In this work we focus on two different methods to deal with parametrized partial differential equations in an efficient and accurate way. Starting from high fidelity approximations built via the hierarchical model reduction discretization, we consider two approaches, both based on a projection model reduction technique. The two methods differ for the algorithm employed during the construction of the reduced basis. In particular, the former employs the proper orthogonal decomposition, while the latter relies on a greedy algorithm according to the certified reduced basis technique. The two approaches are preliminarily compared on two-dimensional scalar and vector test cases.

}, doi = {10.1137/19M1285330}, author = {Matteo Zancanaro and F. Ballarin and Simona Perotto and Gianluigi Rozza} } @article {2021, title = {Hull Shape Design Optimization with Parameter Space and Model Reductions, and Self-Learning Mesh Morphing}, journal = {Journal of Marine Science and Engineering}, volume = {9}, year = {2021}, pages = {185}, abstract = {In the field of parametric partial differential equations, shape optimization represents a challenging problem due to the required computational resources. In this contribution, a data-driven framework involving multiple reduction techniques is proposed to reduce such computational burden. Proper orthogonal decomposition (POD) and active subspace genetic algorithm (ASGA) are applied for a dimensional reduction of the original (high fidelity) model and for an efficient genetic optimization based on active subspace property. The parameterization of the shape is applied directly to the computational mesh, propagating the generic deformation map applied to the surface (of the object to optimize) to the mesh nodes using a radial basis function (RBF) interpolation. Thus, topology and quality of the original mesh are preserved, enabling application of POD-based reduced order modeling techniques, and avoiding the necessity of additional meshing steps. Model order reduction is performed coupling POD and Gaussian process regression (GPR) in a data-driven fashion. The framework is validated on a benchmark ship.

}, issn = {2077-1312}, doi = {10.3390/jmse9020185}, url = {https://www.mdpi.com/2077-1312/9/2/185}, author = {Nicola Demo and Marco Tezzele and Andrea Mola and Gianluigi Rozza} } @article {ZancanaroMrosekStabileOthmerRozza2021, title = {Hybrid Neural Network Reduced Order Modelling for Turbulent Flows with Geometric Parameters}, journal = {Fluids}, volume = {6}, number = {8}, year = {2021}, pages = {296}, publisher = {MDPI AG}, doi = {10.3390/fluids6080296}, url = {https://doi.org/10.3390/fluids6080296}, author = {Matteo Zancanaro and Markus Mrosek and Giovanni Stabile and Carsten Othmer and Gianluigi Rozza} } @article {2020, title = {A hybrid reduced order method for modelling turbulent heat transfer problems}, journal = {Computers \& Fluids}, volume = {208}, year = {2020}, pages = {104615}, abstract = {A parametric, hybrid reduced order model approach based on the Proper Orthogonal Decomposition with both Galerkin projection and interpolation based on Radial Basis Functions method is presented. This method is tested against a case of turbulent non-isothermal mixing in a T-junction pipe, a common ow arrangement found in nuclear reactor cooling systems. The reduced order model is derived from the 3D unsteady, incompressible Navier-Stokes equations weakly coupled with the energy equation. For high Reynolds numbers, the eddy viscosity and eddy diffusivity are incorporated into the reduced order model with a Proper Orthogonal Decomposition (nested and standard) with Interpolation (PODI), where the interpolation is performed using Radial Basis Functions. The reduced order solver, obtained using a k-ω SST URANS full order model, is tested against the full order solver in a 3D T-junction pipe with parametric velocity inlet boundary conditions.

}, doi = {10.1016/j.compfluid.2020.104615}, url = {https://arxiv.org/abs/1906.08725}, author = {Sokratia Georgaka and Giovanni Stabile and Kelbij Star and Gianluigi Rozza and Michael J. Bluck} } @article {CANGIANI20193090, title = {hp-adaptive discontinuous Galerkin methods for non-stationary convection{\textendash}diffusion problems}, journal = {Computers \& Mathematics with Applications}, volume = {78}, number = {9}, year = {2019}, note = {Applications of Partial Differential Equations in Science and Engineering}, pages = {3090-3104}, abstract = {An a posteriori error estimator for the error in the (L2(H1)+L$\infty$(L2))-type norm for an interior penalty discontinuous Galerkin (dG) spatial discretisation and backward Euler temporal discretisation of linear non-stationary convection{\textendash}diffusion initial/boundary value problems is derived, allowing for anisotropic elements. The proposed error estimator is used to drive an hp-space{\textendash}time adaptive algorithm wherein directional mesh refinement is employed to give rise to highly anisotropic elements able to accurately capture layers. The performance of the hp-space{\textendash}time adaptive algorithm is assessed via a number of standard test problems characterised by sharp and/or moving layers.