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 $Re=O(10^5)$.

%B Journal of Computational Physics %V 416 %P 109513 %G eng %U https://arxiv.org/abs/1907.09909 %R 10.1016/j.jcp.2020.109513 %0 Journal Article %J Computers and Mathematics with Applications %D 2020 %T The deal.II finite element library: Design, features, and insights %A Daniel Arndt %A Wolfgang Bangerth %A Denis Davydov %A Timo Heister %A Luca Heltai %A Martin Kronbichler %A Matthias Maier %A Jean-Paul Pelteret %A Bruno Turcksin %A David Wells %B Computers and Mathematics with Applications %G eng %U https://doi.org/10.1016/j.camwa.2020.02.022 %0 Journal Article %J Advances in Computational Mathematics %D 2020 %T Efficient computation of bifurcation diagrams with a deflated approach to reduced basis spectral element method %A Pintore, Moreno %A Pichi, Federico %A Hess, Martin %A Rozza, Gianluigi %A Canuto, Claudio %XThe 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.

%B Advances in Computational Mathematics %G eng %U https://arxiv.org/abs/1912.06089 %0 Report %D 2020 %T On the gauge group of Galois objects %A Xiao Han %A Giovanni Landi %X We study the Ehresmann--Schauenburg bialgebroid of a noncommutative principal bundle as a quantization of the classical gauge groupoid of a principal bundle. When the base algebra is in the centre of the total space algebra, the gauge group of the noncommutative principal bundle is isomorphic to the group of bisections of the bialgebroid. In particular we consider Galois objects (non-trivial noncommutative bundles over a point in a sense) for which the bialgebroid is a Hopf algebra. For these we give a crossed module structure for the bisections and the automorphisms of the bialgebroid. Examples include Galois objects of group Hopf algebras and of Taft algebras. %8 03/2020 %G eng %U https://arxiv.org/abs/2002.06097 %0 Journal Article %J Mathematics in Engineering %D 2020 %T MicroMotility: State of the art, recent accomplishments and perspectives on the mathematical modeling of bio-motility at microscopic scales %A Daniele Agostinelli %A Roberto Cerbino %A Del Alamo, Juan C %A Antonio DeSimone %A Stephanie Höhn %A Cristian Micheletti %A Giovanni Noselli %A Eran Sharon %A Julia Yeomans %K active matter %K adhesive locomotion %K cell motility %K cell sheet folding %K knotted DNA %K topological defects %K unicellular swimmers %K unjamming transition %XMathematical modeling and quantitative study of biological motility (in particular, of motility at microscopic scales) is producing new biophysical insight and is offering opportunities for new discoveries at the level of both fundamental science and technology. These range from the explanation of how complex behavior at the level of a single organism emerges from body architecture, to the understanding of collective phenomena in groups of organisms and tissues, and of how these forms of swarm intelligence can be controlled and harnessed in engineering applications, to the elucidation of processes of fundamental biological relevance at the cellular and sub-cellular level. In this paper, some of the most exciting new developments in the fields of locomotion of unicellular organisms, of soft adhesive locomotion across scales, of the study of pore translocation properties of knotted DNA, of the development of synthetic active solid sheets, of the mechanics of the unjamming transition in dense cell collectives, of the mechanics of cell sheet folding in volvocalean algae, and of the self-propulsion of topological defects in active matter are discussed. For each of these topics, we provide a brief state of the art, an example of recent achievements, and some directions for future research.

%B Mathematics in Engineering %V 2 %P 230 %G eng %U http://dx.doi.org/10.3934/mine.2020011 %9 Perspective %R 10.3934/mine.2020011 %0 Journal Article %J Computers & Structures %D 2020 %T Multiscale modeling of fiber reinforced materials via non-matching immersed methods %A Giovanni Alzetta %A Luca Heltai %B Computers & Structures %G eng %U https://arxiv.org/abs/1906.03881 %0 Conference Paper %B QUIET Selected Contributions %D 2020 %T Non-Intrusive Polynomial Chaos Method Applied to Problems in Computational Fluid Dynamics with a Comparison to Proper Orthogonal Decomposition %A Saddam Hijazi %A Giovanni Stabile %A Andrea Mola %A Gianluigi Rozza %E van Brummelen, Harald %E Corsini, Alessandro %E Perotto, Simona %E Rozza, Gianluigi %XIn 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.

%B QUIET Selected Contributions %I Springer International Publishing %G eng %U https://arxiv.org/abs/1901.02285 %& Non-Intrusive Polynomial Chaos Method Applied to Problems in Computational Fluid Dynamics [...] %0 Journal Article %J Journal of Convex Analysis %D 2020 %T A numerical study of the jerky crack growth in elastoplastic materials with localized plasticity %A Gianni Dal Maso %A Luca Heltai %B Journal of Convex Analysis %G eng %U https://arxiv.org/abs/2004.12705 %0 Journal Article %J Numerische Mathematik %D 2020 %T A priori error estimates of regularized elliptic problems %A Luca Heltai %A Wenyu Lei %B Numerische Mathematik %G eng %0 Journal Article %J International Journal of Computational Fluid Dynamics %D 2020 %T Reduced Basis Model Order Reduction for Navier-Stokes equations in domains with walls of varying curvature %A Hess, Martin %A Quaini, Annalisa %A Rozza, Gianluigi %XWe 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.

%B International Journal of Computational Fluid Dynamics %V 34 %P 119-126 %G eng %U https://arxiv.org/abs/1901.03708 %R 10.1080/10618562.2019.1645328 %0 Conference Paper %B IUTAM Symposium on Model Order Reduction of Coupled Systems, Stuttgart, Germany, May 22–25, 2018 %D 2020 %T A Reduced Order Approach for the Embedded Shifted Boundary FEM and a Heat Exchange System on Parametrized Geometries %A Efthymios N. Karatzas %A Giovanni Stabile %A Nabib Atallah %A Guglielmo Scovazzi %A Gianluigi Rozza %E Fehr, Jörg %E Haasdonk, Bernard %XA 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.

%B IUTAM Symposium on Model Order Reduction of Coupled Systems, Stuttgart, Germany, May 22–25, 2018 %I Springer International Publishing %G eng %U https://arxiv.org/abs/1807.07753 %R 10.1007/978-3-030-21013-7_8 %0 Report %D 2020 %T Twisted Ehresmann Schauenburg bialgebroids %A Xiao Han %X We construct an invertible normalised 2 cocycle on the Ehresmann Schauenburg bialgebroid of a cleft Hopf Galois extension under the condition that the corresponding Hopf algebra is cocommutative and the image of the unital cocycle corresponding to this cleft Hopf Galois extension belongs to the centre of the coinvariant subalgebra. Moreover, we show that any Ehresmann Schauenburg bialgebroid of this kind is isomorphic to a 2-cocycle twist of the Ehresmann Schauenburg bialgebroid corresponding to a Hopf Galois extension without cocycle, where comodule algebra is an ordinary smash product of the coinvariant subalgebra and the Hopf algebra (i.e. $\C(B/#_{\sigma}H, H)\simeq \C(B\#H, H)^{\tilde{\sigma}}$). We also study the theory in the case of a Galois object where the base is trivial but without requiring the Hopf algebra to be cocommutative. %8 09/2020 %G eng %U https://arxiv.org/abs/2009.02764 %0 Journal Article %J Journal of Numerical Mathematics %D 2019 %T The deal.II Library, Version 9.1 %A Arndt, Daniel %A Bangerth, Wolfgang %A Clevenger, Thomas C. %A Davydov, Denis %A Fehling, Marc %A Garcia-Sanchez, Daniel %A Harper, Graham %A Heister, Timo %A Heltai, Luca %A Kronbichler, Martin %A Maguire Kynch, Ross %A Maier, Matthias %A Pelteret, Jean Paul %A Turcksin, Bruno %A Wells, David %X This paper provides an overview of the new features of the finite element library deal.II, version 9.1. %B Journal of Numerical Mathematics %G eng %R 10.1515/jnma-2019-0064 %0 Unpublished Work %D 2019 %T Discontinuous Galerkin Model Order Reduction of Geometrically Parametrized Stokes Equation %A Nirav Vasant Shah %A Martin Hess %A Gianluigi Rozza %XThe 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.

%G eng %U https://arxiv.org/abs/1912.09787 %0 Journal Article %J Computers & Mathematics with Applications %D 2019 %T Error estimates in weighted Sobolev norms for finite element immersed interface methods %A Luca Heltai %A Nella Rotundo %B Computers & Mathematics with Applications %I Elsevier BV %V 78 %P 3586–3604 %G eng %U https://doi.org/10.1016/j.camwa.2019.05.029 %R 10.1016/j.camwa.2019.05.029 %0 Journal Article %J Computer Methods in Applied Mechanics and Engineering %D 2019 %T A Localized Reduced-Order Modeling Approach for PDEs with Bifurcating Solutions %A Hess, Martin %A Alla, Alessandro %A Quaini, Annalisa %A Rozza, Gianluigi %A Gunzburger, Max %XReduced-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.

