@article {LS20a, title = {Rigidity and trace properties of divergence-measure vector fields}, journal = {Adv. Calc. Var.}, year = {In Press}, doi = {10.1515/acv-2019-0094}, author = {Leonardi, G. P. and Saracco, G.} } @booklet {2023, title = {Relaxed area of $0$-homogeneous maps in the strict $BV$-convergence}, year = {2023}, author = {Carano, Simone} } @booklet {2023, title = {Relaxed area of graphs of piecewise Lipschitz maps in the strict $BV$-convergence}, year = {2023}, author = {Bellettini, Giovanni and Carano, Simone and Scala, Riccardo} } @article {2022, title = {The relaxed area of $S^1$-valued singular maps in the strict $BV$-convergence}, journal = {ESAIM: Control, Optimization and Calculus of Variations}, volume = {28}, year = {2022}, pages = {38}, url = {http://cvgmt.sns.it/paper/5440/}, author = {Bellettini, Giovanni and Carano, Simone and Scala, Riccardo} } @article {2022, title = {Rigidities of Isoperimetric inequality under nonnegative Ricci curvature}, year = {2022}, abstract = {

The sharp isoperimetric inequality for non-compact Riemannian manifolds with non-negative Ricci curvature and Euclidean volume growth has been obtained in increasing generality with different approaches in a number of contributions [arXiv:1812.05022,\ arXiv:2012.09490,\ arXiv:2009.13717,\ arXiv:2103.08496] culminated by Balogh and Kristaly [arXiv:2012.11862] covering also m.m.s.{\textquoteright}s verifying the non-negative Ricci curvature condition in the synthetic sense of Lott, Sturm and Villani. In sharp contrast with the compact case of positive Ricci curvature, for a large class of spaces including weighted Riemannian manifolds, no complete characterisation of the equality cases is present in the literature.
The scope of this note is to settle this problem by proving, in the same generality of [arXiv:2012.11862], that the equality in the isoperimetric inequality can be attained only by metric balls. Whenever this happens the space is forced, in a measure theoretic sense, to be a cone.
Our result applies to different frameworks yielding as corollaries new rigidity results: it extend to weighted Riemannian manifold the rigidity results of [arXiv:2009.13717], it extend to general\ RCD\ spaces the rigidity results of [arXiv:2201.04916] and finally applies also to the Euclidean setting by proving that that optimisers in the anisotropic and weighted isoperimetric inequality for Euclidean cones are necessarily the Wulff shapes.

}, author = {Cavalletti, Fabio and Manini, Davide} } @article {2021, title = {Rectifiability of the free boundary for varifolds}, journal = {Indiana Univ. Math. J.}, volume = {70}, year = {2021}, pages = {2603{\textendash}2651}, issn = {0022-2518}, author = {Luigi De Masi} } @article {2021, title = {A Reduced Order Cut Finite Element method for geometrically parametrized steady and unsteady Navier{\textendash}Stokes problems}, journal = {Computer \& Mathematics With Applications}, year = {2021}, month = {2021/08/12/}, abstract = {

We focus on steady and unsteady Navier{\textendash}Stokes flow systems in a reduced-order modeling framework based on Proper Orthogonal Decomposition within a levelset geometry description and discretized by an unfitted mesh Finite Element Method. This work extends the approaches of [1], [2], [3] to nonlinear CutFEM discretization. We construct and investigate a unified and geometry independent reduced basis which overcomes many barriers and complications of the past, that may occur whenever geometrical morphings are taking place. By employing a geometry independent reduced basis, we are able to avoid remeshing and transformation to reference configurations, and we are able to handle complex geometries. This combination of a fixed background mesh in a fixed extended background geometry with reduced order techniques appears beneficial and advantageous in many industrial and engineering applications, which could not be resolved efficiently in the past.

}, keywords = {Cut Finite Element Method, Navier{\textendash}Stokes equations, Parameter{\textendash}dependent shape geometry, Reduced Order Models, Unfitted mesh}, isbn = {0898-1221}, url = {https://www.sciencedirect.com/science/article/pii/S0898122121002790}, author = {Efthymios N Karatzas and Monica Nonino and F. Ballarin and Gianluigi Rozza} } @conference { 20.500.11767_123375, title = {Reduced Order Methods for Parametrized Non-linear and Time Dependent Optimal Flow Control Problems, Towards Applications in Biomedical and Environmental Sciences}, booktitle = {Numerical Mathematics and Advanced Applications ENUMATH 2019}, volume = {139}, year = {2021}, pages = {841{\textendash}850}, publisher = {Springer}, organization = {Springer}, isbn = {978-3-030-55873-4}, doi = {10.1007/978-3-030-55874-1_83}, url = {https://arxiv.org/abs/1912.07886}, author = {Maria Strazzullo and Zakia Zainib and F. Ballarin and Gianluigi Rozza} } @conference {2021, title = {Reduced Order Methods for Parametrized Non-linear and Time Dependent Optimal Flow Control Problems, Towards Applications in Biomedical and Environmental Sciences}, booktitle = {Numerical Mathematics and Advanced Applications ENUMATH 2019}, year = {2021}, month = {2021//}, publisher = {Springer International Publishing}, organization = {Springer International Publishing}, address = {Cham}, abstract = {

We introduce reduced order methods as an efficient strategy to solve parametrized non-linear and time dependent optimal flow control problems governed by partial differential equations. Indeed, the optimal control problems require a huge computational effort in order to be solved, most of all in physical and/or geometrical parametrized settings. Reduced order methods are a reliable and 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 employ a POD-Galerkin reduction approach over a parametrized optimality system, derived from the Karush-Kuhn-Tucker conditions. The methodology presented is tested on two boundary control problems, governed respectively by (1) time dependent Stokes equations and (2) steady non-linear Navier-Stokes equations.

}, isbn = {978-3-030-55874-1}, doi = {https://doi.org/10.1007/978-3-030-55874-1_83}, url = {https://www.springerprofessional.de/en/reduced-order-methods-for-parametrized-non-linear-and-time-depen/19122676}, author = {Maria Strazzullo and Zakia Zainib and F. Ballarin and Gianluigi Rozza}, editor = {Fred J Vermolen and Cornelis Vuik} } @article {StarSanderseStabileRozzaDegroote2020, title = {Reduced order models for the incompressible Navier-Stokes equations on collocated grids using a {\textquoteleft}discretize-then-project{\textquoteright} approach}, journal = {International Journal for Numerical Methods in Fluids}, volume = {93}, number = {8}, year = {2021}, pages = {2694{\textendash}2722}, publisher = {Wiley}, doi = {10.1002/fld.4994}, url = {https://doi.org/10.1002/fld.4994}, author = {Kelbij Star and Benjamin Sanderse and Giovanni Stabile and Gianluigi Rozza and Joris Degroote} } @booklet {2021, title = {A remark on two notions of flatness for sets in the Euclidean space}, year = {2021}, abstract = {

In this note we compare two ways of measuring the\ n-dimensional "flatness" of a set\ SRd, where\ nN\ and\ d\>n. The first one is to consider the classical Reifenberg-flat numbers\ α(x,r)\ (xS,\ r\>0), which measure the minimal scaling-invariant Hausdorff distances in\ Br(x)\ between\ S\ and\ n-dimensional affine subspaces of\ Rd. The second is an {\textquoteleft}intrinsic{\textquoteright} approach in which we view the same set\ S\ as a metric space (endowed with the induced Euclidean distance). Then we consider numbers\ a(x,r){\textquoteright}s, that are the scaling-invariant Gromov-Hausdorff distances between balls centered at\ x\ of radius\ r\ in\ S\ and the\ n-dimensional Euclidean ball of the same radius. As main result of our analysis we make rigorous a phenomenon, first noted by David and Toro, for which the numbers\ a(x,r){\textquoteright}s behaves as the square of the numbers\ α(x,r){\textquoteright}s. Moreover we show how this result finds application in extending the Cheeger-Colding intrinsic-Reifenberg theorem to the biLipschitz case. As a by-product of our arguments, we deduce analogous results also for the Jones{\textquoteright} numbers\ β{\textquoteright}s (i.e. the one-sided version of the numbers\ α{\textquoteright}s).

}, author = {Ivan Yuri Violo} } @booklet {2021, title = {Rigidity and almost rigidity of Sobolev inequalities on compact spaces with lower Ricci curvature bounds}, year = {2021}, abstract = {

We prove that ifMis a closedn-dimensional Riemannian manifold,n>=3, withRic>=n-1and for which the optimal constant in the critical Sobolev inequality equals the one of then-dimensional sphereSn, thenMis isometric toSn. An almost-rigidity result is also established, saying that if equality is almost achieved, thenMis close in the measure Gromov-Hausdorff sense to a spherical suspension. These statements are obtained in theRCD-setting of (possibly non-smooth) metric measure spaces satisfying synthetic lower Ricci curvature bounds.
An independent result of our analysis is the characterization of the best constant in the Sobolev inequality on any compactCDspace, extending to the non-smooth setting a classical result by Aubin. Our arguments are based on a new concentration compactness result for mGH-converging sequences ofRCDspaces and on a Polya-Szego inequality of Euclidean-type inCDspaces.
As an application of the technical tools developed we prove both an existence result for the Yamabe equation and the continuity of the generalized Yamabe constant under measure Gromov-Hausdorff convergence, in theRCD-setting.
}, author = {Francesco Nobili and Ivan Yuri Violo} } @article {2020, title = {Reduced Basis Model Order Reduction for Navier-Stokes equations in domains with walls of varying curvature}, journal = {International Journal of Computational Fluid Dynamics}, volume = {34}, year = {2020}, pages = {119-126}, abstract = {

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.

}, doi = {10.1080/10618562.2019.1645328}, url = {https://arxiv.org/abs/1901.03708}, author = {Martin W. Hess and Annalisa Quaini and Gianluigi Rozza} } @article {2020, title = {Reduced basis model order reduction for Navier{\textendash}Stokes equations in domains with walls of varying curvature}, journal = {International Journal of Computational Fluid Dynamics}, volume = {34}, year = {2020}, pages = {119-126}, abstract = {

We consider the Navier{\textendash}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.

}, doi = {10.1080/10618562.2019.1645328}, url = {https://www.scopus.com/inward/record.uri?eid=2-s2.0-85085233294\&doi=10.1080\%2f10618562.2019.1645328\&partnerID=40\&md5=e2ed8f24c66376cdc8b5485aa400efb0}, author = {Martin W. Hess and Annalisa Quaini and Gianluigi Rozza} } @conference {2020, title = {A Reduced Order Approach for the Embedded Shifted Boundary FEM and a Heat Exchange System on Parametrized Geometries}, booktitle = {IUTAM Symposium on Model Order Reduction of Coupled Systems, Stuttgart, Germany, May 22{\textendash}25, 2018}, year = {2020}, publisher = {Springer International Publishing}, organization = {Springer International Publishing}, abstract = {

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.