}, keywords = {A posteriori error estimation, Adaptive finite element methods, Anisotropic meshes, Discontinuous Galerkin, Unsteady convection{\textendash}diffusion}, issn = {0898-1221}, doi = {https://doi.org/10.1016/j.camwa.2019.04.002}, url = {https://www.sciencedirect.com/science/article/pii/S0898122119302007}, author = {Andrea Cangiani and E.H. Georgoulis and Stefano Giani and S. Metcalfe} } @article {agostiniani2018heterogeneous, title = {Heterogeneous elastic plates with in-plane modulation of the target curvature and applications to thin gel sheets}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, year = {2018}, publisher = {EDP Sciences}, doi = {https://doi.org/10.1051/cocv/2018046}, author = {Virginia Agostiniani and Alessandro Lucantonio and Danka Lu{\v c}i{\'c}} } @article {2018, title = {Hydrogenoid Spectra with Central Perturbations}, number = {SISSA;34/2018/MATE}, year = {2018}, note = {Mathematics Subject Classification (2010) 34L10 . 34L15 . 34L16 . 47B15 . 47B25 . 47N20 . 81Q10 . 81Q80}, abstract = {Through the Krein-Vi{\v s}ik-Birman extension scheme, unlike the previous classical analysis based on von Neumann{\textquoteright}s theory, we reproduce the construction and classification of all self-adjoint realisations of two intimately related models: the three-dimensional hydrogenoid-like Hamiltonians with singular perturbation supported at the centre (the nucleus), and the Sch{\"o}rdinger operators on the halfline with Coulomb potentials centred at the origin. These two problems are technically equivalent, albeit sometimes treated by their own in the the literature. Based on such scheme, we then recover the formula to determine the eigenvalues of each self-adjoint extension, which are corrections to the non-relativistic hydrogenoid energy levels.We discuss in which respect the Krein-Vi{\v s}ik-Birman scheme is somehow more natural in yielding the typical boundary condition of self-adjointness at the centre of the perturbation and in identifying the eigenvalues of each extension.}, url = {http://preprints.sissa.it/handle/1963/35321}, author = {Matteo Gallone and Alessandro Michelangeli} } @article {Agrachev2017, title = {Homotopically invisible singular curves}, journal = {Calculus of Variations and Partial Differential Equations}, volume = {56}, number = {4}, year = {2017}, month = {Jul}, pages = {105}, issn = {1432-0835}, doi = {10.1007/s00526-017-1203-z}, url = {https://doi.org/10.1007/s00526-017-1203-z}, author = {Andrei A. Agrachev and Francesco Boarotto and Antonio Lerario} } @article {boarotto2017homotopy, title = {Homotopy properties of horizontal path spaces and a theorem of Serre in subriemannian geometry}, journal = {Communications in Analysis and Geometry}, volume = {25}, number = {2}, year = {2017}, pages = {269{\textendash}301}, publisher = {International Press of Boston}, doi = {10.4310/CAG.2017.v25.n2.a1}, author = {Francesco Boarotto and Antonio Lerario} } @book {MR3729265, title = {$hp$-version discontinuous Galerkin methods on polygonal and polyhedral meshes}, series = {SpringerBriefs in Mathematics}, year = {2017}, pages = {viii+131}, publisher = {Springer, Cham}, organization = {Springer, Cham}, isbn = {978-3-319-67671-5; 978-3-319-67673-9}, author = {Andrea Cangiani and Dong, Zhaonan and E.H. Georgoulis and Houston, Paul} } @article {MR3672375, title = {$hp$-version space-time discontinuous Galerkin methods for parabolic problems on prismatic meshes}, journal = {SIAM J. Sci. Comput.}, volume = {39}, number = {4}, year = {2017}, pages = {A1251{\textendash}A1279}, issn = {1064-8275}, doi = {10.1137/16M1073285}, url = {https://doi.org/10.1137/16M1073285}, author = {Andrea Cangiani and Dong, Zhaonan and E.H. Georgoulis} } @article {BertoBothBalogh, title = {Hankel determinant approach to generalized Vorob{\textquoteright}ev-Yablonski polynomials and their roots}, journal = {Constr. Approx.}, volume = {44}, number = {3}, year = {2016}, pages = {417{\textendash}453}, issn = {0176-4276}, doi = {10.1007/s00365-016-9328-4}, url = {http://dx.doi.org/10.1007/s00365-016-9328-4}, author = {Ferenc Balogh and Marco Bertola and Thomas Bothner} } @article {MR3507270, title = {$hp$-version discontinuous Galerkin methods for advection-diffusion-reaction problems on polytopic meshes}, journal = {ESAIM Math. Model. Numer. Anal.}, volume = {50}, number = {3}, year = {2016}, pages = {699{\textendash}725}, issn = {0764-583X}, doi = {10.1051/m2an/2015059}, url = {https://doi.org/10.1051/m2an/2015059}, author = {Andrea Cangiani and Dong, Zhaonan and E.H. Georgoulis and Houston, Paul} } @article {2015, title = {Hilbert schemes of points of OP1(-n) as quiver varieties}, year = {2015}, publisher = {arXiv:1504.02987 [math.AG]}, abstract = {Relying on a representation of framed torsion-free sheaves on Hirzebruch surfaces in terms of monads, we construct ADHM data for the Hilbert scheme of points of the total space of the line bundle $\mathcal O(-n)$ on $\mathbb P^1$. This ADHM description is then used to realize these Hilbert schemes as quiver varieties.