%B Computer Methods in Applied Mechanics and Engineering %V 351 %P 379-403 %G eng %U https://arxiv.org/abs/1807.08851 %R 10.1016/j.cma.2019.03.050 %0 Journal Article %J International Journal for Numerical Methods in Biomedical Engineering %D 2019 %T Multiscale modeling of vascularized tissues via non-matching immersed methods %A Luca Heltai %A Alfonso Caiazzo %B International Journal for Numerical Methods in Biomedical Engineering %V 35 %P e3264 %G eng %U https://doi.org/10.1002/cnm.3264 %0 Book Section %B Numerical Mathematics and Advanced Applications - ENUMATH 2017 %D 2019 %T A Spectral Element Reduced Basis Method in Parametric CFD %A Hess, Martin W. %A Rozza, Gianluigi %E Radu, Florin Adrian %E Kumar, Kundan %E Berre, Inga %E Nordbotten, Jan Martin %E Pop, Iuliu Sorin %XWe consider the Navier-Stokes equations in a channel with varying Reynolds numbers. The model is discretized with high-order spectral element ansatz functions, resulting in 14 259 degrees of freedom. The steady-state snapshot solu- tions define a reduced order space, which allows to accurately evaluate the steady- state solutions for varying Reynolds number with a reduced order model within a fixed-point iteration. In particular, we compare different aspects of implementing the reduced order model with respect to the use of a spectral element discretization. It is shown, how a multilevel static condensation in the pressure and velocity boundary degrees of freedom can be combined with a reduced order modelling approach to enhance computational times in parametric many-query scenarios.

%B Numerical Mathematics and Advanced Applications - ENUMATH 2017 %I Springer International Publishing %V 126 %G eng %U https://arxiv.org/abs/1712.06432 %& A Spectral Element Reduced Basis Method in Parametric CFD %R 10.1007/978-3-319-96415-7_64 pages = 693–701 %0 Journal Article %J NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS %D 2018 %T Accelerating the iterative solution of convection-diffusion problems using singular value decomposition %A Giuseppe Pitton %A Luca Heltai %B NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS %P 1–21 %G eng %U https://arxiv.org/abs/1807.09467 %R 10.1002/nla.2211 %0 Journal Article %J JOURNAL OF INDUSTRIAL TEXTILES %D 2018 %T An authenticated theoretical modeling of electrified fluid jet in core–shell nanofibers production %A Rafiei, S. %A Noroozi, B. %A Luca Heltai %A Haghi, A. K. %B JOURNAL OF INDUSTRIAL TEXTILES %V 47 %P 1791–1811 %G eng %R 10.1177/1528083717710711 %0 Journal Article %J Journal of Scientific Computing %D 2018 %T Certified Reduced Basis Approximation for the Coupling of Viscous and Inviscid Parametrized Flow Models %A Immanuel Martini %A Bernard Haasdonk %A Gianluigi Rozza %B Journal of Scientific Computing %V 74 %P 197-219 %G eng %U https://www.scopus.com/inward/record.uri?eid=2-s2.0-85017156114&doi=10.1007%2fs10915-017-0430-y&partnerID=40&md5=023ef0bb95713f4442d1fa374c92a964 %R 10.1007/s10915-017-0430-y %0 Journal Article %J SOFTWAREX %D 2018 %T deal2lkit: A toolkit library for high performance programming in deal.II %A Alberto Sartori %A Nicola Giuliani %A Mauro Bardelloni %A Luca Heltai %B SOFTWAREX %V 7 %P 318–327 %G eng %R 10.1016/j.softx.2018.09.004 %0 Journal Article %J JOURNAL OF NUMERICAL MATHEMATICS %D 2018 %T The deal.II Library, Version 9.0 %A Giovanni Alzetta %A Arndt, Daniel %A W. Bangerth %A Boddu, Vishal %A Brands, Benjamin %A Denis Davydov %A Gassmöller, Rene %A Timo Heister %A Luca Heltai %A Kormann, Katharina %A Martin Kronbichler %A Matthias Maier %A Pelteret, Jean-Paul %A B. Turcksin %A David Wells %B JOURNAL OF NUMERICAL MATHEMATICS %G eng %U https://doi.org/10.1515/jnma-2018-0054 %R 10.1515/jnma-2018-0054 %0 Book Section %B Mathematical and Numerical Modeling of the Cardiovascular System and Applications %D 2018 %T A distributed lagrange formulation of the finite element immersed boundary method for fluids interacting with compressible solids %A Boffi, Daniele %A Gastaldi, Lucia %A Luca Heltai %B Mathematical and Numerical Modeling of the Cardiovascular System and Applications %I Springer International Publishing %C Cham %V 16 %P 1–21 %G eng %U https://arxiv.org/abs/1712.02545v1 %R 10.1007/978-3-319-96649-6_1 %0 Generic %D 2018 %T The Effort of Increasing Reynolds Number in Projection-Based Reduced Order Methods: from Laminar to Turbulent Flows %A Saddam Hijazi %A Shafqat Ali %A Giovanni Stabile %A Francesco Ballarin %A Gianluigi Rozza %G eng %0 Journal Article %J ASTRONOMY & ASTROPHYSICS %D 2018 %T Iterative map-making with two-level preconditioning for polarized cosmic microwave background data sets. A worked example for ground-based experiments %A Puglisi, Giuseppe %A Poletti, Davide %A Fabbian, Giulio %A Baccigalupi, Carlo %A Luca Heltai %A Stompor, Radek %B ASTRONOMY & ASTROPHYSICS %V 618 %P 1–14 %G eng %U https://arxiv.org/abs/1801.08937 %R 10.1051/0004-6361/201832710 %0 Journal Article %J COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING %D 2018 %T NURBS-SEM: A hybrid spectral element method on NURBS maps for the solution of elliptic PDEs on surfaces %A Giuseppe Pitton %A Luca Heltai %B COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING %V 338 %P 440–462 %G eng %U https://arxiv.org/abs/1804.08271 %R 10.1016/j.cma.2018.04.039 %0 Journal Article %J SOFT ROBOTICS %D 2018 %T Predicting and Optimizing Microswimmer Performance from the Hydrodynamics of Its Components: The Relevance of Interactions %A Nicola Giuliani %A Luca Heltai %A Antonio DeSimone %B SOFT ROBOTICS %V 5 %P 410–424 %G eng %U https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6094362/ %R 10.1089/soro.2017.0099 %0 Journal Article %J Reviews in Mathematical Physics %D 2018 %T Principal fibrations over noncommutative spheres %A Michel Dubois-Violette %A Xiao Han %A Giovanni Landi %X We present examples of noncommutative four-spheres that are base spaces of $SU(2)$-principal bundles with noncommutative seven-spheres as total spaces. The noncommutative coordinate algebras of the four-spheres are generated by the entries of a projection which is invariant under the action of $SU(2)$. We give conditions for the components of the Connes–Chern character of the projection to vanish but the second (the top) one. The latter is then a non-zero Hochschild cycle that plays the role of the volume form for the noncommutative four-spheres. %B Reviews in Mathematical Physics %V 30 %P 1850020 %G eng %U https://arxiv.org/abs/1804.07032 %R 10.1142/S0129055X18500204 %0 Book Section %B Numerical Methods for PDEs %D 2018 %T Reduced Basis Approximation and A Posteriori Error Estimation: Applications to Elasticity Problems in Several Parametric Settings %A Huynh, D. B. P. %A Pichi, Federico %A Rozza, Gianluigi %B Numerical Methods for PDEs %V 15 %G eng %U https://link.springer.com/chapter/10.1007/978-3-319-94676-4_8 %R https://doi.org/10.1007/978-3-319-94676-4_8 %0 Journal Article %J Advances in Engineering Software %D 2018 %T π-BEM : A flexible parallel implementation for adaptive , geometry aware , and high order boundary element methods %A Nicola Giuliani %A Andrea Mola %A Luca Heltai %B Advances in Engineering Software %V 121 %P 39–58 %G eng %0 Journal Article %J Communication in Applied Industrial Mathematics %D 2017 %T Advances in Reduced order modelling for CFD: vortex shedding around a circular cylinder using a POD-Galerkin method %A Giovanni Stabile %A Saddam Hijazi %A Stefano Lorenzi %A Andrea Mola %A Gianluigi Rozza %K finite volume, CFD %K Reduced order methods %XVortex shedding around circular cylinders is a well known and studied phenomenon that appears in many engineering fields. In this work a Reduced Order Model (ROM) of the incompressible flow around a circular cylinder, built performing a Galerkin projection of the governing equations onto a lower dimensional space is presented. The reduced basis space is generated using a Proper Orthogonal Decomposition (POD) approach. In particular the focus is into (i) the correct reproduction of the pressure field, that in case of the vortex shedding phenomenon, is of primary importance for the calculation of the drag and lift coefficients; (ii) for this purpose the projection of the Governing equations (momentum equation and Poisson equation for pressure) is performed onto different reduced basis space for velocity and pressure, respectively; (iii) all the relevant modifications necessary to adapt standard finite element POD-Galerkin methods to a finite volume framework are presented. The accuracy of the reduced order model is assessed against full order results.

