone Montangero A1 - Pietro Silvi A1 - Vittorio Giovannetti A1 - Rosario Fazio AB -In this paper, we review the properties of homogeneous multiscale entanglement renormalization ansatz (MERA) to describe quantum critical systems.We discuss in more detail our results for one-dimensional (1D) systems (the Ising and Heisenberg models) and present new data for the 2D Ising model. Together with the results for the critical exponents, we provide a detailed description of the numerical algorithm and a discussion of new optimization\\nstrategies. The relation between the critical properties of the system and the tensor structure of the MERA is expressed using the formalism of quantum channels, which we review and extend.

PB - IOP Publishing UR - http://hdl.handle.net/1963/4067 U1 - 335 U2 - Physics U3 - Condensed Matter Theory ER - TY - JOUR T1 - Projective Reeds-Shepp car on $S^2$ with quadratic cost JF - ESAIM COCV 16 (2010) 275-297 Y1 - 2010 A1 - Ugo Boscain A1 - Francesco Rossi AB - Fix two points $x,\\\\bar{x}\\\\in S^2$ and two directions (without orientation) $\\\\eta,\\\\bar\\\\eta$ of the velocities in these points. In this paper we are interested to the problem of minimizing the cost $$ J[\\\\gamma]=\\\\int_0^T g_{\\\\gamma(t)}(\\\\dot\\\\gamma(t),\\\\dot\\\\gamma(t))+\\nK^2_{\\\\gamma(t)}g_{\\\\gamma(t)}(\\\\dot\\\\gamma(t),\\\\dot\\\\gamma(t)) ~dt$$ along all smooth curves starting from $x$ with direction $\\\\eta$ and ending in $\\\\bar{x}$ with direction $\\\\bar\\\\eta$. Here $g$ is the standard Riemannian metric on $S^2$ and $K_\\\\gamma$ is the corresponding geodesic curvature.\\nThe interest of this problem comes from mechanics and geometry of vision. It can be formulated as a sub-Riemannian problem on the lens space L(4,1).\\nWe compute the global solution for this problem: an interesting feature is that some optimal geodesics present cusps. The cut locus is a stratification with non trivial topology. UR - http://hdl.handle.net/1963/2668 U1 - 1429 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - The reductions of the dispersionless 2D Toda hierarchy and their Hamiltonian structures JF - J. Phys. A 43 (2010) 045201 Y1 - 2010 A1 - Guido Carlet A1 - Paolo Lorenzoni A1 - Andrea Raimondo AB - We study finite-dimensional reductions of the dispersionless 2D Toda hierarchy showing that the consistency conditions for such reductions are given by a system of radial Loewner equations. We then construct their Hamiltonian structures, following an approach proposed by Ferapontov. PB - IOP Publishing UR - http://hdl.handle.net/1963/3846 U1 - 863 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - Characterization of the time course of changes of the evoked electrical activity in a model of a chemically-induced neuronal plasticity JF - BMC Research Notes (2009) 2:13 Y1 - 2009 A1 - Frederic D. Broccard A1 - Silvia Pegoraro A1 - Maria Elisabetta Ruaro A1 - Claudio Altafini A1 - Vincent Torre AB - BACKGROUND: Neuronal plasticity is initiated by transient elevations of neuronal networks activity leading to changes of synaptic properties and providing the basis for memory and learning 1. An increase of electrical activity can be caused by electrical stimulation 2 or by pharmacological manipulations: elevation of extracellular K+ 3, blockage of inhibitory pathways 4 or by an increase of second messengers intracellular concentrations 5. Neuronal plasticity is mediated by several biochemical pathways leading to the modulation of synaptic strength, density of ionic channels and morphological changes of neuronal arborisation 6. On a time scale of a few minutes, neuronal plasticity is mediated by local protein trafficking 7 while, in order to sustain modifications beyond 2-3 h, changes of gene expression are required 8. FINDINGS: In the present manuscript we analysed the time course of changes of the evoked electrical activity during neuronal plasticity and we correlated it with a transcriptional analysis of the underlying changes of gene expression. Our investigation shows that treatment for 30 min. with the GABAA receptor antagonist gabazine (GabT) causes a potentiation of the evoked electrical activity occurring 2-4 hours after GabT and the concomitant up-regulation of 342 genes. Inhibition of the ERK1/2 pathway reduced but did not abolish the potentiation of the evoked response caused by GabT. In fact not all the genes analysed were blocked by ERK1/2 inhibitors. CONCLUSION: These results are in agreement with the notion that