}, url = {http://urania.sissa.it/xmlui/handle/1963/34487}, author = {Ugo Bruzzo} } @article {2015, title = {Homogenization problems in the Calculus of Variations: an overview}, number = {SISSA;13/2015/MATE}, year = {2015}, note = {DEDICATED TO PROF. ORLANDO LOPES}, institution = {SISSA}, abstract = {In this note we present a brief overview of variational methods to solve homogenization problems. The purpose is to give a first insight on the subject by presenting some fundamental theoretical tools, both classical and modern. We conclude by mentioning some open problems.}, url = {http://urania.sissa.it/xmlui/handle/1963/34455}, author = {Jose Matias and Marco Morandotti} } @article {https://doi.org/10.1002/nme.4854, title = {Hourglass stabilization and the virtual element method}, journal = {International Journal for Numerical Methods in Engineering}, volume = {102}, number = {3-4}, year = {2015}, pages = {404-436}, abstract = {SummaryIn this paper, we establish the connections between the virtual element method (VEM) and the hourglass control techniques that have been developed since the early 1980s to stabilize underintegrated C0 Lagrange finite element methods. In the VEM, the bilinear form is decomposed into two parts: a consistent term that reproduces a given polynomial space and a correction term that provides stability. The essential ingredients of -continuous VEMs on polygonal and polyhedral meshes are described, which reveals that the variational approach adopted in the VEM affords a generalized and robust means to stabilize underintegrated finite elements. We focus on the heat conduction (Poisson) equation and present a virtual element approach for the isoparametric four-node quadrilateral and eight-node hexahedral elements. In addition, we show quantitative comparisons of the consistency and stabilization matrices in the VEM with those in the hourglass control method of Belytschko and coworkers. Numerical examples in two and three dimensions are presented for different stabilization parameters, which reveals that the method satisfies the patch test and delivers optimal rates of convergence in the L2 norm and the H1 seminorm for Poisson problems on quadrilateral, hexahedral, and arbitrary polygonal meshes. Copyright {\textcopyright} 2015 John Wiley \& Sons, Ltd.}, keywords = {consistency matrix, hourglass control, polygonal and polyhedral finite elements, stabilization matrix, underintegration, virtual element method}, doi = {https://doi.org/10.1002/nme.4854}, url = {https://onlinelibrary.wiley.com/doi/abs/10.1002/nme.4854}, author = {Andrea Cangiani and Manzini, G. and Russo, A. and Sukumar, N.} } @article {MR3340083, title = {Hourglass stabilization and the virtual element method}, journal = {Internat. J. Numer. Methods Engrg.}, volume = {102}, number = {3-4}, year = {2015}, pages = {404{\textendash}436}, issn = {0029-5981}, doi = {10.1002/nme.4854}, url = {https://doi.org/10.1002/nme.4854}, author = {Andrea Cangiani and Manzini, G. and Russo, A. and Sukumar, N.} } @article {prandi_2014, title = {H{\"o}lder equivalence of the value function for control-affine systems}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, volume = {20}, number = {4}, year = {2014}, pages = {1224{\textendash}1248}, publisher = {EDP Sciences}, doi = {10.1051/cocv/2014014}, author = {Dario Prandi} } @mastersthesis {2014, title = {Holomorphically symplectic varieties with Prym Lagrangian fibrations}, year = {2014}, school = {SISSA}, abstract = {The thesis presents a construction of singular holomorphically symplectic varieties as Lagrangian fibrations. They are relative compactified Prym varieties associated to curves on symplectic surfaces with an antisymplectic involution. They are identified with the fixed locus of a symplectic involution on singular moduli spaces of sheaves of dimension 1. An explicit example, giving a singular irreducible symplectic 6-fold without symplectic resolutions, is described for a K3 surface which is the double cover of a cubic surface. In the case of abelian surfaces, a variation of this construction is studied to get irreducible symplectic varieties: relative compactified 0-Prym varieties. A partial classification result is obtained for involutions without fixed points: either the 0-Prym variety is birational to an irreducible symplectic variety of K3[n]-type, or it does not admit symplectic resolutions.}, keywords = {Holomorphically symplectic varieties}, url = {http://urania.sissa.it/xmlui/handle/1963/7434}, author = {Tommaso Matteini} } @article {2014, title = {Homogenization of functional with linear growth in the context of A-quasiconvexity}, number = {SISSA;49/2014/MATE}, year = {2014}, institution = {SISSA}, abstract = {This work deals with the homogenization of functionals with linear growth in the context of A-quasiconvexity. A representation theorem is proved, where the new integrand function is obtained by solving a cell problem where the coupling between homogenization and the A-free condition plays a crucial role. This result extends some previous work to the linear case, thus allowing for concentration effects.