%B Communication in Applied Industrial Mathematics %G eng %U https://arxiv.org/abs/1701.03424 %9 reviewed %0 Journal Article %J JOURNAL OF NUMERICAL MATHEMATICS %D 2017 %T The deal.II Library, Version 8.5 %A Arndt, Daniel %A W. Bangerth %A Denis Davydov %A Timo Heister %A Luca Heltai %A Martin Kronbichler %A Matthias Maier %A Pelteret, Jean-Paul %A B. Turcksin %A David Wells %B JOURNAL OF NUMERICAL MATHEMATICS %V 25 %P 137–145 %G eng %U https://www.dealii.org/deal85-preprint.pdf %R 10.1515/jnma-2017-0058 %0 Book Section %B Encyclopedia of Computational Mechanics Second Edition %D 2017 %T Model Reduction Methods %A Francisco Chinesta %A Antonio Huerta %A Gianluigi Rozza %A Karen Willcox %XThis chapter presents an overview of model order reduction – a new paradigm in the field of simulation-based engineering sciences, and one that can tackle the challenges and leverage the opportunities of modern ICT technologies. Despite the impressive progress attained by simulation capabilities and techniques, a number of challenging problems remain intractable. These problems are of different nature, but are common to many branches of science and engineering. Among them are those related to high-dimensional problems, problems involving very different time scales, models defined in degenerate domains with at least one of the characteristic dimensions much smaller than the others, model requiring real-time simulation, and parametric models. All these problems represent a challenge for standard mesh-based discretization techniques; yet the ability to solve these problems efficiently would open unexplored routes for real-time simulation, inverse analysis, uncertainty quantification and propagation, real-time optimization, and simulation-based control – critical needs in many branches of science and engineering. Model order reduction offers new simulation alternatives by circumventing, or at least alleviating, otherwise intractable computational challenges. In the present chapter, we revisit three of these model reduction techniques: proper orthogonal decomposition, proper generalized decomposition, and reduced basis methodologies.} preprint = {http://preprints.sissa.it/xmlui/bitstream/handle/1963/35194/ECM_MOR.pdf?sequence=1&isAllowed=y

%B Encyclopedia of Computational Mechanics Second Edition %I John Wiley & Sons %P 1-36 %G eng %& Model Reduction Methods %R 10.1002/9781119176817.ecm2110 %0 Journal Article %J COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING %D 2017 %T A natural framework for isogeometric fluid-structure interaction based on BEM-shell coupling %A Luca Heltai %A Kiendl, J. %A Antonio DeSimone %A Alessandro Reali %B COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING %V 316 %P 522–546 %G eng %U http://cdsads.u-strasbg.fr/abs/2017CMAME.316..522H %R 10.1016/j.cma.2016.08.008 %0 Journal Article %J Communications in Applied and Industrial Mathematics %D 2017 %T POD-Galerkin reduced order methods for CFD using Finite Volume Discretisation: vortex shedding around a circular cylinder %A Giovanni Stabile %A Saddam Hijazi %A Andrea Mola %A Stefano Lorenzi %A Gianluigi Rozza %B Communications in Applied and Industrial Mathematics %I Walter de Gruyter {GmbH} %V 8 %P 210–236 %8 dec %G eng %U https://doi.org/10.1515/caim-2017-0011 %R 10.1515/caim-2017-0011 %0 Report %D 2017 %T Time quasi-periodic gravity water waves in finite depth %A P Baldi %A Massimiliano Berti %A Emanuele Haus %A Riccardo Montalto %X We prove the existence and the linear stability of Cantor families of small amplitude time quasi-periodic standing water wave solutions - namely periodic and even in the space variable x - of a bi-dimensional ocean with finite depth under the action of pure gravity. Such a result holds for all the values of the depth parameter in a Borel set of asymptotically full measure. This is a small divisor problem. The main difficulties are the quasi-linear nature of the gravity water waves equations and the fact that the linear frequencies grow just in a sublinear way at infinity. We overcome these problems by first reducing the linearized operators obtained at each approximate quasi-periodic solution along the Nash-Moser iteration to constant coefficients up to smoothing operators, using pseudo-differential changes of variables that are quasi-periodic in time. Then we apply a KAM reducibility scheme which requires very weak Melnikov non-resonance conditions (losing derivatives both in time and space), which we are able to verify for most values of the depth parameter using degenerate KAM theory arguments. %G en %U http://preprints.sissa.it/handle/1963/35296 %1 35602 %2 Mathematics %$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2017-09-27T15:46:10Z No. of bitstreams: 1 1708.01517.pdf: 1909445 bytes, checksum: 71fc741593666cacf41187c00b092503 (MD5) %0 Journal Article %J Journal of Ship Research %D 2017 %T Wet and Dry Transom Stern Treatment for Unsteady and Nonlinear Potential Flow Model for Naval Hydrodynamics Simulations %A Andrea Mola %A Luca Heltai %A Antonio DeSimone %XWe present a model for the fast evaluation of the total drag of ship hulls operating in both wet and dry transom stern conditions, in calm or wavy water, based on the combination of an unsteady semi-Lagrangian potential flow formulation with fully nonlinear free-surface treatment, experimental correlations, and simplified viscous drag modeling. The implementation is entirely based on open source libraries. The spatial discretization is solved using a streamline upwind Petrov‐Galerkin stabilization of an iso-parametric, collocation based, boundary element method, implemented using the open source library deal.II. The resulting nonlinear differential-algebraic system is integrated in time using implicit backward differentiation formulas, implemented in the open source library SUNDIALS. The Open CASCADE library is used to interface the model directly with computer-aided design data structures. The model accounts automatically for hulls with a transom stern, both in wet and dry regimes, by using a specific treatment of the free-surface nodes on the stern edge that automatically detects when the hull advances at low speeds. In this case, the transom stern is partially immersed, and a pressure patch is applied on the water surface detaching from the transom stern, to recover the gravity effect of the recirculating water on the underlying irrotational flow domain. The parameters of the model used to impose the pressure patch are approximated from experimental relations found in the literature. The test cases considered are those of the U.S. Navy Combatant DTMB-5415 and the National Physical Laboratory hull. Comparisons with experimental data on quasi-steady test cases for both water elevation and total hull drag are presented and discussed. The quality of the results obtained on quasi-steady simulations suggests that this model can represent a promising alternative to current unsteady solvers for simulations with Froude numbers below 0.35.

%B Journal of Ship Research %V 61 %P 1–14 %G eng %R https://doi.org/10.5957/JOSR.61.1.160016 %0 Journal Article %J ARCHIVE OF NUMERICAL SOFTWARE %D 2016 %T The deal.II Library, Version 8.3 %A W. Bangerth %A Timo Heister %A Luca Heltai %A G. Kanschat %A Martin Kronbichler %A Matthias Maier %A B. Turcksin %B ARCHIVE OF NUMERICAL SOFTWARE %V 4 %P 1–11 %G eng %U http://nbn-resolving.de/urn:nbn:de:bsz:16-ans-231226 %R 10.11588/ans.2016.100.23122 %0 Journal Article %J JOURNAL OF NUMERICAL MATHEMATICS %D 2016 %T The deal.II library, Version 8.4 %A W. Bangerth %A Denis Davydov %A Timo Heister %A Luca Heltai %A G. Kanschat %A Martin Kronbichler %A Matthias Maier %A B. Turcksin %A David Wells %B JOURNAL OF NUMERICAL MATHEMATICS %V 24 %P 135–141 %G eng %U https://www.math.clemson.edu/ heister/preprints/deal84-preprint.pdf %R 10.1515/jnma-2016-1045 %0 Journal Article %J JOURNAL OF SCIENTIFIC COMPUTING %D 2016 %T Error Estimates of B-spline based finite-element method for the wind-driven ocean circulation %A Rotundo, N. %A Kim, T. -Y. %A Jiang, W. %A Luca Heltai %A Fried, E. %B JOURNAL OF SCIENTIFIC COMPUTING %V 69 %P 430–459 %G eng %R 10.1007/s10915-016-0201-1 %0 Report %D 2016 %T Isogeometric analysis-based reduced order modelling for incompressible linear viscous flows in parametrized shapes %A Filippo Salmoiraghi %A Francesco Ballarin %A Luca Heltai %A Gianluigi Rozza %X In this work we provide a combination of isogeometric analysis with reduced order modelling techniques, based on proper orthogonal decomposition, to guarantee computational reduction for the numerical model, and with free-form deformation, for versatile geometrical parametrization. We apply it to computational fluid dynamics problems considering a Stokes flow model. The proposed reduced order model combines efficient shape deformation and accurate and stable velocity and pressure approximation for incompressible viscous flows, computed with a reduced order method. Efficient offine-online computational decomposition is guaranteed in view of repetitive calculations for parametric design and optimization problems. Numerical test cases show the efficiency and accuracy of the proposed reduced order model. %I Springer, AMOS Advanced Modelling and Simulation in Engineering Sciences %G en %U http://urania.sissa.it/xmlui/handle/1963/35199 %1 35493 %2 Mathematics %4 1 %# MAT/08 %$ Submitted by Gianluigi Rozza (grozza@sissa.it) on 2016-06-20T23:55:25Z No. of bitstreams: 1 MAIN_IGA_FFD_ROM.pdf: 1631282 bytes, checksum: b050e472f11943b95d8307915a50b6da (MD5) %0 Journal Article %J COMPUTERS & MATHEMATICS WITH APPLICATIONS %D 2016 %T LinearOperator – a generic, high-level expression syntax for linear algebra %A Matthias Maier %A Mauro Bardelloni %A Luca Heltai %B COMPUTERS & MATHEMATICS WITH APPLICATIONS %V 72 %P 1–24 %G eng %R 10.1016/j.camwa.2016.04.024 %0 Book Section %B Wiley Encyclopedia of Computational Mechanics, 2016 %D 2016 %T Model Order Reduction: a survey %A Francisco Chinesta %A Antonio Huerta %A Gianluigi Rozza %A Karen Willcox %B Wiley Encyclopedia of Computational Mechanics, 2016 %I Wiley %G en %U http://urania.sissa.it/xmlui/handle/1963/35194 %1 35470 %2 Mathematics %4 1 %# MAT/08 %$ Submitted by Gianluigi Rozza (grozza@sissa.it) on 2016-05-27T00:41:12Z No. of bitstreams: 1 ECM_MOR.pdf: 746542 bytes, checksum: 93d6252fe8a175c4378b96bd4192712c (MD5) %0 Conference Paper %B The 26th International Ocean and Polar Engineering Conference %D 2016 %T Ship Sinkage and Trim Predictions Based on a CAD Interfaced Fully Nonlinear Potential Model %A Andrea Mola %A Luca Heltai %A Antonio DeSimone %A Massimiliano Berti %B The 26th International Ocean and Polar Engineering Conference %I International Society of Offshore and Polar Engineers %V 3 %P 511–518 %G eng %0 Journal Article %J Phys. Rev. D %D 2016 %T Towards a gauge theory interpretation of the real topological string %A Hayashi, Hirotaka %A Nicolò Piazzalunga %A Uranga, Angel M. %XWe consider the real topological string on certain noncompact toric Calabi-Yau three-folds $\mathbb{X}$, in its physical realization describing an orientifold of type IIA on $\mathbb{X}$ with an O4-plane and a single D4-brane stuck on top. The orientifold can be regarded as a new kind of surface operator on the gauge theory with 8 supercharges arising from the singular geometry. We use the M-theory lift of this system to compute the real Gopakumar-Vafa invariants (describing wrapped M2-brane Bogomol’nyi-Prasad-Sommerfield (BPS) states) for diverse geometries. We show that the real topological string amplitudes pick up certain signs across flop transitions, in a well-defined pattern consistent with continuity of the real BPS invariants. We further give some preliminary proposals of an intrinsically gauge theoretical description of the effect of the surface operator in the gauge theory partition function.