}, doi = {10.1007/978-3-030-21013-7_8}, url = {https://arxiv.org/abs/1807.07753}, author = {Efthymios N Karatzas and Giovanni Stabile and Nabib Atallah and Guglielmo Scovazzi and Gianluigi Rozza}, editor = {Fehr, J{\"o}rg and Bernard Haasdonk} } @article {2020, title = {Reduced order isogeometric analysis approach for pdes in parametrized domains}, journal = {Lecture Notes in Computational Science and Engineering}, volume = {137}, year = {2020}, pages = {153-170}, abstract = {

In this contribution, we coupled the isogeometric analysis to a reduced order modelling technique in order to provide a computationally efficient solution in parametric domains. In details, we adopt the free-form deformation method to obtain the parametric formulation of the domain and proper orthogonal decomposition with interpolation for the computational reduction of the model. This technique provides a real-time solution for any parameter by combining several solutions, in this case computed using isogeometric analysis on different geometrical configurations of the domain, properly mapped into a reference configuration. We underline that this reduced order model requires only the full-order solutions, making this approach non-intrusive. We present in this work the results of the application of this methodology to a heat conduction problem inside a deformable collector pipe.

}, doi = {10.1007/978-3-030-48721-8_7}, url = {https://www.scopus.com/inward/record.uri?eid=2-s2.0-85089615035\&doi=10.1007\%2f978-3-030-48721-8_7\&partnerID=40\&md5=7b15836ae65fa28dcfe8733788d7730c}, author = {Fabrizio Garotta and Nicola Demo and Marco Tezzele and Massimo Carraturo and Alessandro Reali and Gianluigi Rozza} } @article {2020, title = {Reduced order methods for parametric optimal flow control in coronary bypass grafts, toward patient-specific data assimilation}, journal = {International Journal for Numerical Methods in Biomedical EngineeringInternational Journal for Numerical Methods in Biomedical EngineeringInt J Numer Meth Biomed Engng}, volume = {n/a}, year = {2020}, month = {2020/05/27}, pages = {e3367}, abstract = {

Abstract Coronary artery bypass grafts (CABG) surgery is an invasive procedure performed to circumvent partial or complete blood flow blockage in coronary artery disease. In this work, we apply a numerical optimal flow control model to patient-specific geometries of CABG, reconstructed from clinical images of real-life surgical cases, in parameterized settings. The aim of these applications is to match known physiological data with numerical hemodynamics corresponding to different scenarios, arisen by tuning some parameters. Such applications are an initial step toward matching patient-specific physiological data in patient-specific vascular geometries as best as possible. Two critical challenges that reportedly arise in such problems are: (a) lack of robust quantification of meaningful boundary conditions required to match known data as best as possible and (b) high computational cost. In this work, we utilize unknown control variables in the optimal flow control problems to take care of the first challenge. Moreover, to address the second challenge, we propose a time-efficient and reliable computational environment for such parameterized problems by projecting them onto a low-dimensional solution manifold through proper orthogonal decomposition-Galerkin.

}, keywords = {coronary artery bypass grafts, data assimilation, flow control, Galerkin methods, hemodynamics modeling, Optimization, patient-specific simulations, Proper orthogonal decomposition, reduced order methods}, isbn = {2040-7939}, doi = {https://doi.org/10.1002/cnm.3367}, url = {https://onlinelibrary.wiley.com/doi/10.1002/cnm.3367?af=R}, author = {Zakia Zainib and F. Ballarin and Stephen E. Fremes and Piero Triverio and Laura Jim{\'e}nez-Juan and Gianluigi Rozza} } @article {2020, title = {A reduced order modeling technique to study bifurcating phenomena: Application to the gross-pitaevskii equation}, journal = {SIAM Journal on Scientific Computing}, year = {2020}, abstract = {

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{\textquoteright}s method with a reduced order modeling (ROM) technique, suitably supplemented with a hyperreduction method. To demonstrate the effectiveness of our ROM approach, we trace the steady solution branches of a nonlinear Schr{\"o}dinger equation, called the 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.

}, doi = {10.1137/20M1313106}, url = {https://www.scopus.com/inward/record.uri?eid=2-s2.0-85096768803\&doi=10.1137\%2f20M1313106\&partnerID=40\&md5=47d6012d10854c2f9a04b9737f870592}, author = {Federico Pichi and Annalisa Quaini and Gianluigi Rozza} } @article {2020, title = {A Reduced Order technique to study bifurcating phenomena: application to the Gross-Pitaevskii equation}, journal = {SIAM Journal on Scientific Computing}, year = {2020}, abstract = {

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{\textquoteright}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 Schr{\"o}dinger 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.

}, doi = {https://doi.org/10.1137/20M1313106}, url = {https://arxiv.org/abs/1907.07082}, author = {Federico Pichi and Annalisa Quaini and Gianluigi Rozza} } @article {2020, title = {A reduced-order shifted boundary method for parametrized incompressible Navier{\textendash}Stokes equations}, journal = {Computer Methods in Applied Mechanics and Engineering}, volume = {370}, year = {2020}, abstract = {

We investigate a projection-based reduced order model of the steady incompressible Navier{\textendash}Stokes equations for moderate Reynolds numbers. In particular, we construct an {\textquotedblleft}embedded{\textquotedblright} reduced basis space, by applying proper orthogonal decomposition to the Shifted Boundary Method, a high-fidelity embedded method recently developed. We focus on the geometrical parametrization through level-set geometries, using a fixed Cartesian background geometry and the associated mesh. This approach avoids both remeshing and the development of a reference domain formulation, as typically done in fitted mesh finite element formulations. Two-dimensional computational examples for one and three parameter dimensions are presented to validate the convergence and the efficacy of the proposed approach.

}, doi = {10.1016/j.cma.2020.113273}, url = {https://www.scopus.com/inward/record.uri?eid=2-s2.0-85087886522\&doi=10.1016\%2fj.cma.2020.113273\&partnerID=40\&md5=d864e4808190b682ecb1c8b27cda72d8}, author = {Efthymios N Karatzas and Giovanni Stabile and Leo Nouveau and Guglielmo Scovazzi and Gianluigi Rozza} } @article {Beltr{\'a}n2019, title = {The Real Polynomial Eigenvalue Problem is Well Conditioned on the Average}, journal = {Foundations of Computational Mathematics}, year = {2019}, month = {May}, abstract = {

We study the average condition number for polynomial eigenvalues of collections of matrices drawn from some random matrix ensembles. In particular, we prove that polynomial eigenvalue problems defined by matrices with random Gaussian entries are very well conditioned on the average.

}, issn = {1615-3383}, doi = {10.1007/s10208-019-09414-2}, url = {https://doi.org/10.1007/s10208-019-09414-2}, author = {Carlos Beltr{\'a}n and Khazhgali Kozhasov} } @article {2019, title = {A reduced basis approach for PDEs on parametrized geometries based on the shifted boundary finite element method and application to a Stokes flow}, journal = {Computer Methods in Applied Mechanics and Engineering}, volume = {347}, year = {2019}, pages = {568-587}, abstract = {

We propose a model order reduction technique integrating the Shifted Boundary Method (SBM) with a POD-Galerkin strategy. This approach allows to deal with 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. This unfitted boundary method permits to avoid remeshing and the tedious handling of cut cells by introducing an approximate surrogate boundary. Second, the computational effort is 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.

}, doi = {10.1016/j.cma.2018.12.040}, url = {https://www.scopus.com/inward/record.uri?eid=2-s2.0-85060107322\&doi=10.1016\%2fj.cma.2018.12.040\&partnerID=40\&md5=1a3234f0cb000c91494d946428f8ebef}, author = {Efthymios N Karatzas and Giovanni Stabile and Leo Nouveau and Guglielmo Scovazzi and Gianluigi Rozza} } @article {2019, title = {Reduced basis approaches for parametrized bifurcation problems held by non-linear Von K{\'a}rm{\'a}n equations}, volume = {81}, year = {2019}, pages = {112{\textendash}135}, abstract = {

This work focuses on the computationally efficient detection of the buckling phenomena and bifurcation analysis of the parametric Von K{\'a}rm{\'a}n 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

}, doi = {10.1007/s10915-019-01003-3}, url = {https://arxiv.org/abs/1804.02014}, author = {Federico Pichi and Gianluigi Rozza} } @article {2019, title = {Reduced Basis Approaches for Parametrized Bifurcation Problems held by Non-linear Von K{\'a}rm{\'a}n Equations}, journal = {Journal of Scientific Computing}, volume = {81}, year = {2019}, pages = {112-135}, abstract = {

This work focuses on the computationally efficient detection of the buckling phenomena and bifurcation analysis of the parametric Von K{\'a}rm{\'a}n plate equations based on reduced order methods and spectral analysis. The computational complexity{\textemdash}due to the fourth order derivative terms, the non-linearity and the parameter dependence{\textemdash}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.

}, doi = {10.1007/s10915-019-01003-3}, url = {https://www.scopus.com/inward/record.uri?eid=2-s2.0-85068973907\&doi=10.1007\%2fs10915-019-01003-3\&partnerID=40\&md5=a09af83ce45183d6965cdb79d87a919b}, author = {Federico Pichi and Gianluigi Rozza} } @article {2019, title = {A reduced order variational multiscale approach for turbulent flows}, journal = {Advances in Computational Mathematics}, volume = {45}, year = {2019}, pages = {2349-2368}, abstract = {

The purpose of this work is to present different reduced order model strategies starting from full order simulations stabilized using a residual-based variational multiscale (VMS) approach. The focus is on flows with moderately high Reynolds numbers. The reduced order models (ROMs) presented in this manuscript are based on a POD-Galerkin approach. Two different reduced order models are presented, which differ on the stabilization used during the Galerkin projection. In the first case, the VMS stabilization method is used at both the full order and the reduced order levels. In the second case, the VMS stabilization is used only at the full order level, while the projection of the standard Navier-Stokes equations is performed instead at the reduced order level. The former method is denoted as consistent ROM, while the latter is named non-consistent ROM, in order to underline the different choices made at the two levels. Particular attention is also devoted to the role of inf-sup stabilization by means of supremizers in ROMs based on a VMS formulation. Finally, the developed methods are tested on a numerical benchmark.