}, url = {http://urania.sissa.it/xmlui/handle/1963/7436}, author = {Jose Matias and Marco Morandotti and Pedro M. Santos} } @article {2014, title = {Homology computation for a class of contact structures on T3}, number = {Calculus of variations and partial differential equations;volume 50; issue 3; pages 599-614;}, year = {2014}, publisher = {Springer}, abstract = {We consider a family of tight contact forms on the three-dimensional torus and we compute the relative Contact Homology by using the variational theory of critical points at infinity. We will also show local stability.}, doi = {10.1007/s00526-013-0648-y}, url = {http://urania.sissa.it/xmlui/handle/1963/34649}, author = {Ali Maalaoui and Vittorio Martino} } @article {MR3211116, title = {$hp$-version discontinuous Galerkin methods on polygonal and polyhedral meshes}, journal = {Math. Models Methods Appl. Sci.}, volume = {24}, number = {10}, year = {2014}, pages = {2009{\textendash}2041}, issn = {0218-2025}, doi = {10.1142/S0218202514500146}, url = {https://doi.org/10.1142/S0218202514500146}, author = {Andrea Cangiani and E.H. Georgoulis and Houston, Paul} } @article {2012, title = {On the Hausdorff volume in sub-Riemannian geometry}, journal = {Calculus of Variations and Partial Differential Equations. Volume 43, Issue 3-4, March 2012, Pages 355-388}, number = {arXiv:1005.0540;}, year = {2012}, publisher = {SISSA}, abstract = {For a regular sub-Riemannian manifold we study the Radon-Nikodym derivative\r\nof the spherical Hausdorff measure with respect to a smooth volume. We prove\r\nthat this is the volume of the unit ball in the nilpotent approximation and it\r\nis always a continuous function. We then prove that up to dimension 4 it is\r\nsmooth, while starting from dimension 5, in corank 1 case, it is C^3 (and C^4\r\non every smooth curve) but in general not C^5. These results answer to a\r\nquestion addressed by Montgomery about the relation between two intrinsic\r\nvolumes that can be defined in a sub-Riemannian manifold, namely the Popp and\r\nthe Hausdorff volume. If the nilpotent approximation depends on the point (that\r\nmay happen starting from dimension 5), then they are not proportional, in\r\ngeneral.}, doi = {10.1007/s00526-011-0414-y}, url = {http://hdl.handle.net/1963/6454}, author = {Andrei A. Agrachev and Davide Barilari and Ugo Boscain} } @article {2012, title = {Hybridization in nanostructured DNA monolayers probed by AFM: theory versus experiment}, journal = {Nanoscale. 2012 Mar; 4(5):1734-41}, number = {PMID:22301788;}, year = {2012}, publisher = {Royal Society of Chemistry}, abstract = {Nanografted monolayers (NAMs) of DNA show novel physico-chemical properties that make them ideally suited for advanced biosensing applications. In comparison with alternative solid-phase techniques for diagnostic DNA detection, NAMs have the advantage of combining a small size with a high homogeneity of the DNA surface coverage. These two properties favour the extreme miniaturization and ultrasensitivity in high-throughput biosensing devices. The systematic use of NAMs for quantitative DNA (and protein) detection has so far suffered from the lack of a control on key fabrication parameters, such as the ss- or ds-DNA surface coverage. Here we report on a combined experimental-computational study that allows us to estimate the surface density of the grafted DNA by analyzing the sample mechanical response, that is the DNA patch height vs. applied tip load curves. It is shown that the same analysis scheme can be used to detect the occurrence of hybridization with complementary strands in solution and estimate its efficiency. Thanks to these quantitative relationships it is possible to use a single AFM-based setup to: (i) fabricate a DNA NAM, (ii) control the DNA surface coverage, and (iii) characterize its level of hybridization helping the design of NAMs with pre-determined fabrication parameters.}, doi = {10.1039/c2nr11662f}, author = {Alessandro Bosco and Fouzia Bano and Pietro Parisse and Loredana Casalis and Antonio DeSimone and Cristian Micheletti} } @article {2011, title = {Holomorphic Cartan geometry on manifolds with numerically effective tangent bundle}, journal = {Differential Geometry and its Applications 29 (2011) 147-153}, number = {SISSA;05/2010/FM}, year = {2011}, publisher = {Elsevier}, doi = {10.1016/j.difgeo.2011.02.001}, url = {http://hdl.handle.net/1963/3830}, author = {Indranil Biswas and Ugo Bruzzo} } @mastersthesis {2011, title = {Homology invariants of quadratic maps}, year = {2011}, school = {SISSA}, abstract = {Given a real projective algebraic set X we could hope that the equations describing it can give some information on its topology, e.g. on the number of its connected components. Unfortunately in the general case this hope is too vague and there is no direct way to extract such information from the algebraic description of X: Even the problem to decide whether X is empty or not is far from an easy visualization and requires some complicated algebraic machinery. A fi rst step observation is that as long as we are interested only in the topology of X, we can replace, using some Veronese embedding, the original ambient space with a much bigger RPn and assume that X is cut by quadratic equations. The price for this is the increase of the number of equations de ning our set; the advantage is that quadratic polynomials are easier to handle and our hope becomes more concrete...