%B Phys. Rev. D %I American Physical Society %V 93 %P 066001 %8 Mar %G eng %U https://link.aps.org/doi/10.1103/PhysRevD.93.066001 %R 10.1103/PhysRevD.93.066001 %0 Journal Article %J Computers and Mathematics with Applications 69 (2015) 1167–1188 %D 2015 %T Benchmarking the Immersed Finite Element Method for Fluid-Structure Interaction Problems %A Roy Saswati %A Luca Heltai %A Francesco Costanzo %X We present an implementation of a fully variational formulation of an immersed methods for fluid-structure interaction problems based on the finite element method. While typical implementation of immersed methods are characterized by the use of approximate Dirac delta distributions, fully variational formulations of the method do not require the use of said distributions. In our implementation the immersed solid is general in the sense that it is not required to have the same mass density and the same viscous response as the surrounding fluid. We assume that the immersed solid can be either viscoelastic of differential type or hyperelastic. Here we focus on the validation of the method via various benchmarks for fluid-structure interaction numerical schemes. This is the first time that the interaction of purely elastic compressible solids and an incompressible fluid is approached via an immersed method allowing a direct comparison with established benchmarks. %B Computers and Mathematics with Applications 69 (2015) 1167–1188 %G en %1 34633 %2 Mathematics %4 1 %# MAT/08 %$ Submitted by Luca Heltai (heltai@sissa.it) on 2015-04-20T15:26:58Z No. of bitstreams: 1 rhc_feibm_validation-v3.pdf: 7039131 bytes, checksum: 705f28e17037504f00766fc4e84a10a4 (MD5) %R 10.1016/j.camwa.2015.03.012 %0 Book %B Springer Briefs in Mathematics %D 2015 %T Certified Reduced Basis Methods for Parametrized Partial Differential Equations %A Jan S Hesthaven %A Gianluigi Rozza %A Benjamin Stamm %K a posteriori error bounds %K empirical interpolation %K parametrized partial differential equations %K reduced basis methods, greedy algorithms %XThis book provides a thorough introduction to the mathematical and algorithmic aspects of certified reduced basis methods for parametrized partial differential equations. Central aspects ranging from model construction, error estimation and computational efficiency to empirical interpolation methods are discussed in detail for coercive problems. More advanced aspects associated with time-dependent problems, non-compliant and non-coercive problems and applications with geometric variation are also discussed as examples.

%B Springer Briefs in Mathematics %7 1 %I Springer %C Switzerland %P 135 %@ 978-3-319-22469-5 %G eng %6 1 %0 Journal Article %D 2015 %T Deal2lkit: a Toolkit Library for High Performance Programming in deal.II %A Alberto Sartori %A Nicola Giuliani %A Mauro Bardelloni %A Luca Heltai %X We present version 1.0.0 of the deal2lkit (deal.II ToolKit) library. deal2lkit is a collection of modules and classes for the general purpose finite element library deal.II. Its principal aim is to provide a high level interface, controlled via parameter files, for those steps that are common in all finite element programs: mesh generation, selection of the finite element type, application of boundary conditions and many others. Each module can be used as a building block independently on the others, and can be integrated in existing finite element codes based on deal.II, drastically reducing the size of programs, rendering their use automatically parametrised, and reducing the overall time-to-market of finite element programming. Moreover, deal2lkit features interfaces with the SUNDIALS (SUite of Nonlinear and DIfferential/ALgebraic equation Solvers) and ASSIMP (Open Asset Import Library) libraries. Some examples are provided which show the aim and scopes of deal2lkit. The deal2lkit library is released under the GNU Lesser General Public License (LGPL) and can be retrieved from the deal2lkit repository https://github.com/mathLab/deal2lkit. %I SISSA %G en %U http://urania.sissa.it/xmlui/handle/1963/35006 %1 35235 %2 Mathematics %4 1 %# MAT/08 %$ Submitted by asartori@sissa.it (asartori@sissa.it) on 2015-11-13T13:12:29Z No. of bitstreams: 1 paper_deal2lkit.pdf: 1085486 bytes, checksum: 1dc7e9b9790fe38d2ced7b5328633cd8 (MD5) %0 Journal Article %J Archive of Numerical Software, vol. 3, n. 100, (2015), pages : 1-8 %D 2015 %T The deal.II Library, Version 8.2 %A W. Bangerth %A Timo Heister %A Luca Heltai %A G. Kanschat %A Martin Kronbichler %A Matthias Maier %A B. Turcksin %A T. D. Young %X This paper provides an overview of the new features of the finite element library deal.II version 8.2 %B Archive of Numerical Software, vol. 3, n. 100, (2015), pages : 1-8 %G en %U http://urania.sissa.it/xmlui/handle/1963/34464 %1 34637 %2 Mathematics %4 1 %# MAT/08 %$ Submitted by Luca Heltai (heltai@sissa.it) on 2015-04-20T15:40:46Z No. of bitstreams: 1 18031-44567-1-PB (1).pdf: 137014 bytes, checksum: 9a5fddeccfa389c32ca89a466e653074 (MD5) %R 10.11588/ans.2015.100.18031 %0 Journal Article %J Engineering Analysis with Boundary Elements 59 (2015), pp. 8-22 %D 2015 %T FEM SUPG stabilisation of mixed isoparametric BEMs: application to linearised free surface flows %A Nicola Giuliani %A Andrea Mola %A Luca Heltai %A L. Formaggia %XIn finite element formulations, transport dominated problems are often stabilised through the Streamline-Upwind-Petrov–Galerkin (SUPG) method. Its application is straightforward when the problem at hand is solved using Galerkin methods. Applications of boundary integral formulations often resort to collocation techniques which are computationally more tractable. In this framework, the Galerkin method and the stabilisation may still be used to successfully apply boundary conditions and resolve instabilities that are frequently observed in transport dominated problems. We apply this technique to an adaptive collocation boundary element method for the solution of stationary potential flows, where we solve a mixed Poisson problem in boundary integral form, with the addition of linearised free surface boundary conditions. We use a mixed boundary element formulation to allow for different finite dimensional spaces describing the flow potential and its normal derivative, and we validate our method simulating the flow around both a submerged body and a surface piercing body. The coupling of mixed surface finite elements and strongly consistent stabilisation techniques with boundary elements opens up the possibility to use non conformal unstructured grids with local refinement, without introducing the inconsistencies of other stabilisation techniques based on up-winding and finite difference schemes.

%B Engineering Analysis with Boundary Elements 59 (2015), pp. 8-22 %G en_US %U http://urania.sissa.it/xmlui/handle/1963/34466 %1 34640 %2 Mathematics %4 1 %# MAT/08 %$ Submitted by ngiuliani@sissa.it (ngiuliani@sissa.it) on 2015-05-04T12:32:04Z No. of bitstreams: 1 paper-free-surface.pdf: 888763 bytes, checksum: 68366266786e84b78a356acd6de50840 (MD5) %R 10.1016/j.enganabound.2015.04.006 %0 Journal Article %J Advances in Computational Mathematics %D 2015 %T Reduced basis approximation and a-posteriori error estimation for the coupled Stokes-Darcy system %A Immanuel Martini %A Gianluigi Rozza %A Bernard Haasdonk %K Domain decomposition %K Error estimation %K Non-coercive problem %K Porous medium equation %K Reduced basis method %K Stokes flow %XThe coupling of a free flow with a flow through porous media has many potential applications in several fields related with computational science and engineering, such as blood flows, environmental problems or food technologies. We present a reduced basis method for such coupled problems. The reduced basis method is a model order reduction method applied in the context of parametrized systems. Our approach is based on a heterogeneous domain decomposition formulation, namely the Stokes-Darcy problem. Thanks to an offline/online-decomposition, computational times can be drastically reduced. At the same time the induced error can be bounded by fast evaluable a-posteriori error bounds. In the offline-phase the proposed algorithms make use of the decomposed problem structure. Rigorous a-posteriori error bounds are developed, indicating the accuracy of certain lifting operators used in the offline-phase as well as the accuracy of the reduced coupled system. Also, a strategy separately bounding pressure and velocity errors is extended. Numerical experiments dealing with groundwater flow scenarios demonstrate the efficiency of the approach as well as the limitations regarding a-posteriori error estimation.