}, doi = {10.1007/s10444-019-09712-x}, url = {https://www.scopus.com/inward/record.uri?eid=2-s2.0-85068076665\&doi=10.1007\%2fs10444-019-09712-x\&partnerID=40\&md5=af0142e6d13bbc2e88c6f31750aef6ad}, author = {Giovanni Stabile and F. Ballarin and G. Zuccarino and Gianluigi Rozza} } @article {2019, title = {Reducibility for a fast-driven linear Klein{\textendash}Gordon equation}, volume = {198}, year = {2019}, month = {2019/08/01}, pages = {1407 - 1439}, abstract = {

We prove a reducibility result for a linear Klein{\textendash}Gordon equation with a quasi-periodic driving on a compact interval with Dirichlet boundary conditions. No assumptions are made on the size of the driving; however, we require it to be fast oscillating. In particular, provided that the external frequency is sufficiently large and chosen from a Cantor set of large measure, the original equation is conjugated to a time-independent, diagonal one. We achieve this result in two steps. First, we perform a preliminary transformation, adapted to fast oscillating systems, which moves the original equation in a perturbative setting. Then, we show that this new equation can be put to constant coefficients by applying a KAM reducibility scheme, whose convergence requires a new type of Melnikov conditions.

}, isbn = {1618-1891}, url = {https://doi.org/10.1007/s10231-019-00823-2}, author = {Luca Franzoi and Alberto Maspero} } @article {FEOLA2019932, title = {Reducibility of first order linear operators on tori via Moser{\textquoteright}s theorem}, journal = {Journal of Functional Analysis}, volume = {276}, number = {3}, year = {2019}, pages = {932 - 970}, abstract = {

In this paper we prove reducibility of a class of first order, quasi-linear, quasi-periodic time dependent PDEs on the torus∂tu+ζ.∂xu+a(ωt,x).∂xu=0,x∈Td,ζ∈Rd,ω∈Rν. As a consequence we deduce a stability result on the associated Cauchy problem in Sobolev spaces. By the identification between first order operators and vector fields this problem can be formulated as the problem of finding a change of coordinates which conjugates a weakly perturbed constant vector field on Tν+d to a constant diophantine flow. For this purpose we generalize Moser{\textquoteright}s straightening theorem: considering smooth perturbations we prove that the corresponding straightening torus diffeomorphism is smooth, under the assumption that the perturbation is small only in some given Sobolev norm and that the initial frequency belongs to some Cantor-like set. In view of applications in KAM theory for PDEs we provide also tame estimates on the change of variables.

}, keywords = {Hyperbolic PDEs, KAM theory, Nash{\textendash}Moser, Reducibility}, issn = {0022-1236}, doi = {https://doi.org/10.1016/j.jfa.2018.10.009}, url = {http://www.sciencedirect.com/science/article/pii/S0022123618303793}, author = {Roberto Feola and Filippo Giuliani and Riccardo Montalto and Michela Procesi} } @booklet {2019, title = {Regularity of minimizers for a model of charged droplets}, year = {2019}, author = {Guido De Philippis and Jonas Hirsch and Giulia Vescovo} } @article {Bellettini2019, title = {On the relaxed area of the graph of discontinuous maps from the plane to the plane taking three values with no symmetry assumptions}, journal = {Annali di Matematica Pura ed Applicata (1923 -)}, year = {2019}, month = {Jul}, abstract = {

In this paper, we estimate from above the area of the graph of a singular map u taking a disk to three vectors, the vertices of a triangle, and jumping along three $\mathcal{C}^2$-embedded curves that meet transversely at only one point of the disk. We show that the singular part of the relaxed area can be estimated from above by the solution of a Plateau-type problem involving three entangled nonparametric area-minimizing surfaces. The idea is to {\textquoteleft}{\textquoteleft}fill the hole{\textquoteright}{\textquoteright} in the graph of the singular map with a sequence of approximating smooth two-codimensional surfaces of graph-type, by imagining three minimal surfaces, placed vertically over the jump of u, coupled together via a triple point in the target triangle. Such a construction depends on the choice of a target triple point, and on a connection passing through it, which dictate the boundary condition for the three minimal surfaces. We show that the singular part of the relaxed area of u cannot be larger than what we obtain by minimizing over all possible target triple points and all corresponding connections.

}, issn = {1618-1891}, doi = {10.1007/s10231-019-00887-0}, url = {https://doi.org/10.1007/s10231-019-00887-0}, author = {Giovanni Bellettini and Alaa Elshorbagy and Maurizio Paolini and Riccardo Scala} } @article {2018, title = {On real resonances for the three-dimensional, multi-centre point interaction}, number = {SISSA;40/2018/MATE}, year = {2018}, author = {Alessandro Michelangeli and Raffaele Scandone} } @article {Gigli2018, title = {Recognizing the flat torus among RCD*(0,N) spaces via the study of the first cohomology group}, journal = {Calculus of Variations and Partial Differential Equations}, volume = {57}, number = {4}, year = {2018}, month = {Jun}, pages = {104}, abstract = {

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.

}, issn = {1432-0835}, doi = {10.1007/s00526-018-1377-z}, url = {https://doi.org/10.1007/s00526-018-1377-z}, author = {Nicola Gigli and Chiara Rigoni} } @article {2018, title = {Reduced Basis Approximation and A Posteriori Error Estimation: Applications to Elasticity Problems in Several Parametric Settings}, journal = {SEMA SIMAI Springer Series}, volume = {15}, year = {2018}, pages = {203-247}, abstract = {

In this work we consider (hierarchical, Lagrange) reduced basis approximation and a posteriori error estimation for elasticity problems in affinely parametrized geometries. The essential ingredients of the methodology are: a Galerkin projection onto a low-dimensional space associated with a smooth {\textquotedblleft}parametric manifold{\textquotedblright}{\textemdash}dimension reduction; an efficient and effective greedy sampling methods for identification of optimal and numerically stable approximations{\textemdash}rapid convergence; an a posteriori error estimation procedures{\textemdash}rigorous and sharp bounds for the functional outputs related with the underlying solution or related quantities of interest, like stress intensity factor; and Offline-Online computational decomposition strategies{\textemdash}minimum marginal cost for high performance in the real-time and many-query (e.g., design and optimization) contexts. We present several illustrative results for linear elasticity problem in parametrized geometries representing 2D Cartesian or 3D axisymmetric configurations like an arc-cantilever beam, a center crack problem, a composite unit cell or a woven composite beam, a multi-material plate, and a closed vessel. We consider different parametrization for the systems: either physical quantities{\textemdash}to model the materials and loads{\textemdash}and geometrical parameters{\textemdash}to model different geometrical configurations{\textemdash}with isotropic and orthotropic materials working in plane stress and plane strain approximation. We would like to underline the versatility of the methodology in very different problems. As last example we provide a nonlinear setting with increased complexity.

}, doi = {10.1007/978-3-319-94676-4_8}, url = {https://www.scopus.com/inward/record.uri?eid=2-s2.0-85055036627\&doi=10.1007\%2f978-3-319-94676-4_8\&partnerID=40\&md5=e9c07038e7bcc6668ec702c0653410dc}, author = {D.B.P. Huynh and Federico Pichi and Gianluigi Rozza} } @inbook {2018, title = {Reduced Basis Approximation and A Posteriori Error Estimation: Applications to Elasticity Problems in Several Parametric Settings}, booktitle = {Numerical Methods for PDEs}, volume = {15}, number = {SEMA SIMAI}, year = {2018}, doi = {https://doi.org/10.1007/978-3-319-94676-4_8}, url = {https://link.springer.com/chapter/10.1007/978-3-319-94676-4_8}, author = {Huynh, D. B. P. and Federico Pichi and Gianluigi Rozza} } @article {1806.06604, title = {Reducibility for a class of weakly dispersive linear operators arising from the Degasperis Procesi equation}, year = {2018}, author = {Roberto Feola and Filippo Giuliani and Michela Procesi} } @article {doi:10.1142/S0219891618500200, title = {Regularity estimates for scalar conservation laws in one space dimension}, journal = {Journal of Hyperbolic Differential Equations}, volume = {15}, number = {04}, year = {2018}, pages = {623-691}, abstract = {

We deal with the regularizing effect that, in scalar conservation laws in one space dimension, the nonlinearity of the flux function f has on the entropy solution. More precisely, if the set w : f"(w)/=0 is dense, the regularity of the solution can be expressed in terms of BVΦ spaces, where Φ depends on the nonlinearity of f. If moreover the set w : f"(w) = 0 is finite, under the additional polynomial degeneracy condition at the inflection points, we prove that f'o u(t) ∈BV loc(ℝ) for every t \> 0 and that this can be improved to SBVloc(ℝ) regularity except an at most countable set of singular times. Finally, we present some examples that show the sharpness of these results and counterexamples to related questions, namely regularity in the kinetic formulation and a property of the fractional BV spaces.

}, doi = {10.1142/S0219891618500200}, url = {https://doi.org/10.1142/S0219891618500200}, author = {Elio Marconi} } @article {HasLan18, title = {Renormalization analysis for degenerate ground states}, journal = {J. Funct. Anal.}, volume = {275}, number = {1}, year = {2018}, pages = {103{\textendash}148}, issn = {0022-1236}, doi = {10.1016/j.jfa.2018.03.005}, url = {https://doi.org/10.1016/j.jfa.2018.03.005}, author = {Hasler, David and Markus Lange} } @article {doi:10.1098/rspa.2017.0608, title = {Revealing new dynamical patterns in a reaction\&\#x2013;diffusion model with cyclic competition via a novel computational framework}, journal = {Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences}, volume = {474}, number = {2213}, year = {2018}, pages = {20170608}, abstract = {Understanding how patterns and travelling waves form in chemical and biological reaction{\textendash}diffusion models is an area which has been widely researched, yet is still experiencing fast development. Surprisingly enough, we still do not have a clear understanding about all possible types of dynamical regimes in classical reaction{\textendash}diffusion models, such as Lotka{\textendash}Volterra competition models with spatial dependence. In this study, we demonstrate some new types of wave propagation and pattern formation in a classical three species cyclic competition model with spatial diffusion, which have been so far missed in the literature. These new patterns are characterized by a high regularity in space, but are different from patterns previously known to exist in reaction{\textendash}diffusion models, and may have important applications in improving our understanding of biological pattern formation and invasion theory. Finding these new patterns is made technically possible by using an automatic adaptive finite element method driven by a novel a posteriori error estimate which is proved to provide a reliable bound for the error of the numerical method. We demonstrate how this numerical framework allows us to easily explore the dynamical patterns in both two and three spatial dimensions.}, doi = {10.1098/rspa.2017.0608}, url = {https://royalsocietypublishing.org/doi/abs/10.1098/rspa.2017.0608}, author = {Andrea Cangiani and E.H. Georgoulis and Morozov, A. Yu. and Sutton, O. J.} } @article {1711.08253, title = {Random spectrahedra}, year = {2017}, author = {Paul Breiding and Khazhgali Kozhasov and Antonio Lerario} } @article {riccobelli2017rayleigh, title = {Rayleigh{\textendash}Taylor instability in soft elastic layers}, journal = {Phil. Trans. R. Soc. A}, volume = {375}, number = {2093}, year = {2017}, publisher = {The Royal Society}, author = {Davide Riccobelli and Pasquale Ciarletta} } @article {Narain2017, title = {Real topological string amplitudes}, journal = {Journal of High Energy Physics}, volume = {2017}, number = {3}, year = {2017}, month = {Mar}, pages = {80}, abstract = {

We discuss the physical superstring correlation functions in type I theory (or equivalently type II with orientifold) that compute real topological string amplitudes. We consider the correlator corresponding to holomorphic derivative of the real topological amplitude $\mathcal{G_\chi}$, at fixed worldsheet Euler characteristic $\chi$. This corresponds in the low-energy effective action to $\mathcal{N}=2$ Weyl multiplet, appropriately reduced to the orientifold invariant part, and raised to the power $g{\textquoteright}= -\chi+ 1$. We show that the physical string correlator gives precisely the holomorphic derivative of topological amplitude. Finally, we apply this method to the standard closed oriented case as well, and prove a similar statement for the topological amplitude $\mathcal{F}_g$.