}, url = {http://hdl.handle.net/1963/6245}, author = {Antonio Lerario} } @article {2010, title = {Hamiltonian PDEs: deformations, integrability, solutions}, journal = {Journal of Physics A: Mathematical and Theoretical. Volume 43, Issue 43, 29 October 2010, Article number 434002}, year = {2010}, publisher = {SISSA}, abstract = {We review recent classification results on the theory of systems of nonlinear\\r\\nHamiltonian partial differential equations with one spatial dimension, including\\r\\na perturbative approach to the integrability theory of such systems, and discuss\\r\\nuniversality conjectures describing critical behaviour of solutions to such\\r\\nsystems.}, doi = {10.1088/1751-8113/43/43/434002}, url = {http://hdl.handle.net/1963/6469}, author = {Boris Dubrovin} } @article {2010, title = {Hitchin systems, N=2 gauge theories and W-gravity}, journal = {Phys. Lett. B 691 (2010) 111-115}, number = {SISSA;60/2009/EP-FM}, year = {2010}, abstract = {We propose some arguments supporting an M-theory derivation of the duality recently discovered by Alday, Gaiotto and Tachikawa between two-dimensional conformal field theories and N=2 superconformal gauge theories in four dimensions. We find that A_{N-1} Toda field theory is the simplest two-dimensional conformal field theory quantizing the moduli of N M5-branes wrapped on a Riemann surface. This leads us to identify chiral operators of the N=2 gauge theories with W-algebra currents. As a check of this correspondence we study some relevant OPE\\\'s obtaining that Nekrasov\\\'s partition function satisfies W-geometry constraints.}, doi = {10.1016/j.physletb.2010.06.027}, url = {http://hdl.handle.net/1963/3831}, author = {Giulio Bonelli and Alessandro Tanzini} } @article {2010, title = {Homogeneous binary trees as ground states of quantum critical Hamiltonians}, journal = {Phys. Rev. A 81 (2010) 062335}, number = {arXiv.org;0912.0466v2}, year = {2010}, publisher = {American Physical Society}, abstract = {Many-body states whose wave-function admits a representation in terms of a uniform binary-tree tensor decomposition are shown to obey to power-law two-body correlations functions. Any such state can be associated with the ground state of a translational invariant Hamiltonian which, depending on the dimension of the systems sites, involve at most couplings between third-neighboring sites. A detailed analysis of their spectra shows that they admit an exponentially large ground space.

}, doi = {10.1103/PhysRevA.81.062335}, url = {http://hdl.handle.net/1963/3909}, author = {Pietro Silvi and Vittorio Giovannetti and Simone Montangero and Matteo Rizzi and J. Ignacio Cirac and Rosario Fazio} } @article {2010, title = {Homogeneous multiscale entanglement renormalization ansatz tensor networks for quantum critical systems}, journal = {New J. Phys. 12 (2010) 075018}, year = {2010}, publisher = {IOP Publishing}, abstract = {In this paper, we review the properties of homogeneous multiscale entanglement renormalization ansatz (MERA) to describe quantum critical systems.We discuss in more detail our results for one-dimensional (1D) systems (the Ising and Heisenberg models) and present new data for the 2D Ising model. Together with the results for the critical exponents, we provide a detailed description of the numerical algorithm and a discussion of new optimization\\nstrategies. The relation between the critical properties of the system and the tensor structure of the MERA is expressed using the formalism of quantum channels, which we review and extend.

}, doi = {10.1088/1367-2630/12/7/075018}, url = {http://hdl.handle.net/1963/4067}, author = {Matteo Rizzi and Simone Montangero and Pietro Silvi and Vittorio Giovannetti and Rosario Fazio} } @article {2010, title = {Homogenization of fiber reinforced brittle material: the intermediate case}, journal = {Adv. Calc. Var. 3 (2010) 345-370}, number = {SISSA;14/2009/M}, year = {2010}, publisher = {Walter de Gruyter}, abstract = {We derive a cohesive fracture model by homogenizing a periodic composite material whose microstructure is characterized by the presence of brittle inclusions in a reticulated unbreakable elastic structure.}, doi = {10.1515/ACV.2010.011}, url = {http://hdl.handle.net/1963/3607}, author = {Gianni Dal Maso and Caterina Ida Zeppieri} } @inbook {2009, title = {Hamiltonian perturbations of hyperbolic PDEs: from classification results to the properties of solutions}, booktitle = {New Trends in Mathematical Physics : Selected contributions of the XVth International Congress on Mathematical Physics, Springer Netherlands, 2009, pp. 231-276.