neuronal plasticity is mediated by several distinct pathways working in unison. PB - BioMed Central UR - http://hdl.handle.net/1963/3706 U1 - 599 U2 - Neuroscience U3 - Neurobiology ER - TY - JOUR T1 - Equivariant cohomology and localization for Lie algebroids JF - Funct. Anal. Appl. 43 (2009) 18-29 Y1 - 2009 A1 - Ugo Bruzzo A1 - Lucio Cirio A1 - Paolo Rossi A1 - Vladimir Rubtsov AB - Let M be a manifold carrying the action of a Lie group G, and A a Lie algebroid on M equipped with a compatible infinitesimal G-action. Out of these data we construct an equivariant Lie algebroid cohomology and prove for compact G a related localization formula. As an application we prove a Bott-type formula. SN - 978-981-270-377-4 UR - http://hdl.handle.net/1963/1724 U1 - 2427 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - Gauged Laplacians on quantum Hopf bundles JF - Comm. Math. Phys. 287 (2009) 179-209 Y1 - 2009 A1 - Giovanni Landi A1 - Cesare Reina A1 - Alessandro Zampini AB - We study gauged Laplacian operators on line bundles on a quantum 2-dimensional sphere. Symmetry under the (co)-action of a quantum group allows for their complete diagonalization. These operators describe `excitations moving on the quantum sphere\\\' in the field of a magnetic monopole. The energies are not invariant under the exchange monopole/antimonopole, that is under inverting the direction of the magnetic field. There are potential applications to models of quantum Hall effect. PB - Springer UR - http://hdl.handle.net/1963/3540 U1 - 1161 U2 - Mathematics U3 - Mathematical Physics ER - TY - RPRT T1 - Holomorphic equivariant cohomology of Atiyah algebroids and localization Y1 - 2009 A1 - Ugo Bruzzo A1 - Vladimir Rubtsov UR - http://hdl.handle.net/1963/3774 U1 - 551 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - The intrinsic hypoelliptic Laplacian and its heat kernel on unimodular Lie groups JF - J. Funct. Anal. 256 (2009) 2621-2655 Y1 - 2009 A1 - Andrei A. Agrachev A1 - Ugo Boscain A1 - Jean-Paul Gauthier A1 - Francesco Rossi AB - We present an invariant definition of the hypoelliptic Laplacian on sub-Riemannian structures with constant growth vector, using the Popp\\\'s volume form introduced by Montgomery. This definition generalizes the one of the Laplace-Beltrami operator in Riemannian geometry. In the case of left-invariant problems on unimodular Lie groups we prove that it coincides with the usual sum of squares.\\nWe then extend a method (first used by Hulanicki on the Heisenberg group) to compute explicitly the kernel of the hypoelliptic heat equation on any unimodular Lie group of type I. The main tool is the noncommutative Fourier transform. We then study some relevant cases: SU(2), SO(3), SL(2) (with the metrics inherited by the Killing form), and the group SE(2) of rototranslations of the plane.\\nOur study is motivated by some recent results about the cut and conjugate loci on these sub-Riemannian manifolds. The perspective is to understand how singularities of the sub-Riemannian distance reflect on the kernel of the corresponding hypoelliptic heat equation. UR - http://hdl.handle.net/1963/2669 U1 - 1428 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Fulde-Ferrell-Larkin-Ovchinnikov pairing in one-dimensional optical lattices JF - Phys. Rev. B 77 (2008) 245105 Y1 - 2008 A1 - Matteo Rizzi A1 - Marco Polini A1 - Miguel A. Cazalilla A1 - M.R. Bakhtiari A1 - Mario P. Tosi A1 - Rosario Fazio AB -Spin-polarized attractive Fermi gases in one-dimensional (1D) optical lattices are expected to be remarkably good candidates for the observation of the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase. We model these systems with an attractive Hubbard model with population imbalance. By means of the density-matrix renormalization-group method we compute the pairing correlations as well as the static spin and charge structure factors in the whole range from weak to strong coupling. We demonstrate that pairing correlations exhibit quasi-long range order and oscillations at the wave number expected from FFLO theory. However, we also show by numerically computing the mixed spin-charge static structure factor that charge and spin degrees of freedom appear to be coupled already for small imbalance. We discuss the consequences of this coupling for the observation of the FFLO phase, as well as for the stabilization of the quasi-long range order into long-range order by coupling many identical 1D systems, as in quasi-1D optical lattices.