%B Advances in Computational Mathematics %V special issue for MoRePaS 2012 %G eng %N in press %R 10.1007/s10444-014-9396-6 %0 Journal Article %J Comput Methods Appl Mech Eng. 2015;284:1147–1180 %D 2015 %T Reduced Basis Isogeometric Methods (RB-IGA) for the real-time simulation of potential flows about parametrized NACA airfoils %A Andrea Manzoni %A Filippo Salmoiraghi %A Luca Heltai %X We present a Reduced Basis (RB) method based on Isogeometric Analysis (IGA) for the rapid and reliable evaluation of PDE systems characterized by complex geometrical features. At the current state of the art, this is the first case of coupling between RB and IGA methods. The construction of the RB method relies on an Isogeometric Boundary Element Method (IGA-BEM) as the high-fidelity technique, allowing a direct interface with Computer Aided Design (CAD) tools. A suitable Empirical Interpolation Method (EIM) ensures an efficient offline/online decomposition between the construction and the evaluation of the RB method. We consider the real-time simulation of potential flows past airfoils, parametrized with respect to the angle of attack and the NACA number identifying their shape, and we provide a validation of our methodology with respect to experimental data and reference numerical codes, showing in both cases a very good agreement.We present a Reduced Basis (RB) method based on Isogeometric Analysis (IGA) for the rapid and reliable evaluation of PDE systems characterized by complex geometrical features. At the current state of the art, this is the first case of coupling between RB and IGA methods. The construction of the RB method relies on an Isogeometric Boundary Element Method (IGA-BEM) as the high-fidelity technique, allowing a direct interface with Computer Aided Design (CAD) tools. A suitable Empirical Interpolation Method (EIM) ensures an efficient offline/online decomposition between the construction and the evaluation of the RB method. We consider the real-time simulation of potential flows past airfoils, parametrized with respect to the angle of attack and the NACA number identifying their shape, and we provide a validation of our methodology with respect to experimental data and reference numerical codes, showing in both cases a very good agreement. %B Comput Methods Appl Mech Eng. 2015;284:1147–1180 %G en %1 34587 %2 Mathematics %4 1 %# MAT/08 %$ Submitted by Luca Heltai (heltai@sissa.it) on 2015-03-03T16:15:31Z No. of bitstreams: 1 RB_IGA_BEM.pdf: 725470 bytes, checksum: e2fe6fabfdbe6118e9ef07fb185113af (MD5) %R 10.1016/j.cma.2014.11.037 %0 Journal Article %D 2014 %T Finite dimensional Kadomtsev-Petviashvili τ-functions. I. Finite Grassmannians %A Ferenc Balogh %A Tiago Fonseca %A John P. Harnad %X We study τ-functions of the Kadomtsev-Petviashvili hierarchy in terms of abelian group actions on finite dimensional Grassmannians, viewed as subquotients of the Hilbert space Grassmannians of Sato, Segal, and Wilson. A determinantal formula of Gekhtman and Kasman involving exponentials of finite dimensional matrices is shown to follow naturally from such reductions. All reduced flows of exponential type generated by matrices with arbitrary nondegenerate Jordan forms are derived, both in the Grassmannian setting and within the fermionic operator formalism. A slightly more general determinantal formula involving resolvents of the matrices generating the flow, valid on the big cell of the Grassmannian, is also derived. An explicit expression is deduced for the Plücker coordinates appearing as coefficients in the Schur function expansion of the τ-function. %I American Institute of Physics Inc. %G en %U http://urania.sissa.it/xmlui/handle/1963/34952 %1 35153 %2 Mathematics %4 1 %$ Submitted by gfeltrin@sissa.it (gfeltrin@sissa.it) on 2015-11-04T14:31:01Z No. of bitstreams: 1 preprint2014.pdf: 403949 bytes, checksum: 01d5c482538ca09d858c0586f648b196 (MD5) %R 10.1063/1.4890818 %0 Conference Proceedings %B Proceedings of the 24th International Ocean and Polar Engineering Conference, Busan, 2014 %D 2014 %T A fully nonlinear potential model for ship hydrodynamics directly interfaced with CAD data structures %A Andrea Mola %A Luca Heltai %A Antonio DeSimone %K ship hydrodynamics %X We present a model for ship hydrodynamics simulations currently under development at SISSA. The model employs potential flow theory and fully nonlinear free surface boundary conditions. The spatial discretization of the equations is performed by means of a collocation BEM. This gives rise to a Differential Algbraic Equations (DAE) system, solved using an implicit BDF scheme to time advance the solution. The model has been implemented into a C++ software able to automatically generate the computational grids from the CAD geometry of the hull. Numerical results on Kriso KCS and KVLCC2 hulls are presented and discussed. %B Proceedings of the 24th International Ocean and Polar Engineering Conference, Busan, 2014 %I SISSA %G en %1 7357 %2 Mathematics %4 1 %# MAT/08 ANALISI NUMERICA %$ Submitted by Luca Heltai (heltai@sissa.it) on 2014-03-22T15:00:37Z No. of bitstreams: 1 MolaHeltaiDeSimone2014.pdf: 1241007 bytes, checksum: 1a56d909a08f1f470c7efc14018e37f8 (MD5) %0 Journal Article %D 2014 %T Nonsingular Isogeometric Boundary Element Method for Stokes Flows in 3D %A Luca Heltai %A Marino Arroyo %A Antonio DeSimone %K Isogeometric Analysis %X Isogeometric analysis (IGA) is emerging as a technology bridging Computer Aided Geometric Design (CAGD), most commonly based on Non-Uniform Rational B-Splines (NURBS) surfaces, and engineering analysis. In finite element and boundary element isogeometric methods (FE-IGA and IGA-BEM), the NURBS basis functions that de- scribe the geometry define also the approximation spaces. In the FE-IGA approach, the surfaces generated by the CAGD tools need to be extended to volumetric descriptions, a major open problem in 3D. This additional passage can be avoided in principle when the partial differential equations to be solved admit a formulation in terms of bound- ary integral equations, leading to Boundary Element Isogeometric Analysis (IGA-BEM). The main advantages of such an approach are given by the dimensionality reduction of the problem (from volumetric-based to surface-based), by the fact that the interface with CAGD tools is direct, and by the possibility to treat exterior problems, where the computational domain is infinite. By contrast, these methods produce system matrices which are full, and require the integration of singular kernels. In this paper we address the second point and propose a nonsingular formulation of IGA-BEM for 3D Stokes flows, whose convergence is carefully tested numerically. Standard Gaussian quadrature rules suffice to integrate the boundary integral equations, and carefully chosen known exact solutions of the interior Stokes problem are used to correct the resulting matrices, extending the work by Klaseboer et al. [27] to IGA-BEM. %I Elsevier %G en %U http://hdl.handle.net/1963/6326 %1 6250 %2 Mathematics %4 1 %# MAT/08 ANALISI NUMERICA %$ Submitted by Luca Heltai (heltai@sissa.it) on 2012-12-03T08:19:28Z No. of bitstreams: 1 isogeo_stokes.pdf: 732320 bytes, checksum: 224ef57de6c49f03db0afaccdcb234f9 (MD5) %R 10.1016/j.cma.2013.09.017 %0 Conference Paper %B The 24th International Ocean and Polar Engineering Conference %D 2014 %T Potential Model for Ship Hydrodynamics Simulations Directly Interfaced with CAD Data Structures %A Andrea Mola %A Luca Heltai %A Antonio DeSimone %A Massimiliano Berti %B The 24th International Ocean and Polar Engineering Conference %I International Society of Offshore and Polar Engineers %V 4 %P 815–822 %G eng %0 Report %D 2013 %T The deal.II Library, Version 8.1 %A W. Bangerth %A Timo Heister %A Luca Heltai %A G. Kanschat %A Martin Kronbichler %A Matthias Maier %A B. Turcksin %A T. D. Young %X This paper provides an overview of the new features of the finite element library deal.II version 8.0. %I SISSA %G en %U http://hdl.handle.net/1963/7236 %1 7272 %2 Mathematics %4 1 %# MAT/08 ANALISI NUMERICA %$ Submitted by Luca Heltai (heltai@sissa.it) on 2013-12-10T10:43:08Z No. of bitstreams: 1 1312.2266v1.pdf: 245445 bytes, checksum: e00fcd6d7d9ac6e2076208958b07eef3 (MD5) %0 Journal Article %J Numerische Mathematik, 2013 %D 2013 %T Reduced basis approximation and a posteriori error estimation for Stokes flows in parametrized geometries: roles of the inf-sup stability constants %A Gianluigi Rozza %A Phuong Huynh %A Andrea Manzoni %K parametrized Stokes equations %X In this paper we review and we extend the reduced basis approximation and a posteriori error estimation for steady Stokes flows in a ffinely parametrized geometries, focusing on the role played by the Brezzi\\\'s and Babu ska\\\'s stability constants. The crucial ingredients of the methodology are a Galerkin projection onto a low-dimensional space of basis functions properly selected, an a ne parametric dependence enabling to perform competitive Off ine-Online splitting in the computational\\r\\nprocedure and a rigorous a posteriori error estimation on eld variables.\\r\\nThe combination of these three factors yields substantial computational savings which are at the basis of an e fficient model order reduction, ideally suited for real-time simulation and many-query contexts (e.g. optimization, control or parameter identi cation). In particular, in this work we focus on i) the stability of the reduced basis approximation based on the Brezzi\\\'s saddle point theory and the introduction of a supremizer operator on the pressure terms, ii) a rigorous a posteriori error estimation procedure for velocity and pressure elds based on the Babu ska\\\'s inf-sup constant (including residuals calculations), iii) the computation of a lower bound of the stability constant, and iv) di erent options for the reduced basis spaces construction. We present some illustrative results for both\\r\\ninterior and external steady Stokes flows in parametrized geometries representing two parametrized classical Poiseuille and Couette \\r\\nflows, a channel contraction and a simple flow control problem around a curved obstacle. %B Numerische Mathematik, 2013 %I Springer %G en %U http://hdl.handle.net/1963/6339 %1 6269 %2 Mathematics %4 1 %# MAT/08 ANALISI NUMERICA %$ Submitted by Gianluigi Rozza (grozza@sissa.it) on 2012-12-13T17:19:05Z\\nNo. of bitstreams: 1\\nStokes_HMR10_SISSA_report.pdf: 4641267 bytes, checksum: 82a163163a62d0b9deef8d5d6590f8d7 (MD5) %0 Journal Article %J Engineering Analysis with Boundary Elements, 37(1):128 – 143, 2013. %D 2013 %T A stable and adaptive semi-Lagrangian potential model for unsteady and nonlinear ship-wave interactions %A Andrea Mola %A Luca Heltai %A Antonio DeSimone %K Unsteady ship-wave interaction %XWe present an innovative numerical discretization of the equations of inviscid potential flow for the simulation of three dimensional unsteady and nonlinear water waves generated by a ship hull advancing in water. The equations of motion are written in a semi-Lagrangian framework, and the resulting integro-diff erential equations are discretized in space via an adaptive iso-parametric collocation Boundary Element Method, and in time via adaptive implicit Backward Di erentiation Formulas (BDF) with variable step and variable order. When the velocity of the advancing ship hull is non-negligible, the semi-Lagrangian formulation (also known as Arbitrary Lagrangian Eulerian formulation, or ALE) of the free surface equations contains dominant transport terms which are stabilized with a Streamwise Upwind Petrov-Galerkin (SUPG) method. The SUPG stabilization allows automatic and robust adaptation of the spatial discretization with unstructured quadrilateral grids. Preliminary results are presented where we compare our numerical model with experimental results on the case of a Wigley hull advancing in calm water with fi xed sink and trim.