}, year = {2009}, publisher = {SISSA}, organization = {SISSA}, abstract = {We begin with presentation of classi cation results in the theory of Hamiltonian\\r\\nPDEs with one spatial dimension depending on a small parameter. Special\\r\\nattention is paid to the deformation theory of integrable hierarchies, including an\\r\\nimportant subclass of the so-called integrable hierarchies of the topological type\\r\\nassociated with semisimple Frobenius manifolds. Many well known equations of\\r\\nmathematical physics, such as KdV, NLS, Toda, Boussinesq etc., belong to this\\r\\nsubclass, but there are many new integrable PDEs, some of them being of interest\\r\\nfor applications. Connections with the theory of Gromov{Witten invariants\\r\\nand random matrices are outlined. We then address the problem of comparative\\r\\nstudy of singularities of solutions to the systems of first order quasilinear\\r\\nPDEs and their Hamiltonian perturbations containing higher derivatives. We\\r\\nformulate Universality Conjectures describing different types of critical behavior\\r\\nof perturbed solutions near the point of gradient catastrophe of the unperturbed\\r\\none.}, isbn = {978-90-481-2810-5}, url = {http://hdl.handle.net/1963/6470}, author = {Boris Dubrovin} } @article {2009, title = {Hardy-Sobolev-Maz\\\'ja inequalities: symmetry and breaking symmetry of extremal functions}, journal = {Commun. Contemp. Math. 11 (2009) 993-1007}, number = {SISSA;06/2008/M}, year = {2009}, doi = {10.1142/S0219199709003636}, url = {http://hdl.handle.net/1963/2569}, author = {Marita Gazzini and Roberta Musina} } @article {2009, title = {A higher order model for image restoration: the one dimensional case}, journal = {SIAM J. Math. Anal. 40 (2009) 2351-2391}, number = {SISSA;66/2008/M}, year = {2009}, abstract = {The higher order total variation-based model for image restoration proposed by Chan, Marquina, and Mulet in [6] is analyzed in one dimension. A suitable functional framework in which the minimization problem is well posed is being proposed and it is proved analytically that the\\nhigher order regularizing term prevents the occurrence of the staircase effect. The generalized version of the model considered here includes, as particular cases, some curvature dependent functionals.}, doi = {10.1137/070697823}, url = {http://hdl.handle.net/1963/3174}, author = {Gianni Dal Maso and Irene Fonseca and Giovanni Leoni and Massimiliano Morini} } @article {2009, title = {Holomorphic equivariant cohomology of Atiyah algebroids and localization}, number = {SISSA;65/2009/FM}, year = {2009}, url = {http://hdl.handle.net/1963/3774}, author = {Ugo Bruzzo and Vladimir Rubtsov} } @article {2009, title = {Homogenization of fiber reinforced brittle materials: the extremal cases}, journal = {SIAM J. Math. Anal. 41 (2009) 1874-1889}, number = {SISSA;49/2008/M}, year = {2009}, publisher = {SIAM}, abstract = {We analyze the behavior of a fragile material reinforced by a reticulated elastic unbreakable structure in the case of antiplane shear. The microscopic geometry of this material is described by means of two small parameters: the period $\\\\varepsilon$ of the grid and the ratio $\\\\delta$ between the thickness of the fibers and the period $\\\\varepsilon$. We show that the asymptotic behavior as $\\\\varepsilon\\\\to0^+$ and $\\\\delta\\\\to0^+$ depends dramatically on the relative size of these parameters. Indeed, in the two cases considered, i.e., $\\\\varepsilon\\\\ll\\\\delta$ and $\\\\varepsilon\\\\gg\\\\delta$, we obtain two different limit models: a perfectly elastic model and an elastic model with macroscopic cracks, respectively.}, url = {http://hdl.handle.net/1963/2705}, author = {Marco Barchiesi and Gianni Dal Maso} } @article {2008, title = {Hamiltonian partial differential equations and Frobenius manifolds}, journal = {Russian Mathematical Surveys. Volume 63, Issue 6, 2008, Pages 999-1010}, year = {2008}, publisher = {SISSA}, abstract = {In the first part of this paper the theory of Frobenius manifolds\\r\\nis applied to the problem of classification of Hamiltonian systems of partial\\r\\ndifferential equations depending on a small parameter. Also developed is\\r\\na deformation theory of integrable hierarchies including the subclass of\\r\\nintegrable hierarchies of topological type. Many well-known examples\\r\\nof integrable hierarchies, such as the Korteweg{\textendash}de Vries, non-linear\\r\\nSchr{\textasciidieresis}odinger, Toda, Boussinesq equations, and so on, belong to this\\r\\nsubclass that also contains new integrable hierarchies. Some of these new\\r\\nintegrable hierarchies may be important for applications. Properties of the\\r\\nsolutions to these equations are studied in the second part. Consideration\\r\\nis given to the comparative study of the local properties of perturbed and\\r\\nunperturbed solutions near a point of gradient catastrophe. A Universality\\r\\nConjecture is formulated describing the various types of critical behaviour\\r\\nof solutions to perturbed Hamiltonian systems near the point of gradient\\r\\ncatastrophe of the unperturbed solution.