UR - http://hdl.handle.net/1963/2694 U1 - 1406 U2 - Physics U3 - Condensed Matter Theory ER - TY - JOUR T1 - Invariant Carnot-Caratheodory metrics on S3, SO(3), SL(2) and Lens Spaces JF - SIAM J. Control Optim. 47 (2008) 1851-1878 Y1 - 2008 A1 - Ugo Boscain A1 - Francesco Rossi AB - In this paper we study the invariant Carnot-Caratheodory metrics on SU(2) \\\' S3,\\nSO(3) and SL(2) induced by their Cartan decomposition. Beside computing explicitly geodesics and conjugate loci, we compute the cut loci (globally) and we give the expression of the Carnot-Caratheodory distance as the inverse of an elementary function. We then prove that the metric\\ngiven on SU(2) projects on the so called Lens Spaces L(p; q). Also for Lens Spaces, we compute\\nthe cut loci (globally). UR - http://hdl.handle.net/1963/2144 U1 - 2099 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Multiple bound states for the Schroedinger-Poisson problem JF - Commun. Contemp. Math. 10 (2008) 391-404 Y1 - 2008 A1 - Antonio Ambrosetti A1 - David Ruiz UR - http://hdl.handle.net/1963/2679 U1 - 1421 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Noncommutative families of instantons JF - Int. Math. Res. Not. vol. 2008, Article ID rnn038 Y1 - 2008 A1 - Giovanni Landi A1 - Chiara Pagani A1 - Cesare Reina A1 - Walter van Suijlekom AB - We construct $\\\\theta$-deformations of the classical groups SL(2,H) and Sp(2). Coacting on the basic instanton on a noncommutative four-sphere $S^4_\\\\theta$, we construct a noncommutative family of instantons of charge 1. The family is parametrized by the quantum quotient of $SL_\\\\theta(2,H)$ by $Sp_\\\\theta(2)$. PB - Oxford University Press UR - http://hdl.handle.net/1963/3417 U1 - 918 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - Solitons of linearly coupled systems of semilinear non-autonomous equations on Rn JF - J. Funct. Anal. 254 (2008) 2816-2845 Y1 - 2008 A1 - Antonio Ambrosetti A1 - Giovanna Cerami A1 - David Ruiz AB - Using concentration compactness type arguments, we prove some results about the existence of positive ground and bound state of linearly coupled systems of nonlinear Schrödinger equations. UR - http://hdl.handle.net/1963/2175 U1 - 2069 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Luther-Emery Phase and Atomic-Density Waves in a Trapped Fermion Gas JF - Phys. Rev. Lett. 98 (2007) 030404 Y1 - 2007 A1 - Gao Xianlong A1 - Matteo Rizzi A1 - Marco Polini A1 - Rosario Fazio A1 - Mario P. Tosi A1 - Vivaldo L. Jr. Campo A1 - Klaus Capelle AB -The Luther-Emery liquid is a state of matter that is predicted to occur in one-dimensional systems of interacting fermions and is characterized by a gapless charge spectrum and a gapped spin spectrum. In this Letter we discuss a realization of the Luther-Emery phase in a trapped cold-atom gas. We study by means of the density-matrix renormalization-group technique a two-component atomic Fermi gas with attractive interactions subject to parabolic trapping inside an optical lattice. We demonstrate how this system exhibits compound phases characterized by the coexistence of spin pairing and atomic-density waves. A smooth crossover occurs with increasing magnitude of the atom-atom attraction to a state in which tightly bound spin-singlet dimers occupy the center of the trap. The existence of atomic-density waves could be detected in the elastic contribution to the light-scattering diffraction pattern.