%B Engineering Analysis with Boundary Elements, 37(1):128 – 143, 2013. %I SISSA %G en %U http://hdl.handle.net/1963/5669 %1 5457 %2 Mathematics %3 Functional Analysis and Applications %4 -1 %$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2012-03-16T11:19:51Z\\nNo. of bitstreams: 1\\nMolaHeltaiDeSimone-2012-aa.pdf: 1981527 bytes, checksum: 8a82da009242da0cf884e2c7f04009a6 (MD5) %0 Journal Article %J Journal fur die Reine und Angewandte Mathematik, Issue 671, October 2012, Pages 131-198 %D 2012 %T A Codazzi-like equation and the singular set for C1 smooth surfaces in the Heisenberg group. %A Andrea Malchiodi %A Paul Yang %A Jih-Hsin Cheng %A JennFang Hwang %X In this paper, we study the structure of the singular set for a C 1 smooth surface in the 3-dimensional Heisenberg group ℍ 1. We discover a Codazzi-like equation for the p-area element along the characteristic curves on the surface. Information obtained from this ordinary differential equation helps us to analyze the local configuration of the singular set and the characteristic curves. In particular, we can estimate the size and obtain the regularity of the singular set. We understand the global structure of the singular set through a Hopf-type index theorem. We also justify the Codazzi-like equation by proving a fundamental theorem for local surfaces in ℍ 1 %B Journal fur die Reine und Angewandte Mathematik, Issue 671, October 2012, Pages 131-198 %I SISSA %G en %U http://hdl.handle.net/1963/6556 %1 6490 %2 Mathematics %4 1 %# MAT/05 ANALISI MATEMATICA %$ Approved for entry into archive by Lucio Lubiana (lubiana@sissa.it) on 2013-03-25T14:33:32Z (GMT) No. of bitstreams: 0 %R 10.1515/CRELLE.2011.159 %0 Book Section %B Natural locomotion in fluids and on surfaces : swimming, flying, and sliding / editors Stephen Childress, Anette Hosoi, William W. Schultz, and Z. Jane Wang, editors, %D 2012 %T Computing optimal strokes for low reynolds number swimmers %A Antonio DeSimone %A Luca Heltai %A François Alouges %A Lefebvre-Lepot Aline %K Numerical analysis. %XWe discuss connections between low-Reynolds-number swimming and geometric control theory, and present a general algorithm for the numerical computation of energetically optimal strokes. As an illustration of our approach, we show computed motility maps and optimal strokes for two model swimmers.

%B Natural locomotion in fluids and on surfaces : swimming, flying, and sliding / editors Stephen Childress, Anette Hosoi, William W. Schultz, and Z. Jane Wang, editors, %I Springer %@ 9781461439967 %G en %U http://hdl.handle.net/1963/6445 %1 6381 %2 Mathematics %4 1 %# MAT/08 ANALISI NUMERICA %$ Submitted by Luca Heltai (heltai@sissa.it) on 2013-02-01T17:33:50Z\\nNo. of bitstreams: 0 %R 10.1007/978-1-4614-3997-4_13 %0 Report %D 2012 %T A Fully Coupled Immersed Finite Element Method for Fluid Structure Interaction via the Deal.II Library %A Luca Heltai %A Saswati Roy %A Francesco Costanzo %K Finite Element Method %K Immersed Boundary Method %K Immersed Finite Element Method %X 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. %I SISSA %G en %U http://hdl.handle.net/1963/6255 %1 6172 %2 Mathematics %3 Functional Analysis and Applications %4 1 %# MAT/08 ANALISI NUMERICA %$ Submitted by Luca Heltai (heltai@sissa.it) on 2012-09-28T11:06:13Z\\nNo. of bitstreams: 1\\n1209.2811v1.pdf: 4766431 bytes, checksum: a46803f7f8daf3195359f65c9161b944 (MD5) %0 Journal Article %J Proceedings of the National Academy of Sciences of the United States of America. Volume 109, Issue 44, 30 October 2012, Pages 17874-17879 %D 2012 %T Reverse engineering the euglenoid movement %A Marino Arroyo %A Luca Heltai %A Daniel Millán %A Antonio DeSimone %K microswimmers %X Euglenids exhibit an unconventional motility strategy amongst unicellular eukaryotes, consisting of large-amplitude highly concerted deformations of the entire body (euglenoid movement or metaboly). A plastic cell envelope called pellicle mediates these deformations. Unlike ciliary or flagellar motility, the biophysics of this mode is not well understood, including its efficiency and molecular machinery. We quantitatively examine video recordings of four euglenids executing such motions with statistical learning methods. This analysis reveals strokes of high uniformity in shape and pace. We then interpret the observations in the light of a theory for the pellicle kinematics, providing a precise understanding of the link between local actuation by pellicle shear and shape control. We systematically understand common observations, such as the helical conformations of the pellicle, and identify previously unnoticed features of metaboly. While two of our euglenids execute their stroke at constant body volume, the other two exhibit deviations of about 20% from their average volume, challenging current models of low Reynolds number locomotion. We find that the active pellicle shear deformations causing shape changes can reach 340%, and estimate the velocity of the molecular motors. Moreover, we find that metaboly accomplishes locomotion at hydrodynamic efficiencies comparable to those of ciliates and flagellates. Our results suggest new quantitative experiments, provide insight into the evolutionary history of euglenids, and suggest that the pellicle may serve as a model for engineered active surfaces with applications in microfluidics. %B Proceedings of the National Academy of Sciences of the United States of America. Volume 109, Issue 44, 30 October 2012, Pages 17874-17879 %G en %U http://hdl.handle.net/1963/6444 %1 6380 %2 Mathematics %4 1 %# MAT/08 ANALISI NUMERICA %$ Submitted by Luca Heltai (heltai@sissa.it) on 2013-02-01T17:29:31Z\\nNo. of bitstreams: 0 %R 10.1073/pnas.1213977109 %0 Conference Paper %B 17th Int. Conf. Ships Shipp. Res. %D 2012 %T A stable semi-lagrangian potential method for the simulation of ship interaction with unsteady and nonlinear waves %A Andrea Mola %A Luca Heltai %A Antonio DeSimone %B 17th Int. Conf. Ships Shipp. Res. %G eng %0 Journal Article %J Computer Methods in Applied Mechanics and Engineering. Volume 229-232, 1 July 2012, Pages 110-127 %D 2012 %T Variational implementation of immersed finite element methods %A Luca Heltai %A Francesco Costanzo %K Turbulent flow %XDirac-delta distributions are often crucial components of the solid-fluid coupling operators in immersed solution methods for fluid-structure interaction (FSI) problems. This is certainly so for methods like the Immersed Boundary Method (IBM) or the Immersed Finite Element Method (IFEM), where Dirac-delta distributions are approximated via smooth functions. By contrast, a truly variational formulation of immersed methods does not require the use of Dirac-delta distributions, either formally or practically. This has been shown in the Finite Element Immersed Boundary Method (FEIBM), where the variational structure of the problem is exploited to avoid Dirac-delta distributions at both the continuous and the discrete level. In this paper, we generalize the FEIBM to the case where an incompressible Newtonian fluid interacts with a general hyperelastic solid. Specifically, we allow (i) the mass density to be different in the solid and the fluid, (ii) the solid to be either viscoelastic of differential type or purely elastic, and (iii) the solid to be and either compressible or incompressible. At the continuous level, our variational formulation combines the natural stability estimates of the fluid and elasticity problems. In immersed methods, such stability estimates do not transfer to the discrete level automatically due to the non- matching nature of the finite dimensional spaces involved in the discretization. After presenting our general mathematical framework for the solution of FSI problems, we focus in detail on the construction of natural interpolation operators between the fluid and the solid discrete spaces, which guarantee semi-discrete stability estimates and strong consistency of our spatial discretization.