}, issn = {1029-8479}, doi = {10.1007/JHEP03(2017)080}, url = {https://doi.org/10.1007/JHEP03(2017)080}, author = {Narain, K. S. and Nicol{\`o} Piazzalunga and Alessandro Tanzini} } @article {2017, title = {Reduced Basis Methods for Uncertainty Quantification}, journal = {SIAM/ASA Journal on Uncertainty Quantification}, volume = {5}, year = {2017}, month = {08/2017}, pages = {869}, type = {reviewed}, chapter = {813}, abstract = {

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. Babu{\v s}ka, 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

}, author = {Peng Chen and Alfio Quarteroni and Gianluigi Rozza} } @article {2017, title = {A reduced order model for investigating the dynamics of the Gen-IV LFR coolant pool}, journal = {Applied Mathematical Modelling}, volume = {46}, year = {2017}, pages = {263-284}, doi = {10.1016/j.apm.2017.01.066}, url = {https://www.scopus.com/inward/record.uri?eid=2-s2.0-85020006623\&doi=10.1016\%2fj.apm.2017.01.066\&partnerID=40\&md5=f6e5715037eb0ef2ecb9ae03f373294f}, author = {Stefano Lorenzi and Antonio Cammi and Lelio Luzzi and Gianluigi Rozza} } @inbook {BallarinRozzaMaday2017, title = {Reduced-order semi-implicit schemes for fluid-structure interaction problems}, booktitle = {Model Reduction of Parametrized Systems}, year = {2017}, pages = {149{\textendash}167}, publisher = {Springer International Publishing}, organization = {Springer International Publishing}, chapter = {Reduced-order semi-implicit schemes for fluid-structure interaction problems}, abstract = {

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

}, doi = {10.1007/978-3-319-58786-8_10}, author = {F. Ballarin and Gianluigi Rozza and Yvon Maday}, editor = {Peter Benner and Mario Ohlberger and Anthony Patera and Gianluigi Rozza and Karsten Urban} } @article {2017, title = {Regularity estimates for scalar conservation laws in one space dimension}, number = {SISSA;37/2017/MATE}, year = {2017}, abstract = {In this paper we deal with the regularizing effect that, in a scalar conservation laws in one space dimension, the nonlinearity of the flux function {\textflorin} has on the entropy solution. More precisely, if the set ⟨w : {\textflorin} " (w) /= 0⟩ is dense, the regularity of the solution can be expressed in terms of BV Ф spaces, where Ф depends on the nonlinearity of {\textflorin}. If moreover the set ⟨w : {\textflorin} " (w) = 0⟩ is finite, under the additional polynomial degeneracy condition at the inflection points, we prove that {\textflorin}{\textquoteright} 0 u(t) ∈ BVloc (R) for every t > 0 and that this can be improved to SBVloc (R) regularity except an at most countable set of singular times. Finally we present some examples that shows the sharpness of these results and counterexamples to related questions, namely regularity in the kinetic formulation and a property of the fractional BV spaces.}, url = {http://preprints.sissa.it/handle/1963/35291}, author = {Elio Marconi} } @inbook {Olgiati2017, title = {Remarks on the Derivation of Gross-Pitaevskii Equation with Magnetic Laplacian}, booktitle = {Advances in Quantum Mechanics: Contemporary Trends and Open Problems}, year = {2017}, pages = {257{\textendash}266}, publisher = {Springer International Publishing}, organization = {Springer International Publishing}, address = {Cham}, abstract = {

The effective dynamics for a Bose-Einstein condensate in the regime of high dilution and subject to an external magnetic field is governed by a magnetic Gross-Pitaevskii equation. We elucidate the steps needed to adapt to the magnetic case the proof of the derivation of the Gross-Pitaevskii equation within the {\textquoteleft}{\textquoteleft}projection counting{\textquoteright}{\textquoteright} scheme.

}, isbn = {978-3-319-58904-6}, doi = {10.1007/978-3-319-58904-6_15}, url = {https://doi.org/10.1007/978-3-319-58904-6_15}, author = {Alessandro Olgiati}, editor = {Alessandro Michelangeli and Gianfausto Dell{\textquoteright}Antonio} } @article {11967, title = {A Reduced Basis Approach for Modeling the Movement of Nuclear Reactor Control Rods}, journal = {NERS-14-1062; ASME J of Nuclear Rad Sci, 2, 2 (2016) 021019}, volume = {2}, number = {Journal of Nuclear Engineering and Radiation Science;2}, year = {2016}, note = {8 pages}, month = {02/2016}, pages = {8}, publisher = {ASME}, abstract = {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 {\textquotedblleft}classical{\textquotedblright} 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.}, doi = {10.1115/1.4031945}, url = {http://urania.sissa.it/xmlui/handle/1963/35192}, author = {Alberto Sartori and Antonio Cammi and Lelio Luzzi and Gianluigi Rozza} } @article {2015, title = {Reduced basis approaches in time-dependent noncoercive settings for modelling the movement of nuclear reactor control rods}, journal = {Communications in Computational Physics}, year = {2016}, month = {2016}, publisher = {SISSA}, abstract = {

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 {\textquotedblleft}staircase{\textquotedblright} 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 {\textquotedblleft}truth{\textquotedblright} solutions. At the same time, the computational speed-up in the Online phase, with respect to the fine {\textquotedblleft}truth{\textquotedblright} 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.

}, url = {http://urania.sissa.it/xmlui/handle/1963/34963}, author = {Alberto Sartori and Antonio Cammi and Lelio Luzzi and Gianluigi Rozza} } @article {2015, title = {Reduced basis method and domain decomposition for elliptic problems in networks and complex parametrized geometries}, journal = {Computers and Mathematics with Applications }, volume = {71}, year = {2016}, month = {01/2016}, pages = {430}, publisher = {Elsevier}, chapter = {408}, abstract = {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.}, author = {Laura Iapichino and Alfio Quarteroni and Gianluigi Rozza} } @article {gottsche2016refined, title = {Refined node polynomials via long edge graphs}, journal = {Communications in Number Theory and Physics}, volume = {10}, number = {2}, year = {2016}, pages = {193{\textendash}234}, publisher = {International Press of Boston}, abstract = {

The generating functions of the Severi degrees for sufficiently ample line bundles on algebraic surfaces are multiplicative in the topological invariants of the surface and the line bundle. Recently new proofs of this fact were given for toric surfaces by Block, Colley, Kennedy and Liu, Osserman, using tropical geometry and in particular the combinatorial tool of long-edged graphs. In the first part of this paper these results are for $\mathbb{P}^2$ and rational ruled surfaces generalised to refined Severi degrees. In the second part of the paper we give a number of mostly conjectural generalisations of this result to singular surfaces, and curves with prescribed multiple points. The formulas involve modular forms and theta functions.

}, doi = {10.4310/CNTP.2016.v10.n2.a2}, url = {http://dx.doi.org/10.4310/CNTP.2016.v10.n2.a2}, author = {Lothar G{\"o}ttsche and Benjamin Kipkirui Kikwai} } @article {doi:10.1137/15M1007380, title = {Renormalization for Autonomous Nearly Incompressible BV Vector Fields in Two Dimensions}, journal = {SIAM Journal on Mathematical Analysis}, volume = {48}, number = {1}, year = {2016}, pages = {1-33}, abstract = {

Given a bounded autonomous vector field $b \colon \mathbb{R}^d \to \mathbb{R}^d$, we study the uniqueness of bounded solutions to the initial value problem for the related transport equation \begin{equation*} \partial_t u + b \cdot \nabla u= 0. \end{equation*} We are interested in the case where $b$ is of class BV and it is nearly incompressible. Assuming that the ambient space has dimension $d=2$, we prove uniqueness of weak solutions to the transport equation. The starting point of the present work is the result which has been obtained in [7] (where the steady case is treated). Our proof is based on splitting the equation onto a suitable partition of the plane: this technique was introduced in [3], using the results on the structure of level sets of Lipschitz maps obtained in [1]. Furthermore, in order to construct the partition, we use Ambrosio{\textquoteright}s superposition principle [4].

}, doi = {10.1137/15M1007380}, url = {https://doi.org/10.1137/15M1007380}, author = {Stefano Bianchini and Paolo Bonicatto and N.A. Gusev} } @inbook {MR3585793, title = {Review of discontinuous Galerkin finite element methods for partial differential equations on complicated domains}, booktitle = {Building bridges: connections and challenges in modern approaches to numerical partial differential equations}, series = {Lect. Notes Comput. Sci. Eng.}, volume = {114}, year = {2016}, pages = {279{\textendash}308}, publisher = {Springer, [Cham]}, organization = {Springer, [Cham]}, author = {Antonietti, Paola F. and Andrea Cangiani and Collis, Joe and Dong, Zhaonan and E.H. Georgoulis and Stefano Giani and Houston, Paul} } @article {MR3592329, title = {Rogue waves in multiphase solutions of the focusing nonlinear Schr{\"o}dinger equation}, journal = {Proc. A.}, volume = {472}, number = {2194}, year = {2016}, pages = {20160340, 12}, issn = {1364-5021}, doi = {10.1098/rspa.2016.0340}, url = {http://dx.doi.org/10.1098/rspa.2016.0340}, author = {Marco Bertola and Gennady El and Alexander Tovbis} } @article {2015, title = {Reduced basis approximation and a-posteriori error estimation for the coupled Stokes-Darcy system}, journal = {Advances in Computational Mathematics}, volume = {special issue for MoRePaS 2012}, year = {2015}, abstract = {

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.