UR - http://hdl.handle.net/1963/2056 U1 - 2140 U2 - Physics U3 - Condensed Matter Theory ER - TY - RPRT T1 - Multi-bump solitons to linearly coupled systems of nonlinear Schrödinger equations Y1 - 2007 A1 - Antonio Ambrosetti A1 - Eduardo Colorado A1 - David Ruiz JF - Calc. Var. Partial Differential Equations 30 (2007) 85-112 UR - http://hdl.handle.net/1963/1835 U1 - 2381 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Viscosity solutions of Hamilton-Jacobi equations with discontinuous coefficients JF - J. Hyperbolic Differ. Equ. 4 (2007) 771-795 Y1 - 2007 A1 - Giuseppe Maria Coclite A1 - Nils Henrik Risebro AB - We consider Hamilton--Jacobi equations, where the Hamiltonian depends discontinuously on both the spatial and temporal location. Our main results are the existence and well--posedness of a viscosity solution to the Cauchy problem. We define a viscosity solution by treating the discontinuities in the coefficients analogously to \\\"internal boundaries\\\". By defining an appropriate penalization function, we prove that viscosity solutions are unique. The existence of viscosity solutions is established by showing that a sequence of front tracking approximations is compact in $L^\\\\infty$, and that the limits are viscosity solutions. PB - World Scientific UR - http://hdl.handle.net/1963/2907 U1 - 1793 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - 4e-condensation in a fully frustrated Josephson junction diamond chain JF - Phys. Rev. B 73 (2006) 100502(R) Y1 - 2006 A1 - Matteo Rizzi A1 - Vittorio Cataudella A1 - Rosario Fazio AB -Fully frustrated one-dimensional diamond Josephson chains have been shown [B. Dou\\\\c{c}ot and J. Vidal, Phys. Rev. Lett. {\\\\bf 88}, 227005 (2002)] to posses a remarkable property: The superfluid phase occurs through the condensation of pairs of Cooper pairs. By means of Monte Carlo simulations we analyze quantitatively the Insulator to $4e$-Superfluid transition. We determine the location of the critical point and discuss the behaviour of the phase-phase correlators. For comparison we also present the case of a diamond chain at zero and 1/3 frustration where the standard $2e$-condensation is observed.