%B Computer Methods in Applied Mechanics and Engineering. Volume 229-232, 1 July 2012, Pages 110-127 %I Elsevier %G en %U http://hdl.handle.net/1963/6462 %1 6389 %2 Mathematics %4 1 %# MAT/05 ANALISI MATEMATICA %$ Submitted by Lucio Lubiana (lubiana@sissa.it) on 2013-02-04T16:21:56Z\\nNo. of bitstreams: 0 %R 10.1016/j.cma.2012.04.001 %0 Journal Article %J Sports Engineering %D 2011 %T Multi-physics modelling and sensitivity analysis of olympic rowing boat dynamics %A Andrea Mola %A Mehdi Ghommem %A Muhammad R. Hajj %B Sports Engineering %I Springer Nature %V 14 %P 85–94 %8 nov %G eng %U https://doi.org/10.1007/s12283-011-0075-2 %R 10.1007/s12283-011-0075-2 %0 Journal Article %J Mathematical Models and Methods in Applied Sciences 21 (2011) 361-387 %D 2011 %T Numerical Strategies for Stroke Optimization of Axisymmetric Microswimmers %A François Alouges %A Antonio DeSimone %A Luca Heltai %K Optimal swimming %X We propose a computational method to solve optimal swimming problems, based on the boundary integral formulation of the hydrodynamic interaction between swimmer and surrounding fluid and direct constrained minimization of the energy consumed by the swimmer. We apply our method to axisymmetric model examples. We consider a classical model swimmer (the three-sphere swimmer of Golestanian et al.) as well as a novel axisymmetric swimmer inspired by the observation of biological micro-organisms. %B Mathematical Models and Methods in Applied Sciences 21 (2011) 361-387 %I World Scientific %G en_US %U http://hdl.handle.net/1963/3657 %1 648 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2009-06-23T17:30:51Z\\r\\nNo. of bitstreams: 1\\r\\nSissa33_2009M.pdf: 708341 bytes, checksum: 6134bd52f083488620fb5bb24bcf9b93 (MD5) %R 10.1142/S0218202511005088 %0 Report %D 2010 %T Optimally swimming Stokesian Robots %A François Alouges %A Antonio DeSimone %A Luca Heltai %A Aline Lefebvre %A Benoit Merlet %X We study self propelled stokesian robots composed of assemblies of balls, in dimen-\\nsions 2 and 3, and prove that they are able to control their position and orientation. This is a result of controllability, and its proof relies on applying Chow\\\'s theorem in an analytic framework, similarly to what has been done in [3] for an axisymmetric system swimming along the axis of symmetry. However, we simplify drastically\\nthe analyticity result given in [3] and apply it to a situation where more complex swimmers move either in a plane or in three-dimensional space, hence experiencing also rotations. We then focus our attention on energetically optimal strokes, which we are able to compute numerically. Some examples of computed optimal strokes are discussed in detail. %G en_US %U http://hdl.handle.net/1963/3929 %1 472 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2010-07-29T11:02:56Z\\nNo. of bitstreams: 1\\nHeltai_54M_2010.pdf: 993075 bytes, checksum: 77225380bab031438ac940e694ea0e6c (MD5) %0 Conference Paper %B IUTAM Symposium on Variational Concepts with Applications to the Mechanics of Materials %D 2010 %T A Phase Field Approach to Wetting and Contact Angle Hysteresis Phenomena %A Antonio DeSimone %A Livio Fedeli %A Turco, Alessandro %E Hackl, Klaus %XWe discuss a phase field model for the numerical simulation of contact angle hysteresis phenomena in wetting. The performance of the model is assessed by comparing its predictions with experimental data on the critical size of drops that can stick on a vertical glass plate.