}, keywords = {Domain decomposition, Error estimation, Non-coercive problem, Porous medium equation, Reduced basis method, Stokes flow}, issn = {1019-7168}, doi = { 10.1007/s10444-014-9396-6}, author = {Immanuel Martini and Gianluigi Rozza and Bernard Haasdonk} } @article {PacciariniRozza2015, title = {Reduced basis approximation of parametrized advection-diffusion PDEs with high P{\'e}clet number}, journal = {Lecture Notes in Computational Science and Engineering}, volume = {103}, year = {2015}, pages = {419{\textendash}426}, abstract = {

In this work we show some results about the reduced basis approximation of advection dominated parametrized problems, i.e. advection-diffusion problems with high P{\'e}clet 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.

}, doi = {10.1007/978-3-319-10705-9__41}, author = {Pacciarini, P. and Gianluigi Rozza} } @article {NegriManzoniRozza2015, title = {Reduced basis approximation of parametrized optimal flow control problems for the Stokes equations}, journal = {Computers and Mathematics with Applications}, volume = {69}, number = {4}, year = {2015}, pages = {319{\textendash}336}, abstract = {

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.

}, doi = {10.1016/j.camwa.2014.12.010}, author = {Federico Negri and Andrea Manzoni and Gianluigi Rozza} } @article {2015, title = {Reduced Basis Isogeometric Methods (RB-IGA) for the real-time simulation of potential flows about parametrized NACA airfoils}, journal = {Comput Methods Appl Mech Eng. 2015;284:1147{\textendash}1180}, number = {;284}, year = {2015}, abstract = {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.}, doi = {10.1016/j.cma.2014.11.037}, author = {Andrea Manzoni and Filippo Salmoiraghi and Luca Heltai} } @conference {StabileMatthiesBorri2015, title = {A Reduced Order Model for the Simulation of Mooring Cable Dynamics}, booktitle = {Computational Methods in Marine Engineering VI {\textendash} MARINE2015}, year = {2015}, pages = {387{\textendash}400}, publisher = {Salvatore, Francesco; Broglia, Riccardo; Muscari, Roberto}, organization = {Salvatore, Francesco; Broglia, Riccardo; Muscari, Roberto}, author = {Giovanni Stabile and Hermann G. Matthies and Claudio Borri} } @mastersthesis {2015, title = {The relaxed area of maps from the plane to the plane with a line discontinuity, and the role of semicartesian surfaces.}, year = {2015}, school = {SISSA}, abstract = {In this thesis we study the relaxation of the area functional w.r.t. the L^1 topology of a map from a bounded planar domain with values in the plane and jumping on a segment. We estimate from above the singular contribution of this functional due to the presence of the jump in terms of the infimum of the area among a suitable family of surfaces that we call semicartesian surfaces. In our analysis, we also introduce a different notion of area, namely the relaxation of the area w.r.t. a convergence stronger than the L^1 convergence, whose singular contribution is completely characterized in terms of suitable semicartesian area minimizing problems. We propose also some examples of maps for which the two notions of relaxation are different: these examples underline the highly non-local behaviour of the L^1-relaxation, and justify the introduction of the other functional. Some result about the existence of a semicartesian area-minimizing surface is also provided.}, keywords = {Area functional}, author = {Lucia Tealdi} } @article {2015, title = {Results on the minimization of the Dirichlet functional among semicartesian parametrizations}, year = {2015}, note = {The article is compsed of 18 pages and is recorded in PDF format}, abstract = {

We start to investigate the existence of conformal minimizers for the Dirichlet functional in the setting of the so-called semicartesian parametrizations, adapting to this context some techniques used in solving the classical Plateau{\textquoteright}s problem. The final goal is to find area minimizing semicartesian parametrizations spanning a Jordan curve obtained as union of two graphs; this problem appeared in the study of the relaxed area functional for maps from the plane to the plane jumping on a line.

}, url = {http://urania.sissa.it/xmlui/handle/1963/34488}, author = {Lucia Tealdi and Giovanni Bellettini and Maurizio Paolini} } @article {2015, title = {Rigidity of three-dimensional lattices and dimension reduction in heterogeneous nanowires}, number = {SISSA;04/2015/MATE}, year = {2015}, institution = {SISSA}, abstract = {In the context of nanowire heterostructures we perform a discrete to continuum limit of the corresponding free energy by means of Γ-convergence techniques. Nearest neighbours are identified by employing the notions of Voronoi diagrams and Delaunay triangulations. The scaling of the nanowire is done in such a way that we perform not only a continuum limit but a dimension reduction simultaneously. The main part of the proof is a discrete geometric rigidity result that we announced in an earlier work and show here in detail for a variety of three-dimensional lattices. We perform the passage from discrete to continuum twice: once for a system that compensates a lattice mismatch between two parts of the heterogeneous nanowire without defects and once for a system that creates dislocations. It turns out that we can verify the experimentally observed fact that the nanowires show dislocations when the radius of the specimen is large}, url = {http://urania.sissa.it/xmlui/handle/1963/7494}, author = {Giuliano Lazzaroni and Mariapia Palombaro and Anja Schlomerkemper} } @article {2014, title = {Rate-independent damage in thermo-viscoelastic materials with inertia}, number = {SISSA;52/2014/MATE}, year = {2014}, institution = {SISSA}, abstract = {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.}, url = {http://urania.sissa.it/xmlui/handle/1963/7444}, author = {Giuliano Lazzaroni and Riccarda Rossi and Marita Thomas and Rodica Toader} } @mastersthesis {2014, title = {Rational curves and instantons on the Fano threefold Y_5}, year = {2014}, school = {arXiv preprint}, abstract = {This thesis is an investigation of the moduli spaces of instanton bundles on the Fano threefold Y_5 (a linear section of Gr(2,5)). It contains new proofs of classical facts about lines, conics and cubics on Y_5, and about linear sections of Y_5. The main original results are a Grauert-M{\"u}lich theorem for the splitting type of instantons on conics, a bound to the splitting type of instantons on lines and an SL_2-equivariant description of the moduli space in charge 2 and 3. Using these results we prove the existence of a unique SL_2-equivariant instanton of minimal charge and we show that for all instantons of charge 2 the divisor of jumping lines is smooth. In charge 3, we provide examples of instantons with reducible divisor of jumping lines. Finally, we construct a natural compactification for the moduli space of instantons of charge 3, together with a small resolution of singularities for it.}, keywords = {Moduli space of vector bundles}, url = {http://urania.sissa.it/xmlui/handle/1963/7482}, author = {Giangiacomo Sanna} } @conference {IapichinoQuarteroniRozzaVolkwein2014, title = {Reduced basis method for the Stokes equations in decomposable domains using greedy optimization}, booktitle = {ECMI 2014 proceedings}, year = {2014}, pages = {1{\textendash}7}, author = {Laura Iapichino and Alfio Quarteroni and Gianluigi Rozza and Volkwein, Stefan} } @book {2014, title = {Reduced Order Methods for Modeling and Computational Reduction}, series = {MS\&A}, volume = {9}, year = {2014}, pages = {334}, publisher = {Springer}, organization = {Springer}, edition = {1}, address = {Milano}, abstract = {

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.

}, keywords = {reduced order methods, MOR, ROM, POD, RB, greedy, CFD, Numerical Analysis}, issn = {978-3-319-02089-1}, doi = {10.1007/978-3-319-02090-7}, author = {Alfio Quarteroni and Gianluigi Rozza} } @proceedings {2014, title = {A reduced order model for multi-group time-dependent parametrized reactor spatial kinetics}, number = {International Conference on Nuclear Engineering, Proceedings, ICONE;volume 5;}, year = {2014}, note = {2014 22nd International Conference on Nuclear Engineering, ICONE 2014; Prague; Czech Republic; 7 July 2014 through 11 July 2014; Code 109131;}, month = {07/2014}, pages = {V005T17A048-V005T17A048}, publisher = {American Society of Mechanical Engineers (ASME)}, edition = {American Society Mechanical Engineering}, address = {Prague, Czech Republic}, abstract = {

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.