}, doi = {10.1070/RM2008v063n06ABEH004575}, url = {http://hdl.handle.net/1963/6471}, author = {Boris Dubrovin} } @article {MR2435752, title = {Harish-Chandra integrals as nilpotent integrals}, journal = {Int. Math. Res. Not. IMRN}, number = {16}, year = {2008}, pages = {Art. ID rnn062, 15}, issn = {1073-7928}, author = {Marco Bertola and Ferrer, Aleix Prats} } @article {2007, title = {High-order angles in almost-Riemannian geometry}, number = {SISSA;59/2007/M}, year = {2007}, abstract = {Let X and Y be two smooth vector fields on a two-dimensional manifold M. If X and Y are everywhere linearly independent, then they define a Riemannian metric on M (the metric for which they are orthonormal) and they give to M the structure of metric space. If X and Y become linearly dependent somewhere on M, then the corresponding Riemannian metric has singularities, but under generic conditions the metric structure is still well defined. Metric structures that can be defined locally in this way are called almost-Riemannian structures. The main result of the paper is a generalization to almost-Riemannian structures of the Gauss-Bonnet formula for domains with piecewise-C2 boundary. The main feature of such formula is the presence of terms that play the role of high-order angles at the intersection points with the set of singularities.}, url = {http://hdl.handle.net/1963/1995}, author = {Ugo Boscain and Mario Sigalotti} } @article {2007, title = {The holomorphic anomaly for open string moduli}, journal = {JHEP 10 (2007) 060}, number = {SISSA;55/2007/EP}, year = {2007}, abstract = {We complete the holomorphic anomaly equations for topological strings with their dependence on open moduli. We obtain the complete system by standard path integral arguments generalizing the analysis of BCOV (Commun. Math. Phys. 165 (1994) 311) to strings with boundaries. We study both the anti-holomorphic dependence on open moduli and on closed moduli in presence of Wilson lines. By providing the compactification a\\\' la Deligne-Mumford of the moduli space of Riemann surfaces with boundaries, we show that the open holomorphic anomaly equations are structured on the (real codimension one) boundary components of this space.}, doi = {10.1088/1126-6708/2007/10/060}, url = {http://hdl.handle.net/1963/2113}, author = {Giulio Bonelli and Alessandro Tanzini} } @article {2006, title = {On Hamiltonian perturbations of hyperbolic systems of conservation laws I: quasitriviality of bihamiltonian perturbations}, journal = {Comm. Pure Appl. Math. 59 (2006) 559-615}, number = {arXiv.org;math/0410027}, year = {2006}, abstract = {We study the general structure of formal perturbative solutions to the Hamiltonian perturbations of spatially one-dimensional systems of hyperbolic PDEs. Under certain genericity assumptions it is proved that any bihamiltonian perturbation can be eliminated in all orders of the perturbative expansion by a change of coordinates on the infinite jet space depending rationally on the derivatives. The main tools is in constructing of the so-called quasi-Miura transformation of jet coordinates eliminating an arbitrary deformation of a semisimple bihamiltonian structure of hydrodynamic type (the quasitriviality theorem). We also describe, following \\\\cite{LZ1}, the invariants of such bihamiltonian structures with respect to the group of Miura-type transformations depending polynomially on the derivatives.}, doi = {10.1002/cpa.20111}, url = {http://hdl.handle.net/1963/2535}, author = {Boris Dubrovin and Liu Si-Qi and Zhang Youjin} } @article {2006, title = {On Hamiltonian perturbations of hyperbolic systems of conservation laws, II: universality of critical behaviour}, number = {SISSA;89/2005/FM}, year = {2006}, abstract = {Hamiltonian perturbations of the simplest hyperbolic equation $u_t + a(u) u_x=0$ are studied. We argue that the behaviour of solutions to the perturbed equation near the point of gradient catastrophe of the unperturbed one should be essentially independent on the choice of generic perturbation neither on the choice of generic solution. Moreover, this behaviour is described by a special solution to an integrable fourth order ODE.}, doi = {10.1007/s00220-006-0021-5}, url = {http://hdl.handle.net/1963/1786}, author = {Boris Dubrovin} } @article {2006, title = {Homogeneous polynomial forms for simultaneous stabilizability of families of linear control systems: a tensor product approach}, journal = {IEEE Trans. Automat. Control 51 (2006) 1566-1571}, year = {2006}, abstract = {The paper uses the formalism of tensor products in order to deal with the problem of simultaneous\\nstabilizability of a family of linear control systems by means of Lyapunov functions which are homogeneous polynomial forms. While the feedback synthesis seems to be nonconvex, the simultaneous stability by means of homogeneous polynomial forms of the uncontrollable modes yields (convex) necessary but not sufficient conditions for simultaneous stabilizability.}, doi = {10.1109/TAC.2006.880811}, url = {http://hdl.handle.net/1963/2226}, author = {Claudio Altafini} } @article {2006, title = {A Hopf bundle over a quantum four-sphere from the symplectic group}, journal = {Commun. Math. Phys. 263 (2006) 65-88}, number = {arXiv.org;math/0407342v2}, year = {2006}, abstract = {We construct a quantum version of the SU(2) Hopf bundle $S^7 \\\\to S^4$. The quantum sphere $S^7_q$ arises from the symplectic group $Sp_q(2)$ and a quantum 4-sphere $S^4_q$ is obtained via a suitable self-adjoint idempotent $p$ whose entries generate the algebra $A(S^4_q)$ of polynomial functions over it. This projection determines a deformation of an (anti-)instanton bundle over the classical sphere $S^4$. We compute the fundamental $K$-homology class of $S^4_q$ and pair it with the class of $p$ in the $K$-theory getting the value -1 for the topological charge. There is a right coaction of $SU_q(2)$ on $S^7_q$ such that the algebra $A(S^7_q)$ is a non trivial quantum principal bundle over $A(S^4_q)$ with structure quantum group $A(SU_q(2))$.