UR - http://hdl.handle.net/1963/2400 U1 - 2297 U2 - Physics U3 - Condensed Matter Theory ER - TY - JOUR T1 - Bound states of Nonlinear Schroedinger Equations with Potentials Vanishing at Infinity JF - J. Anal. Math. 98 (2006) 317-348 Y1 - 2006 A1 - Antonio Ambrosetti A1 - Andrea Malchiodi A1 - David Ruiz UR - http://hdl.handle.net/1963/1756 U1 - 2788 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - A Hopf bundle over a quantum four-sphere from the symplectic group JF - Commun. Math. Phys. 263 (2006) 65-88 Y1 - 2006 A1 - Giovanni Landi A1 - Chiara Pagani A1 - Cesare Reina AB - We construct a quantum version of the SU(2) Hopf bundle $S^7 \\\\to S^4$. The quantum sphere $S^7_q$ arises from the symplectic group $Sp_q(2)$ and a quantum 4-sphere $S^4_q$ is obtained via a suitable self-adjoint idempotent $p$ whose entries generate the algebra $A(S^4_q)$ of polynomial functions over it. This projection determines a deformation of an (anti-)instanton bundle over the classical sphere $S^4$. We compute the fundamental $K$-homology class of $S^4_q$ and pair it with the class of $p$ in the $K$-theory getting the value -1 for the topological charge. There is a right coaction of $SU_q(2)$ on $S^7_q$ such that the algebra $A(S^7_q)$ is a non trivial quantum principal bundle over $A(S^4_q)$ with structure quantum group $A(SU_q(2))$. UR - http://hdl.handle.net/1963/2179 U1 - 2065 U2 - Mathematics U3 - Mathematical Physics ER - TY - RPRT T1 - Normal bundles to Laufer rational curves in local Calabi-Yau threefolds Y1 - 2006 A1 - Ugo Bruzzo A1 - Antonio Ricco AB - We prove a conjecture by F. Ferrari. Let X be the total space of a nonlinear deformation of a rank 2 holomorphic vector bundle on a smooth rational curve, such that X has trivial canonical bundle and has sections. Then the normal bundle to such sections is computed in terms of the rank of the Hessian of a suitably defined superpotential at its critical points. JF - Lett. Math. Phys. 76 (2006) 57-63 UR - http://hdl.handle.net/1963/1785 U1 - 2759 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - Radial solutions concentrating on spheres of nonlinear Schrödinger equations with vanishing potentials JF - Proc. Roy. Soc. Edinburgh Sect. A 136 (2006) 889-907 Y1 - 2006 A1 - Antonio Ambrosetti A1 - David Ruiz AB - We prove the existence of radial solutions of 1.2) concentrating at a sphere for potentials which might be zero and might decay to zero at\\r\\ninfinity. The proofs use a perturbation technique in a variational setting, through a Lyapunov-Schmidt reduction. UR - http://hdl.handle.net/1963/1755 U1 - 2789 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Conservation laws with time dependent discontinuous coefficients JF - SIAM J. Math. Anal. 36 (2005) 1293-1309 Y1 - 2005 A1 - Giuseppe Maria Coclite A1 - Nils Henrik Risebro AB - We consider scalar conservation laws where the flux function depends discontinuously on both the spatial and temporal location. Our main results are the existence and well-posedness of an entropy solution to the Cauchy problem. The existence is established by showing that a sequence of front tracking approximations is compact in L1, and that the limits are entropy solutions. Then, using the definition of an entropy solution taken form [11], we show that the solution operator is L1 contractive. These results generalize the corresponding results from [16] and [11]. PB - SISSA Library UR - http://hdl.handle.net/1963/1666 U1 - 2452 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Calculation of impulsively started incompressible viscous flows JF - Int. J. Numer. Meth. Fluids Y1 - 2004 A1 - Marra, Andrea A1 - Andrea Mola A1 - Quartapelle, Luigi A1 - Riviello, Luca VL - 46 ER - TY - JOUR T1 - Quantum spin coverings and statistics JF - J. Phys. A 36 (2003), no. 13, 3829-3840 Y1 - 2003 A1 - Ludwik Dabrowski A1 - Cesare Reina AB - SL_q(2) at odd roots of unity q^l =1 is studied as a quantum cover of the complex rotation group SO(3,C), in terms of the associated Hopf algebras of (quantum) polynomial functions. We work out the irreducible corepresentations, the decomposition of their tensor products and a coquasitriangular structure, with the associated braiding (or statistics). As an example, the case l=3 is discussed in detail. PB - IOP Publishing UR - http://hdl.handle.net/1963/1667 U1 - 2451 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - A note on the super Krichever map JF - J. Geom. Phys. 37 (2001), no. 1-2, 