}, isbn = {978-079184595-0}, doi = {10.1115/ICONE22-30707}, url = {http://urania.sissa.it/xmlui/handle/1963/35123}, author = {Alberto Sartori and Davide Baroli and Antonio Cammi and Lelio Luzzi and Gianluigi Rozza} } @inbook {2013, title = {Reduction on characteristics for continuous of a scalar balance law}, booktitle = {AIMS Series on Applied Mathematics, vol. 8 (2014): 399 - 406}, number = {AIMS Series on Applied Mathematics}, year = {2014}, publisher = {SISSA}, organization = {SISSA}, keywords = {Method of characteristics}, url = {http://hdl.handle.net/1963/6562}, author = {Giovanni Alberti and Stefano Bianchini and Laura Caravenna} } @article {2014, title = {A Review of the Sixth Painlev{\'e} Equation}, number = {Constructive approximation;volume 41; issue 3; pages 495-527;}, year = {2014}, publisher = {Springer}, abstract = {For the Painlev{\'e} VI transcendents, we provide a unitary description of the critical behaviours, the connection formulae, their complete tabulation, and the asymptotic distribution of poles close to a critical point.}, doi = {10.1007/s00365-014-9250-6}, url = {http://urania.sissa.it/xmlui/handle/1963/34658}, author = {Davide Guzzetti} } @article {2014, title = {A robotic crawler exploiting directional frictional interactions: experiments, numerics, and derivation of a reduced model}, journal = {Proceedings of the Royal Society A 470, 20140333 (2014)}, year = {2014}, publisher = {Royal Society Publishing}, abstract = {We present experimental and numerical results for a model crawler which is able to extract net positional changes from reciprocal shape changes, i.e. {\textquoteleft}breathing-like{\textquoteright} deformations, thanks to directional, frictional interactions with a textured solid substrate, mediated by flexible inclined feet. We also present a simple reduced model that captures the essential features of the kinematics and energetics of the gait, and compare its predictions with the results from experiments and from numerical simulations.}, doi = {10.1098/rspa.2014.0333}, author = {Giovanni Noselli and Antonio DeSimone} } @article {2013, title = {Reduced basis approximation and a posteriori error estimation for Stokes flows in parametrized geometries: roles of the inf-sup stability constants}, journal = {Numerische Mathematik, 2013}, number = {SISSA Preprint;43/2012/M}, year = {2013}, publisher = {Springer}, abstract = {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.}, keywords = {parametrized Stokes equations}, url = {http://hdl.handle.net/1963/6339}, author = {Gianluigi Rozza and Phuong Huynh and Andrea Manzoni} } @inbook {2015, title = {Reduced Basis Approximation for the Structural-Acoustic Design based on Energy Finite Element Analysis (RB-EFEA)}, booktitle = {CEMRACS 2013 - Modelling and simulation of complex systems: stochastic and deterministic approaches}, volume = {48}, number = {ESAIM Proceedings}, year = {2013}, pages = {98-115}, doi = { http://dx.doi.org/10.1051/proc/201448004}, author = {Denis Devaud and Gianluigi Rozza} } @article {NegriRozzaManzoniQuarteroni2013, title = {Reduced basis method for parametrized elliptic optimal control problems}, journal = {SIAM Journal on Scientific Computing}, volume = {35}, number = {5}, year = {2013}, pages = {A2316{\textendash}A2340}, abstract = {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.}, doi = {10.1137/120894737}, author = {Federico Negri and Gianluigi Rozza and Andrea Manzoni and Alfio Quarteroni} } @article {2013, title = {A Reduced Computational and Geometrical Framework for Inverse Problems in Haemodynamics}, number = {SISSA preprint;15/2013/MATE}, year = {2013}, institution = {SISSA}, author = {Toni Lassila and Andrea Manzoni and Alfio Quarteroni and Gianluigi Rozza} } @article {2013, title = {A reduced-order strategy for solving inverse Bayesian identification problems in physiological flows}, number = {SISSA preprint;14/2013/MATE}, year = {2013}, institution = {SISSA}, author = {Toni Lassila and Andrea Manzoni and Alfio Quarteroni and Gianluigi Rozza} } @article {2013, title = {Reduction Strategies for Shape Dependent Inverse Problems in Haemodynamics}, number = {SISSA preprint;13/2013/MATE}, year = {2013}, institution = {SISSA}, author = {Toni Lassila and Andrea Manzoni and Gianluigi Rozza} } @article {11613, title = {Remarks on the Moser{\textendash}Trudinger inequality}, journal = {Advances in Nonlinear Analysis}, volume = {2}, number = {Advances in nonlinear analysis;volume 2; issue 4}, year = {2013}, note = {The article is composed of 32 pages inad recorded in PDF format}, pages = {389-425}, publisher = {Advances in Nonlinear Analysis}, abstract = {

We extend the Moser-Trudinger inequality to any Euclidean domain satisfying Poincar{\'e}{\textquoteright}s inequality. We find out that the same equivalence does not hold in general for conformal metrics on the unit ball, showing counterexamples. We also study the existence of extremals for the Moser-Trudinger inequalities for unbounded domains, proving it for the infinite planar strip.

}, doi = {10.1515/anona-2013-0014}, url = {http://edoc.unibas.ch/43974/}, author = {Gabriele Mancini and Luca Battaglia} } @proceedings {2012, title = {Reduction strategies for PDE-constrained oprimization problems in Haemodynamics}, number = {SISSA;41/2012/M}, year = {2012}, abstract = {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.}, keywords = {inverse problems}, url = {http://hdl.handle.net/1963/6338}, author = {Gianluigi Rozza and Andrea Manzoni and Federico Negri} } @article {garrione2012, title = {Resonance at the first eigenvalue for first-order systems in the plane: vanishing Hamiltonians and the Landesman-Lazer condition}, journal = {Differential Integral Equations}, volume = {25}, number = {5/6}, year = {2012}, month = {05}, pages = {505{\textendash}526}, publisher = {Khayyam Publishing, Inc.}, url = {https://projecteuclid.org:443/euclid.die/1356012676}, author = {Maurizio Garrione} } @article {2012, title = {Reverse engineering the euglenoid movement}, journal = {Proceedings of the National Academy of Sciences of the United States of America. Volume 109, Issue 44, 30 October 2012, Pages 17874-17879}, year = {2012}, abstract = {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.}, keywords = {microswimmers}, doi = {10.1073/pnas.1213977109}, url = {http://hdl.handle.net/1963/6444}, author = {Marino Arroyo and Luca Heltai and Daniel Mill{\'a}n and Antonio DeSimone} } @article {2012, title = {A Review on The Sixth Painlev{\'e} Equation}, number = {arXiv:1210.0311;}, year = {2012}, note = {31 pages, 10 figures}, publisher = {SISSA}, abstract = {

For the Painlev\\\'e 6 transcendents, we provide a unitary description of the\r\ncritical behaviours, the connection formulae, their complete tabulation, and\r\nthe asymptotic distribution of the poles close to a critical point.

}, keywords = {Painlev{\'e} equation}, url = {http://hdl.handle.net/1963/6525}, author = {Davide Guzzetti} } @article {BertolaCafasso2, title = {Riemann{\textendash}Hilbert approach to multi-time processes: The Airy and the Pearcey cases}, journal = {Physica D: Nonlinear Phenomena}, volume = {241}, number = {23-24}, year = {2012}, note = {Integrable Systems in Pure and Applied Mathematics}, pages = {2237 - 2245}, keywords = {Integrable kernels}, issn = {0167-2789}, doi = {10.1016/j.physd.2012.01.003}, url = {http://www.sciencedirect.com/science/article/pii/S0167278912000115}, author = {Marco Bertola and Mattia Cafasso} } @article {2012, title = {On robust Lie-algebraic stability conditions for switched linear systems}, journal = {Systems and Control Letters. Volume 61, Issue 2, February 2012, Pages 347-353}, year = {2012}, abstract = {This paper presents new sufficient conditions for exponential stability of switched linear systems under arbitrary switching, which involve the commutators (Lie brackets) among the given matrices generating the switched system. The main novelty feature of these stability criteria is that, unlike their earlier counterparts, they are robust with respect to small perturbations of the system parameters.}, doi = {10.1016/j.sysconle.2011.11.016}, url = {http://hdl.handle.net/1963/6455}, author = {Andrei A. Agrachev and Yurij Baryshnikov and Daniel Liberzon} } @article {garrione2011resonance, title = {Resonance and Landesman-Lazer conditions for first order systems in R^2}, journal = {Le Matematiche}, volume = {66}, number = {1}, year = {2011}, pages = {153{\textendash}160}, abstract = {

The first part of the paper surveys the concept of resonance for $T$-periodic nonlinear problems. In the second part, some new results about existence conditions for nonlinear planar systems are presented. In particular, the Landesman-Lazer conditions are generalized to systems in $\mathbbR^2$ where the nonlinearity interacts with two resonant Hamiltonians. Such results apply to second order equations, generalizing previous theorems by Fabry [4] (for the undamped case), and Frederickson-Lazer [9] (for the case with friction). The results have been obtained with A. Fonda, and have been published in [8].

}, author = {Maurizio Garrione} } @article {BOSCAGGIN20114166, title = {Resonance and rotation numbers for planar Hamiltonian systems: Multiplicity results via the Poincar{\'e}{\textendash}Birkhoff theorem}, journal = {Nonlinear Analysis: Theory, Methods \& Applications}, volume = {74}, number = {12}, year = {2011}, pages = {4166 - 4185}, abstract = {

In the general setting of a planar first order system (0.1)u'=G(t,u),u∈R2, with G:[0,T]{\texttimes}R2{\textrightarrow}R2, we study the relationships between some classical nonresonance conditions (including the Landesman{\textendash}Lazer one) {\textemdash} at infinity and, in the unforced case, i.e. G(t,0)=0, at zero {\textemdash} and the rotation numbers of {\textquotedblleft}large{\textquotedblright} and {\textquotedblleft}small{\textquotedblright} solutions of (0.1), respectively. Such estimates are then used to establish, via the Poincar{\'e}{\textendash}Birkhoff fixed point theorem, new multiplicity results for T-periodic solutions of unforced planar Hamiltonian systems Ju'=∇uH(t,u) and unforced undamped scalar second order equations x"+g(t,x)=0. In particular, by means of the Landesman{\textendash}Lazer condition, we obtain sharp conclusions when the system is resonant at infinity.