%B IUTAM Symposium on Variational Concepts with Applications to the Mechanics of Materials %I Springer Netherlands %C Dordrecht %P 51–63 %@ 978-90-481-9195-6 %G eng %0 Report %D 2010 %T The role of membrane viscosity in the dynamics of fluid membranes %A Marino Arroyo %A Antonio DeSimone %A Luca Heltai %X Fluid membranes made out of lipid bilayers are the fundamental separation structure in eukaryotic cells. Many physiological processes rely on dramatic shape and topological changes (e.g. fusion, fission) of fluid membrane systems. Fluidity is key to the versatility and constant reorganization of lipid bilayers. Here, we study the role of the membrane intrinsic viscosity, arising from the friction of the lipid molecules as they rearrange to accommodate shape changes, in the dynamics of morphological changes of fluid vesicles. In particular, we analyze the competition between the membrane viscosity and the viscosity of the bulk fluid surrounding the vesicle as the dominant dissipative mechanism. We consider the relaxation dynamics of fluid vesicles put in an out-of-equilibrium state, but conclusions can be drawn regarding the kinetics or power consumption in regulated shape changes in the cell. On the basis of numerical calculations, we find that the dynamics arising from the membrane viscosity are qualitatively different from the dynamics arising from the bulk viscosity. When these two dissipation mechanisms are put in competition, we find that for small vesicles the membrane dissipation dominates, with a relaxation time that scales as the size of the vesicle to the power 2. For large vesicles, the bulk dissipation dominates, and the exponent in the relaxation time vs. size relation is 3. %G en_US %U http://hdl.handle.net/1963/3930 %1 471 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2010-07-29T11:13:35Z\\nNo. of bitstreams: 1\\nArroyo_55M.pdf: 4419667 bytes, checksum: ca8a9c175a457ad33996d46a08e85e44 (MD5) %0 Journal Article %J Journal of Structural Engineering %D 2009 %T Low-Frequency Variations of Force Coefficients on Square Cylinders with Sharp and Rounded Corners %A Andrea Mola %A Giancarlo Bordonaro %A Muhammad R. Hajj %B Journal of Structural Engineering %I American Society of Civil Engineers ({ASCE}) %V 135 %P 828–835 %8 jul %G eng %U https://doi.org/10.1061/(asce)st.1943-541x.0000034 %R 10.1061/(asce)st.1943-541x.0000034 %0 Report %D 2009 %T Stratos: a code for 3D free surface flows with floating constraints %A Antonio DeSimone %A B. Bianchi %A Luca Heltai %X This report presents a brief discussion of the theoretical aspects and practical implementation of STRATOS . STRATOS is a 3D code for the simulation\\nof hydrodynamic flows for incompressible fluids, in the presence of a free surface, capable of simulating the interaction between the free surface and a\\nfloating object via Lagrange multipliers...... %G en_US %U http://hdl.handle.net/1963/3701 %1 604 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2009-07-30T14:37:01Z\\nNo. of bitstreams: 1\\nBianchiDeSimoneHeltai.pdf: 899803 bytes, checksum: aff44eb1a39c6fd4acd4b3cb285e2d64 (MD5) %0 Report %D 2009 %T Tools for the Solution of PDEs Defined on Curved Manifolds with deal.II %A Antonio DeSimone %A Luca Heltai %A Cataldo Manigrasso %X The deal.II finite element library was originally designed to solve partial differential equations defined on one, two or three space dimensions, mostly\\nvia the Finite Element Method. In its versions prior to version 6.2, the user could not solve problems defined on curved manifolds embedded in two or\\nthree spacial dimensions. This infrastructure is needed if one wants to solve, for example, Boundary Integral Equations. %G en_US %U http://hdl.handle.net/1963/3700 %1 605 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2009-07-30T14:20:01Z\\nNo. of bitstreams: 1\\nDeSimoneHeltaiManigrasso.pdf: 915137 bytes, checksum: c3144348b6aab67a5f547060049c7847 (MD5) %0 Journal Article %J Comm. Pure Appl. Math. 60 (2007) 1559-1622 %D 2007 %T Asymptotic behaviour of smooth solutions for partially dissipative hyperbolic systems with a convex entropy %A Stefano Bianchini %A Bernard Hanouzet %A Roberto Natalini %X We study the asymptotic time behavior of global smooth solutions to general entropy dissipative hyperbolic systems of balance law in m space dimensions, under the Shizuta-Kawashima condition. %B Comm. Pure Appl. Math. 60 (2007) 1559-1622 %G en_US %U http://hdl.handle.net/1963/1780 %1 2764 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2006-03-30T10:33:17Z\\nNo. of bitstreams: 1\\n83M.pdf: 393359 bytes, checksum: f3f7b61564332b98b8706af95942dd69 (MD5) %R 10.1002/cpa.20195 %0 Journal Article %J J. Math. Phys. 47 (2006) 032104 %D 2006 %T Reflection symmetries for multiqubit density operators %A Claudio Altafini %A Timothy F. Havel %X For multiqubit density operators in a suitable tensorial basis, we show that a number of nonunitary operations used in the detection and synthesis of entanglement are classifiable as reflection symmetries, i.e., orientation changing rotations. While one-qubit reflections correspond to antiunitary symmetries, as is known for example from the partial transposition criterion, reflections on the joint density of two or more qubits are not accounted for by the Wigner Theorem and are well-posed only for sufficiently mixed states. One example of such nonlocal reflections is the unconditional NOT operation on a multiparty density, i.e., an operation yelding another density and such that the sum of the two is the identity operator. This nonphysical operation is admissible only for sufficiently mixed states. %B J. Math. Phys. 47 (2006) 032104 %G en_US %U http://hdl.handle.net/1963/2121 %1 2122 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-09-18T06:51:45Z\\nNo. of bitstreams: 1\\n0405123v2.pdf: 247264 bytes, checksum: c504f911bff46e503941a0298617759a (MD5) %R 10.1063/1.2181827 %0 Journal Article %J Comm. Math. Phys. %D 2006 %T Semiclassical orthogonal polynomials, matrix models and isomonodromic tau functions %A Marco Bertola %A B. Eynard %A Harnad, J. %B Comm. Math. Phys. %V 263 %P 401–437 %G eng %0 Journal Article %J Differential Geom. Appl. 24 (2006) 403-416 %D 2006 %T Semistability vs. nefness for (Higgs) vector bundles %A Ugo Bruzzo %A Daniel Hernandez Ruiperez %X According to Miyaoka, a vector bundle E on a smooth projective curve is semistable if and only if a certain numerical class in the projectivized bundle PE is nef. We establish a similar criterion for the semistability of Higgs bundles: namely, such a bundle is semistable if and only if for every integer s between 0 and the rank of E, a suitable numerical class in the scheme parametrizing the rank s locally-free Higgs quotients of E is nef. We also extend this result to higher-dimensional complex projective varieties by showing that the nefness of the above mentioned classes is equivalent to the semistability of the Higgs bundle E together with the vanishing of the discriminant of E. %B Differential Geom. Appl. 24 (2006) 403-416 %G en_US %U http://hdl.handle.net/1963/2237 %1 2007 %2 Mathematics %3 Mathematical Physics %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-10-16T09:06:35Z\\nNo. of bitstreams: 1\\n0310040v3.pdf: 216788 bytes, checksum: 574f5aac93686c2348a6313bd129cb61 (MD5) %R 10.1016/j.difgeo.2005.12.007 %0 Journal Article %J Annali della Scuola Normale Superiore di Pisa, Classe di Scienze (5), 4 (2005) 129-177. %D 2005 %T Minimal surfaces in pseudohermitian geometry %A Jih-Hsin Cheng %A JennFang Hwang %A Andrea Malchiodi %A Paul Yang %X We consider surfaces immersed in three-dimensional pseudohermitian manifolds. We define the notion of (p-)mean curvature and of the associated (p-)minimal surfaces, extending some concepts previously given for the (flat) Heisenberg group. We interpret the p-mean curvature not only as the tangential sublaplacian of a defining function, but also as the curvature of a characteristic curve, and as a quantity in terms of calibration geometry. As a differential equation, the p-minimal surface equation is degenerate (hyperbolic and elliptic). To analyze the singular set, we formulate some {\em extension} theorems, which describe how the characteristic curves meet the singular set. This allows us to classify the entire solutions to this equation and to solve a Bernstein-type problem (for graphs over the $xy$-plane) in the Heisenberg group $H_1$. In $H_{1}$, identified with the Euclidean space $R^{3}$, the p-minimal surfaces are classical ruled surfaces with the rulings generated by Legendrian lines. We also prove a uniqueness theorem for the Dirichlet problem under a condition on the size of the singular set in two dimensions, and generalize to higher dimensions without any size control condition. We also show that there are no closed, connected, $C^{2}$ smoothly immersed constant p-mean curvature or p-minimal surfaces of genus greater than one in the standard $S^{3}.$ This fact continues to hold when $S^{3}$ is replaced by a general spherical pseudohermitian 3-manifold. %B Annali della Scuola Normale Superiore di Pisa, Classe di Scienze (5), 4 (2005) 129-177. %I Scuola Normale Superiore %G en %U http://hdl.handle.net/1963/4579 %1 4347 %2 Mathematics %3 Functional Analysis and Applications %4 -1 %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2011-10-07T09:39:45Z No. of bitstreams: 1 math_0401136v3.pdf: 476553 bytes, checksum: 85c9b54a1f7e6c159e8fbc2486849a53 (MD5) %R 10.2422/2036-2145.2005.1.05 %0 Journal Article %J J. Math. Anal. Appl. 308 (2005) 221-239 %D 2005 %T Stability of solutions of quasilinear parabolic equations %A Giuseppe Maria Coclite %A Helge Holden %X We bound the difference between solutions $u$ and $v$ of $u_t = a\\\\Delta u+\\\\Div_x f+h$ and $v_t = b\\\\Delta v+\\\\Div_x g+k$ with initial data $\\\\phi$ and $ \\\\psi$, respectively, by $\\\\Vert u(t,\\\\cdot)-v(t,\\\\cdot)\\\\Vert_{L^p(E)}\\\\le A_E(t)\\\\Vert \\\\phi-\\\\psi\\\\Vert_{L^\\\\infty(\\\\R^n)}^{2\\\\rho_p}+ B(t)(\\\\Vert a-b\\\\Vert_{\\\\infty}+ \\\\Vert \\\\nabla_x\\\\cdot f-\\\\nabla_x\\\\cdot g\\\\Vert_{\\\\infty}+ \\\\Vert f_u-g_u\\\\Vert_{\\\\infty} + \\\\Vert h-k\\\\Vert_{\\\\infty})^{\\\\rho_p} \\\\abs{E}^{\\\\eta_p}$. Here all functions $a$, $f$, and $h$ are smooth and bounded, and may depend on $u$, $x\\\\in\\\\R^n$, and $t$. The functions $a$ and $h$ may in addition depend on $\\\\nabla u$. Identical assumptions hold for the functions that determine the solutions $v$. Furthermore, $E\\\\subset\\\\R^n$ is assumed to be a bounded set, and $\\\\rho_p$ and $\\\\eta_p$ are fractions that depend on $n$ and $p$. The diffusion coefficients $a$ and $b$ are assumed to be strictly positive and the initial data are smooth. %B J. Math. Anal. Appl. 308 (2005) 221-239 %I Elsevier %G en_US %U http://hdl.handle.net/1963/2892 %1 1808 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2008-09-10T12:48:15Z\\nNo. of bitstreams: 1\\n0306160v1.pdf: 240791 bytes, checksum: 4654ca0f7a1735bc7ee6666d37a2d57a (MD5) %R 10.1016/j.jmaa.2005.01.026 %0 Journal Article %J J. Geom. Phys. 49 (2004) 272-293 %D 2004 %T Fredholm modules for quantum euclidean spheres %A Eli Hawkins %A Giovanni Landi %X The quantum Euclidean spheres, $S_q^{N-1}$, are (noncommutative) homogeneous spaces of quantum orthogonal groups, $\\\\SO_q(N)$. The *-algebra $A(S^{N-1}_q)$ of polynomial functions on each of these is given by generators and relations which can be expressed in terms of a self-adjoint, unipotent matrix. We explicitly construct complete sets of generators for the K-theory (by nontrivial self-adjoint idempotents and unitaries) and the K-homology (by nontrivial Fredholm modules) of the spheres $S_q^{N-1}$. We also construct the corresponding Chern characters in cyclic homology and cohomology and compute the pairing of K-theory with K-homology. On odd spheres (i. e., for N even) we exhibit unbounded Fredholm modules by means of a natural unbounded operator D which, while failing to have compact resolvent, has bounded commutators with all elements in the algebra $A(S^{N-1}_q)$. %B J. Geom. Phys. 49 (2004) 272-293 %I SISSA Library %G en %U http://hdl.handle.net/1963/1636 %1 2482 %2 Mathematics %3 Mathematical Physics %$ Made available in DSpace on 2004-09-01T13:05:40Z (GMT). No. of bitstreams: 1\\nmath.KT0210139.pdf: 218018 bytes, checksum: 44f0c66ef47d8b2e1cafc955075c7626 (MD5)\\n Previous issue date: 2002 %R 10.1016/S0393-0440(03)00092-5 %0 Journal Article %J Comm. Math. Phys. %D 2003 %T Differential systems for biorthogonal polynomials appearing in 2-matrix models and the associated Riemann-Hilbert problem %A Marco Bertola %A B. Eynard %A Harnad, J. %B Comm. Math. Phys. %V 243 %P 193–240 %G eng %0 Journal Article %J J. Phys. A %D 2003 %T Partition functions for matrix models and isomonodromic tau functions %A Marco Bertola %A B. Eynard %A Harnad, J. %B J. Phys. A %V 36 %P 3067–3083 %G eng %0 Journal Article %J Comm. Math. Phys. %D 2002 %T Duality, biorthogonal polynomials and multi-matrix models %A Marco Bertola %A B. Eynard %A Harnad, J. %B Comm. Math. Phys. %V 229 %P 73–120 %G eng %0 Journal Article %J Math. Nachr. 238 (2002) 23-36 %D 2002 %T Relatively stable bundles over elliptic fibrations %A Claudio Bartocci %A Ugo Bruzzo %A Daniel Hernandez Ruiperez %A Jose M. Munoz Porras %X We consider a relative Fourier-Mukai transform defined on elliptic fibrations over an arbitrary normal base scheme. This is used to construct relative Atiyah sheaves and generalize Atiyah\\\'s and Tu\\\'s results about semistable sheaves over elliptic curves to the case of elliptic fibrations. Moreover we show that this transform preserves relative (semi)stability of sheaves of positive relative degree. %B Math. Nachr. 238 (2002) 23-36 %I Wiley %G en_US %U http://hdl.handle.net/1963/3132 %1 1201 %2 Mathematics %3 Mathematical Physics %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2008-10-17T07:37:19Z\\nNo. of bitstreams: 1\\n0109123v2.pdf: 199719 bytes, checksum: b8757f4871a0f0fd621bc8f115792a5c (MD5)