169-181 Y1 - 2001 A1 - Gregorio Falqui A1 - Cesare Reina A1 - Alessandro Zampa AB - We consider the geometrical aspects of the Krichever map in the context of Jacobian Super KP hierarchy. We use the representation of the hierarchy based\\non the Fa`a di Bruno recursion relations, considered as the cocycle condition for the natural double complex associated with the deformations of super Krichever data. Our approach is based on the construction of the universal super divisor (of degree g), and a local universal family of geometric data which give the map into the Super Grassmannian. PB - SISSA Library UR - http://hdl.handle.net/1963/1494 U1 - 2669 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - 3D superconformal theories from Sasakian seven-manifolds: new nontrivial evidences for AdS_4/CFT_3 JF - Nucl.Phys. B577 (2000) 547-608 Y1 - 2000 A1 - Davide Fabbri A1 - Pietro Fré A1 - Leonardo Gualtieri A1 - Cesare Reina A1 - Alessandro Tomasiello A1 - Alberto Zaffaroni A1 - Alessandro Zampa PB - SISSA Library UR - http://hdl.handle.net/1963/1327 U1 - 3128 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - A(SLq(2)) at roots of unity is a free module over A(SL(2)) JF - Lett. Math. Phys., 2000, 52, 339 Y1 - 2000 A1 - Ludwik Dabrowski A1 - Cesare Reina A1 - Alessandro Zampa PB - SISSA Library UR - http://hdl.handle.net/1963/1500 U1 - 2663 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - Super KP equations and Darboux transformations: another perspective on the Jacobian super KP hierarchy JF - J. Geom. Phys. 35 (2000), no. 2-3, 239-272 Y1 - 2000 A1 - Gregorio Falqui A1 - Cesare Reina A1 - Alessandro Zampa PB - SISSA Library UR - http://hdl.handle.net/1963/1367 U1 - 3088 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - Enhanced gauge symmetries on elliptic K3 JF - Phys.Lett. B452 (1999) 244-250 Y1 - 1999 A1 - Loriano Bonora A1 - Cesare Reina A1 - Alessandro Zampa AB - We show that the geometry of K3 surfaces with singularities of type A-D-E contains enough information to reconstruct a copy of the Lie algebra associated to the given Dynkin diagram. We apply this construction to explain the enhancement of symmetry in F and IIA theories compactified on singular K3\\\'s. PB - Elsevier UR - http://hdl.handle.net/1963/3366 U1 - 964 U2 - Physics U3 - Elementary Particle Theory ER - TY - JOUR T1 - Krichever maps, Faà di Bruno polynomials, and cohomology in KP theory JF - Lett. Math. Phys. 42 (1997) 349-361 Y1 - 1997 A1 - Gregorio Falqui A1 - Cesare Reina A1 - Alessandro Zampa AB - We study the geometrical meaning of the Faa\\\' di Bruno polynomials in the context of KP theory. They provide a basis in a subspace W of the universal Grassmannian associated to the KP hierarchy. When W comes from geometrical data via the Krichever map, the Faa\\\' di Bruno recursion relation turns out to be the cocycle condition for (the Welters hypercohomology group describing) the deformations of the dynamical line bundle on the spectral curve together with the meromorphic sections which give rise to the Krichever map. Starting from this, one sees that the whole KP hierarchy has a similar cohomological meaning. PB - Springer UR - http://hdl.handle.net/1963/3539 U1 - 1162 U2 - Mathematics U3 - Mathematical Physics ER - TY - CHAP T1 - Quantum homogeneous spaces at roots of unity T2 - Quantization, Coherent States and Poisson Structures, Proc. XIVth Workshop on Geometric Methods in Physics, Bialowieza, Poland, 9-15 July 1995, eds. A. Strasburger,\\nS.T. Ali, J.-P. Antoine, J.-P. Gazeau , A. Odzijewicz, Polish Scientific Publisher PWN 1 Y1 - 1995 A1 - Cesare Reina A1 - Alessandro Zampa JF - Quantization, Coherent States and Poisson Structures, Proc. XIVth Workshop on Geometric Methods in Physics, Bialowieza, Poland, 9-15 July 1995, eds. A. Strasburger,\\nS.T. Ali, J.-P. Antoine, J.-P. Gazeau , A. Odzijewicz, Polish Scientific Publisher PWN 1 PB - SISSA Library UR - http://hdl.handle.net/1963/1022 U1 - 2834 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - A Borel-Weil-Bott approach to representations of \rm sl\sb q(2,C) JF - Lett. Math. Phys. 29 (1993) 215-217 Y1 - 1993 A1 - Davide Franco A1 - Cesare Reina AB -We use a quite concrete and simple realization of $\slq$ involving finite difference operators. We interpret them as derivations (in the non-commutative sense) on a suitable graded algebra, which gives rise to the double of the projective line as the non commutative version of the standard homogeneous space.