}, keywords = {Multiple periodic solutions, Poincar{\'e}{\textendash}Birkhoff theorem, Resonance, Rotation number}, issn = {0362-546X}, doi = {https://doi.org/10.1016/j.na.2011.03.051}, url = {http://www.sciencedirect.com/science/article/pii/S0362546X11001817}, author = {Alberto Boscaggin and Maurizio Garrione} } @article {2010, title = {The reductions of the dispersionless 2D Toda hierarchy and their Hamiltonian structures}, journal = {J. Phys. A 43 (2010) 045201}, number = {arXiv.org;0910.1210v1}, year = {2010}, publisher = {IOP Publishing}, abstract = {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.}, doi = {10.1088/1751-8113/43/4/045201}, url = {http://hdl.handle.net/1963/3846}, author = {Guido Carlet and Paolo Lorenzoni and Andrea Raimondo} } @article {2010, title = {Riemann-Roch theorems and elliptic genus for virtually smooth schemes}, journal = {Geom. Topol. 14 (2010) 83-115}, number = {arXiv.org;0706.0988v1}, year = {2010}, publisher = {Mathematical Sciences Publishers}, abstract = {For a proper scheme X with a fixed 1-perfect obstruction theory, we define virtual versions of holomorphic Euler characteristic, chi y-genus, and elliptic genus; they are deformation invariant, and extend the usual definition in the smooth case. We prove virtual versions of the Grothendieck-Riemann-Roch and Hirzebruch-Riemann-Roch theorems. We show that the virtual chi y-genus is a polynomial, and use this to define a virtual topological Euler characteristic. We prove that the virtual elliptic genus satisfies a Jacobi modularity property; we state and prove a localization theorem in the toric equivariant case. We show how some of our results apply to moduli spaces of stable sheaves.}, doi = {10.2140/gt.2010.14.83}, url = {http://hdl.handle.net/1963/3888}, author = {Barbara Fantechi and Lothar G{\"o}ttsche} } @article {2010, title = {The role of membrane viscosity in the dynamics of fluid membranes}, number = {SISSA;55/2010/M}, year = {2010}, abstract = {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.}, url = {http://hdl.handle.net/1963/3930}, author = {Marino Arroyo and Antonio DeSimone and Luca Heltai} } @article {BertolaBalogh, title = {Regularity of a vector potential problem and its spectral curve}, journal = {J. Approx. Theory}, volume = {161}, number = {1}, year = {2009}, pages = {353{\textendash}370}, issn = {0021-9045}, doi = {10.1016/j.jat.2008.10.010}, url = {http://0-dx.doi.org.mercury.concordia.ca/10.1016/j.jat.2008.10.010}, author = {Ferenc Balogh and Marco Bertola} } @article {2009, title = {Relaxation dynamics of fluid membranes}, journal = {Phys. Rev. E 79 (2009) 031915}, year = {2009}, publisher = {American Physical Society}, abstract = {We study the effect of membrane viscosity in the dynamics of liquid membranes-possibly with free or internal boundaries-driven by conservative forces (curvature elasticity and line tension) and dragged by the bulk dissipation of the ambient fluid and the friction occurring when the amphiphilic molecules move relative to each other. To this end, we formulate a continuum model which includes a form of the governing equations for a two-dimensional viscous fluid moving on a curved, time-evolving surface. The effect of membrane viscosity has received very limited attention in previous continuum studies of the dynamics of fluid membranes, although recent coarse-grained discrete simulations suggest its importance. By applying our model to the study of vesiculation and membrane fusion in a simplified geometry, we conclude that membrane viscosity plays a dominant role in the relaxation dynamics of fluid membranes of sizes comparable to those found in eukaryotic cells, and is not negligible in many large synthetic systems of current interest.}, doi = {10.1103/PhysRevE.79.031915}, url = {http://hdl.handle.net/1963/3618}, author = {Marino Arroyo and Antonio DeSimone} } @article {2008, title = {Relaxation of some transversally isotropic energies and applications to smectic A elastomers}, journal = {Math. Models Methods Appl. Sci. 18 (2008) 1-20}, number = {SISSA;01/2007/M}, year = {2008}, doi = {10.1142/S0218202508002693}, url = {http://hdl.handle.net/1963/1912}, author = {James Adams and Sergio Conti and Antonio DeSimone and Georg Dolzmann} } @article {2007, title = {Reciprocal transformations and flat metrics on Hurwitz spaces}, journal = {J. Phys. A 40 (2007) 10769-10790}, number = {arXiv.org;0704.1779v2}, year = {2007}, abstract = {We consider hydrodynamic systems which possess a local Hamiltonian structure of Dubrovin-Novikov type. To such a system there are also associated an infinite number of nonlocal Hamiltonian structures. We give necessary and sufficient conditions so that, after a nonlinear transformation of the independent variables, the reciprocal system still possesses a local Hamiltonian structure of Dubrovin-Novikov type. We show that, under our hypotheses, bi-hamiltonicity is preserved by the reciprocal transformation. Finally we apply such results to reciprocal systems of genus g Whitham-KdV modulation equations.}, doi = {10.1088/1751-8113/40/35/004}, url = {http://hdl.handle.net/1963/2210}, author = {Simonetta Abenda and Tamara Grava} } @article {2007, title = {Reduced density matrices and Bose-Einstein condensation}, number = {SISSA;39/2007/MP}, year = {2007}, abstract = {Emergence and applications of the ubiquitous tool of reduced density matrices in the rigorous analysis of Bose Einstein condensation is reviewed, and new related results are added. The need and the nature of scaling limits of infinitely many particles is discussed, which imposes that a physically meaningful and mathematically well-posed definition of asymptotic condensation is placed at the level of marginals.\\nThe topic of correlations in the condensed state is addressed in order to show their influence at this level of marginals, both in the true condensed state and in the suitable trial functions one introduces to approximate the many-body structure and energy. Complete condensation is shown to be equivalently defined at any fixed k-body level, both for pure and mixed states. Further, it is proven to be equivalent to some other characterizations in terms of asymptotic factorization of the many-body state, which are currently present in the literature.}, url = {http://hdl.handle.net/1963/1986}, author = {Alessandro Michelangeli} } @article {2007, title = {On the reductions and classical solutions of the Schlesinger equations}, journal = {Differential equations and quantum groups, IRMA Lect. Math. Theor. Phys. 9 (2007) 157-187}, year = {2007}, publisher = {SISSA}, abstract = {The Schlesinger equations S(n,m) describe monodromy preserving\\r\\ndeformations of order m Fuchsian systems with n+1 poles. They\\r\\ncan be considered as a family of commuting time-dependent Hamiltonian\\r\\nsystems on the direct product of n copies of m{\texttimes}m matrix algebras\\r\\nequipped with the standard linear Poisson bracket. In this paper we address\\r\\nthe problem of reduction of particular solutions of {\textquotedblleft}more complicated{\textquotedblright}\\r\\nSchlesinger equations S(n,m) to {\textquotedblleft}simpler{\textquotedblright} S(n',m') having n' < n\\r\\nor m' < m.}, url = {http://hdl.handle.net/1963/6472}, author = {Boris Dubrovin and Marta Mazzocco} } @article {2007, title = {On the regularity of weak solutions to H-systems}, journal = {Atti .Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 18 (2007) 209-219}, number = {SISSA;36/2005/M}, year = {2007}, abstract = {Abstract. In this paper we prove that every weak solution to the H-surface equation is locally bounded, provided the prescibed mean curvatore H is asymptotic to a constant at infinity (with a suitable decay rate). No smoothness ssumptions are required on H. We consider also the Dirichlet problem....}, url = {http://hdl.handle.net/1963/1753}, author = {Roberta Musina} } @article {MR2338404, title = {The residual-free-bubble finite element method on anisotropic partitions}, journal = {SIAM J. Numer. Anal.}, volume = {45}, number = {4}, year = {2007}, pages = {1654{\textendash}1678}, issn = {0036-1429}, doi = {10.1137/060658011}, url = {https://doi.org/10.1137/060658011}, author = {Andrea Cangiani and S{\"u}li, Endre} } @article {2007, title = {Role of scaling limits in the rigorous analysis of Bose-Einstein condensation}, journal = {J. Math. Phys. 48 (2007) 102102}, number = {SISSA;14/2007/MP}, year = {2007}, abstract = {In the context of the rigorous analysis of Bose-Einstein condensation, recent achievements have been obtained in the form of asymptotic results when some appropriate scaling is performed in the Hamiltonian, and the limit of infinite number of particles is taken. In particular, two modified thermodynamic limits of infinite dilution turned out to provide an insight in this analysis, the so-\\ncalled Gross-Pitaevskii limit and the related Tomas-Fermi limit. Here such scalings are discussed with respect to their physical and mathematical motivations, and to the currently known results obtained within this framework.}, doi = {10.1063/1.2783114}, url = {http://hdl.handle.net/1963/1984}, author = {Alessandro Michelangeli} } @article {2006, title = {Radial solutions concentrating on spheres of nonlinear Schr{\"o}dinger equations with vanishing potentials}, journal = {Proc. Roy. Soc. Edinburgh Sect. A 136 (2006) 889-907}, number = {SISSA;38/2005/M}, year = {2006}, abstract = {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.}, doi = {10.1017/S0308210500004789}, url = {http://hdl.handle.net/1963/1755}, author = {Antonio Ambrosetti and David Ruiz} } @inbook {2006, title = {Recent analytical developments in micromagnetics}, booktitle = {The science of hysteresis / eds. Giorgio Bertotti, Isaak D. Mayergoyz. - Amsterdam: Elsevier, 2006. Vol.2, 269-381.}, number = {SISSA;76/2004/M}, year = {2006}, isbn = {978-0-12-480874-4}, url = {http://hdl.handle.net/1963/2230}, author = {Antonio DeSimone and Robert V. Kohn and Stefan M{\"u}ller and Felix Otto} } @article {2006, title = {Reflection symmetries for multiqubit density operators}, journal = {J. Math. Phys. 47 (2006) 032104}, number = {arXiv.org;quant-ph/0405123v2}, year = {2006}, abstract = {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.}, doi = {10.1063/1.2181827}, url = {http://hdl.handle.net/1963/2121}, author = {Claudio Altafini and Timothy F. Havel} } @article {2005, title = {Regularity properties of optimal trajectories of single-input control systems in dimension three}, journal = {Journal of Mathematical Sciences 126 (2005) 1561-1573}, year = {2005}, publisher = {Springer}, abstract = {Let q=f(q)+ug(q) be a smooth control system on a three-dimensional manifold. Given a point q 0 of the manifold at which the iterated Lie brackets of f and g satisfy some prescribed independence condition, we analyze the structure of a control function u(t) corresponding to a time-optimal trajectory lying in a neighborhood of q 0. The control turns out to be the concatenation of some bang-bang and some singular arcs. More general optimality criteria than time-optimality are considered. The paper is a step toward to the analysis of generic single-input systems affine in the control in dimension 3. The main techniques used are second-order optimality conditions and, in particular, the index of the second variation of the switching times for bang-bang trajectories.}, doi = {10.1007/s10958-005-0044-z}, url = {http://hdl.handle.net/1963/4794}, author = {Mario Sigalotti} } @article {2004, title = {Reduction by group symmetry of second order variational problems on a semidirect product of Lie groups with positive definite Riemannian metric}, journal = {ESAIM: COCV 10 (2004) 526-548}, year = {2004}, publisher = {EDP Sciences}, abstract = {For a Riemannian structure on a semidirect product of Lie groups, the variational problems can be reduced using the group symmetry. Choosing the Levi-Civita connection of a positive definite metric tensor, instead of any of the canonical connections for the Lie group, simplifies the reduction of the variations but complicates the expression for the Lie algebra valued covariant derivatives. The origin of the discrepancy is in the semidirect product structure, which implies that the Riemannian exponential map and the Lie group exponential map do not coincide. The consequence is that the reduced equations look more complicated than the original ones. The main scope of this paper is to treat the reduction of second order variational problems (corresponding to geometric splines) on such semidirect products of Lie groups. Due to the semidirect structure, a number of extra terms appears in the reduction, terms that are calculated explicitely. The result is used to compute the necessary conditions of an optimal control problem for a simple mechanical control system having invariant Lagrangian equal to the kinetic energy corresponding to the metric tensor. As an example, the case of a rigid body on the Special Euclidean group is considered in detail.}, doi = {10.1051/cocv:2004018}, url = {http://hdl.handle.net/1963/3521}, author = {Claudio Altafini} } @article {2004, title = {Representing multiqubit unitary evolutions via Stokes tensors}, journal = {Phys. Rev. A 70 (2004) 032331}, number = {SISSA;91/2004/M}, year = {2004}, abstract = {For the Stokes tensor parametrization of a multiqubit density operator, we provide an explicit formulation of the corresponding unitary dynamics at the infinitesimal level. The main advantage of this formalism (clearly reminiscent of the ideas of {\textquoteleft}{\textquoteleft}coherences\\\'\\\' and {\textquoteleft}{\textquoteleft}coupling Hamiltonians\\\'\\\' of spin systems) is that the pattern of correlation between qubits and the pattern of infinitesimal correlation are highlighted simultaneously and can be used constructively for qubit manipulation. For example, it allows to compute explicitly a Rodrigues\\\' formula for the one-parameter orbits of nonlocal Hamiltonians. The result is easily generalizable to orbits of Cartan subalgebras and allows to express the Cartan decomposition of unitary propagators as a linear action directly in terms of the infinitesimal generators.}, doi = {10.1103/PhysRevA.70.032331}, url = {http://hdl.handle.net/1963/2307}, author = {Claudio Altafini} } @article {2004, title = {Resonance of minimizers for n-level quantum systems with an arbitrary cost}, journal = {ESAIM COCV 10 (2004) 593-614}, number = {SISSA;58/2003/M}, year = {2004}, publisher = {EDP Sciences}, abstract = {We consider an optimal control problem describing a laser-induced population transfer on a $n$-level quantum system.\\nFor a convex cost depending only on the moduli of controls (i.e. the lasers intensities), we prove that there always exists a minimizer in resonance. This permits to justify some strategies used in experimental physics. It is also quite important because it permits to reduce remarkably the complexity of the problem (and extend some of our previous results for $n=2$ and $n=3$): instead of looking for minimizers on the sphere $S^{2n-1}\\\\subset\\\\C^n$ one is reduced to look just for minimizers on the sphere $S^{n-1}\\\\subset \\\\R^n$. Moreover, for the reduced problem, we investigate on the question of existence of strict abnormal minimizer.}, doi = {10.1051/cocv:2004022}, url = {http://hdl.handle.net/1963/2910}, author = {Ugo Boscain and Gr{\'e}goire Charlot} } @article {2004, title = {The role of the spectrum of the Laplace operator on \\\\S2 in the H-bubble problem}, journal = {J. Anal. Math. 94 (2004) 265-291}, number = {SISSA;30/2003/M}, year = {2004}, publisher = {Hebrew University Magnes Press}, doi = {10.1007/BF02789050}, url = {http://hdl.handle.net/1963/2894}, author = {Roberta Musina} } @article {2004, title = {Rotating Singular Perturbations of the Laplacian}, journal = {Ann. Henri Poincare 5 (2004) 773-808}, number = {SISSA;66/2003/FM}, year = {2004}, publisher = {Springer}, abstract = {We study a system of a quantum particle interacting with a singular time-dependent uniformly rotating potential in 2 and 3 dimensions: in particular we consider an interaction with support on a point (rotating point interaction) and on a set of codimension 1 (rotating blade). We prove the existence of the Hamiltonians of such systems as suitable self-adjoint operators and we give an explicit expression for their unitary semigroups. Moreover we analyze the asymptotic limit of large angular velocity and we prove strong convergence of the time-dependent propagator to some one-parameter unitary group as (\\\\omega \\\\to \\\\infty).}, doi = {10.1007/s00023-004-0182-8}, url = {http://hdl.handle.net/1963/2945}, author = {Michele Correggi and Gianfausto Dell{\textquoteright}Antonio} } @article {2002, title = {On the reachability of quantized control systems}, journal = {IEEE Trans. Automat. Contr. 47 (2002) 546-563}, number = {SISSA;43/00/M}, year = {2002}, publisher = {SISSA Library}, abstract = {In this paper, we study control systems whose input sets are quantized, i.e., finite or regularly distributed on a mesh. We specifically focus on problems relating to the structure of the reachable set of such systems, which may turn out to be either dense or discrete. We report results on the reachable set of linear quantized systems, and on a particular but interesting class of nonlinear systems, i.e., nonholonomic chained-form systems. For such systems, we provide a complete characterization of the reachable set, and, in case the set is discrete, a computable method to completely and succinctly describe its structure. Implications and open problems in the analysis and synthesis of quantized control systems are addressed.}, doi = {10.1109/9.995034}, url = {http://hdl.handle.net/1963/1501}, author = {Antonio Bicchi and Alessia Marigo and Benedetto Piccoli} } @article {2002, title = {The reachable set of a linear endogenous switching system}, journal = {Systems Control Lett. 47 (2002) 343-353}, year = {2002}, publisher = {Elsevier}, abstract = {In this work, switching systems are named endogenous when their switching pattern is controllable. Linear endogenous switching systems can be considered as a particular class of bilinear control systems. The key idea is that both types of systems are equivalent to polysystems, i.e. to systems whose flow is piecewise smooth. The reachable set of a linear endogenous switching system can be studied consequently. The main result is that, in general, it has the structure of a semigroup, even when the Lie algebra rank condition is satisfied since the logic inputs cannot reverse the direction of the flow. The adaptation of existing controllability criteria for bilinear systems is straightforward.}, doi = {10.1016/S0167-6911(02)00228-1}, url = {http://hdl.handle.net/1963/3142}, author = {Claudio Altafini} } @article {2002, title = {Relatively stable bundles over elliptic fibrations}, journal = {Math. Nachr. 238 (2002) 23-36}, number = {arXiv.org;math/0109123v2}, year = {2002}, publisher = {Wiley}, abstract = {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.}, url = {http://hdl.handle.net/1963/3132}, author = {Claudio Bartocci and Ugo Bruzzo and Daniel Hernandez Ruiperez and Jose M. Munoz Porras} } @inbook {2000, title = {Reachability Analysis for a Class of Quantized Control Systems}, booktitle = {Proc. 39th IEEE Int. Conf. on Decision and Control 4 (2000) 3963-3968}, year = {2000}, publisher = {IEEE}, organization = {IEEE}, abstract = {We study control systems whose input sets are quantized. We specifically focus on problems relating to the structure of the reachable set of such systems, which may turn out to be either dense or discrete. We report on some results on the reachable set of linear quantized systems, and study in detail an interesting class of nonlinear systems, forming the discrete counterpart of driftless nonholonomic continuous systems. For such systems, we provide a complete characterization of the reachable set, and, in the case the set is discrete, a computable method to describe its lattice structure.}, doi = {10.1109/CDC.2000.912333}, url = {http://hdl.handle.net/1963/3518}, author = {Alessia Marigo and Benedetto Piccoli and Antonio Bicchi} } @article {2000, title = {Reduction of bi-Hamiltonian systems and separation of variables: an example from the Boussinesq hierarchy}, journal = {Theor. Math. Phys. 122 (2000) 176-192}, number = {SISSA;123/98/FM}, year = {2000}, publisher = {Springer}, abstract = {We discuss the Boussinesq system with $t_5$ stationary, within a general framework for the analysis of stationary flows of n-Gel\\\'fand-Dickey hierarchies. We show how a careful use of its bihamiltonian structure can be used to provide a set of separation coordinates for the corresponding Hamilton--Jacobi equations.}, doi = {10.1007/BF02551195}, url = {http://hdl.handle.net/1963/3219}, author = {Gregorio Falqui and Franco Magri and G. Tondo} } @article {2000, title = {Regular Synthesis and Sufficiency Conditions for Optimality}, journal = {SIAM J. Control Optim. 39 (2000) 359-410}, number = {SISSA;68/96/M}, year = {2000}, publisher = {SIAM}, abstract = {We propose a definition of \\\"regular synthesis\\\" that is more general than those suggested by other authors such as Boltyanskii and Brunovsky, and an even more general notion of \\\"regular presynthesis.\\\" We give a complete proof of the corresponding sufficiency theorem, a slightly weaker version of which had been stated in an earlier article, with only a rough outline of the proof. We illustrate the strength of our result by showing that the optimal synthesis for the famous Fuller problem satisfies our hypotheses. We also compare our concept of synthesis with the simpler notion of a \\\"family of solutions of the closed-loop equation arising from an optimal feedback law,\\\" and show by means of examples why the latter is inadequate, and why the difficulty cannot be resolved byusing other concepts of solution--such as Filippov solutions, or the limits of sample-and-hold solutions recently proposed as feedback solutions by Clarke, Ledyaev, Subbotin and Sontag -for equations with a non-Lipschitz and possibly discontinuous right-hand side.}, doi = {10.1137/S0363012999322031}, url = {http://hdl.handle.net/1963/3213}, author = {Benedetto Piccoli and Hector J. Sussmann} } @article {2000, title = {A Remark on One-Dimensional Many-Body Problems with Point Interactions}, journal = {Int. J. Mod. Phys. B 14 (2000) 721-727}, number = {arXiv.org;quant-ph/0001089v1}, year = {2000}, publisher = {World Scientific}, abstract = {The integrability of one dimensional quantum mechanical many-body problems with general contact interactions is extensively studied. It is shown that besides the pure (repulsive or attractive) $\\\\delta$-function interaction there is another singular point interactions which gives rise to a new one-parameter family of integrable quantum mechanical many-body systems. The bound states and scattering matrices are calculated for both bosonic and fermionic statistics.}, doi = {10.1142/S0217979200000601}, url = {http://hdl.handle.net/1963/3214}, author = {Sergio Albeverio and Ludwik Dabrowski and Shao-Ming Fei} } @article {10445, title = {Recurrent procedure for the determination of the free energy ε^2 expansion in the topological string theory}, number = {arXiv:solv-int/990400;}, year = {1999}, institution = {SISSA}, abstract = {

We present here the iteration procedure for the determination of free energy ǫ2-expansion using the theory of KdV - type equations. In our approach we use the conservation laws for KdV - type equations depending explicitly on times t1, t2, . . . to find the ǫ2-expansion of u(x, t1, t2, . . .) after the infinite number of shifts of u(x, 0, 0, . . .) = x along t1, t2, . . . in recurrent form. The formulas for the free energy expansion are just obtained then as a result of quite simple integration procedure applied to un(x).

}, url = {http://hdl.handle.net/1963/6489}, author = {Boris Dubrovin and Andrei Ya A Maltsev} } @article {1999, title = {Renormalized solutions of elliptic equations with general measure data}, journal = {Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 28 (1999), no. 4, 741-808}, number = {SISSA;22/99/M}, year = {1999}, publisher = {Scuola Normale Superiore di Pisa}, url = {http://hdl.handle.net/1963/1236}, author = {Gianni Dal Maso and Francois Murat and Luigi Orsina and Alain Prignet} } @article {1981, title = {Recent advances in the study of the existence of periodic orbits of Hamiltonian systems}, journal = {Advances in Hamiltonian systems (Rome, 1981), 1--22, Ann. CEREMADE, Birkhauser Boston, Boston, MA, 1983.}, number = {SISSA;8/82/M}, year = {1981}, publisher = {SISSA Library}, url = {http://hdl.handle.net/1963/159}, author = {Antonio Ambrosetti} }