PB - Springer UR - http://hdl.handle.net/1963/3538 U1 - 1163 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - Topological "observables" in semiclassical field theories JF - Phys. Lett. B 297 (1992) 82-88 Y1 - 1992 A1 - Margherita Nolasco A1 - Cesare Reina AB -We give a geometrical set up for the semiclassical approximation to euclidean field theories having families of minima (instantons) parametrized by suitable moduli spaces ${\mathcal{M}}$. The standard examples are of course Yang-Mills theory and non-linear $\sigma$-models. The relevant space here is a family of measure spaces $\tilde{\mathcal{N}} \rightarrow \mathcal{M}$, with standard fibre a distribution space, given by a suitable extension of the normal bundle to $\mathcal{M}$ in the space of smooth fields. Over $\tilde{\mathcal{N}}$ there is a probability measure $d\mu$ given by the twisted product of the (normalized) volume element on $\mathcal{M}$ and the family of gaussian measures with covariance given by the tree propagator $C_\phi$ in the background of an instanton $\phi \in \mathcal{M}$. The space of "observables", i.e. measurable functions on ($\tilde{\mathcal{N}},\, d\mu$), is studied and it is shown to contain a topological sector, corresponding to the intersection theory on $\mathcal{M}$. The expectation value of these topological "observables" does not depend on the covariance; it is therefore exact at all orders in perturbation theory and can moreover be computed in the topological regime by setting the covariance to zero.

PB - Elsevier UR - http://hdl.handle.net/1963/3541 U1 - 1160 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - On systems of ordinary differential equations with measures as controls JF - Differential Integral Equations 4 (1991), no.4, p.739-765. Y1 - 1991 A1 - Gianni Dal Maso A1 - Franco Rampazzo PB - SISSA Library UR - http://hdl.handle.net/1963/840 U1 - 2951 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - N=2 super Riemann surfaces and algebraic geometry JF - J. Math. Phys. 31 (1990), no.4, 948-952 Y1 - 1990 A1 - Cesare Reina A1 - Gregorio Falqui AB - The geometric framework for N=2 superconformal field theories are described by studying susy2 curves-a nickname for N=2 super Riemann surfaces. It is proved that \\\"single\\\'\\\' susy2 curves are actually split supermanifolds, and their local model is a Serre self-dual locally free sheaf of rank two over a smooth algebraic curve. Superconformal structures on these sheaves are then examined by setting up deformation theory as a first step in studying moduli problems. PB - American Institute of Physics UR - http://hdl.handle.net/1963/807 U1 - 2984 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - A note on the global structure of supermoduli spaces JF - Comm.Math.Phys. 31 (1990), no.4, 948 Y1 - 1990 A1 - Cesare Reina A1 - Gregorio Falqui PB - SISSA Library UR - http://hdl.handle.net/1963/806 U1 - 2985 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - On differential systems with vector-valued impulsive controls. JF - Boll. Un. Mat. Ital. B (7) 2 (1988), no. 3, 641-656 Y1 - 1988 A1 - Alberto Bressan A1 - Franco Rampazzo PB - SISSA Library UR - http://hdl.handle.net/1963/535 U1 - 3369 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Susy-curves and supermoduli Y1 - 1988 A1 - Gregorio Falqui A1 - Cesare Reina PB - SISSA Library UR - http://hdl.handle.net/1963/761 U1 - 3030 U2 - Mathematics U3 - Mathematical Physics ER -