}, doi = {10.1007/s00220-005-1494-3}, url = {http://hdl.handle.net/1963/2179}, author = {Giovanni Landi and Chiara Pagani and Cesare Reina} } @article {2005, title = {Hybrid necessary principle}, journal = {SIAM J. Control Optim. 43 (2005) 1867-1887}, number = {SISSA;71/2002/M}, year = {2005}, note = {Proceedings of IFAC Conference on Analysis and Design of Hybrid Systems, Saint Malo, France}, publisher = {SIAM}, abstract = {We consider a hybrid control system and general optimal control problems for this system. We suppose that the switching strategy imposes restrictions on control sets and we provide necessary conditions for an optimal hybrid trajectory, stating a hybrid necessary principle (HNP). Our result generalizes various necessary principles available in the literature.}, doi = {10.1137/S0363012903416219}, url = {http://hdl.handle.net/1963/1641}, author = {Mauro Garavello and Benedetto Piccoli} } @article {2004, title = {H-bubbles in a perturbative setting: the finite-dimensional reduction\\\'s method}, journal = {Duke Math. J. 122 (2004), no. 3, 457--484}, number = {SISSA;36/2002/M}, year = {2004}, publisher = {SISSA Library}, abstract = {Given a regular function $H\\\\colon\\\\mathbb{R}^{3}\\\\to\\\\mathbb{R}$, we look for $H$-bubbles, that is, regular surfaces in $\\\\mathbb{R}^{3}$ parametrized on the sphere $\\\\mathbb{S}+^{2}$ with mean curvature $H$ at every point. Here we study the case of $H(u)=H_{0}+\\\\varepsilon H_{1}(u)=:H_{\\\\varepsilon}(u)$, where $H_{0}$ is a nonzero constant, $\\\\varepsilon$ is the smallness parameter, and $H_{1}$ is any $C^{2}$-function. We prove that if $\\\\bar p\\\\in\\\\mathbb{R}^{3}$ is a {\textquoteleft}{\textquoteleft}good\\\'\\\' stationary point for the Melnikov-type function $\\\\Gamma(p)=-\\\\int_{|q-p|<|H_{0}|^{-1}}H_{1}(q)\\\\,dq$, then for $|\\\\varepsilon|$ small there exists an $H_{\\\\varepsilon}$-bubble $\\\\omega^{\\\\varepsilon}$ that converges to a sphere of radius $|H_{0}|^{-1}$ centered at $\\\\bar p$, as $\\\\varepsilon\\\\to 0$.}, doi = {10.1215/S0012-7094-04-12232-8}, url = {http://hdl.handle.net/1963/1607}, author = {Paolo Caldiroli and Roberta Musina} } @article {2004, title = {Higher order quasiconvexity reduces to quasiconvexity}, journal = {Arch. Ration. Mech. Anal. 171 (2004) 55-81}, number = {SISSA;41/2003/M}, year = {2004}, publisher = {Springer}, abstract = {In this paper it is shown that higher order quasiconvex functions suitable in the variational treatment of problems involving second derivatives may be extended to the space of all matrices as classical quasiconvex functions. Precisely, it is proved that a smooth strictly 2-quasiconvex function with p-growth at infinity, p>1, is the restriction to symmetric matrices of a 1-quasiconvex function with the same growth. As a consequence, lower semicontinuity results for second-order variational problems are deduced as corollaries of well-known first order theorems.}, doi = {10.1007/s00205-003-0278-1}, url = {http://hdl.handle.net/1963/2911}, author = {Gianni Dal Maso and Irene Fonseca and Giovanni Leoni and Massimiliano Morini} } @article {2003, title = {Hybrid optimal control: case study of a car with gears}, journal = {Int. J. Control 76 (2003) 1272-1284}, year = {2003}, publisher = {Taylor and Francis}, abstract = {The purpose of this paper is to show the use of some analytical tools for hybrid optimal control. We illustrate both the hybrid maximum principle and the hybrid necessary principle at work on a simple example of a car with gears. The model is sufficiently rich to generate non-trivial optimization problems and the obtained results match with intuition. Finally, computer simulations confirm the theoretical analysis.}, doi = {10.1080/0020717031000147520}, url = {http://hdl.handle.net/1963/3022}, author = {Ciro D{\textquoteright}Apice and Mauro Garavello and Rosanna Manzo and Benedetto Piccoli} } @article {2000, title = {High-order Averaging and Stability of Time-Varying Systems}, number = {SISSA;7/00/M}, year = {2000}, publisher = {SISSA Library}, url = {http://hdl.handle.net/1963/1465}, author = {Andrey Sarychev} } @article {1999, title = {Hyperbolic Systems of Conservation Laws}, journal = {Rev. Mat. Complut. 12 (1999) 135-200}, year = {1999}, abstract = {This is a survey paper, written in the occasion of an invited talk given by the author at the Universidad Complutense in Madrid, October 1998. Its purpose is to provide an account of some recent advances in the mathematical theory of hyperbolic systems of conservation laws in one space dimension. After a brief review of basic concepts, we describe in detail the method of wave-front tracking approximation and present some of the latest results on uniqueness and stability of entropy weak solutions.}, url = {http://hdl.handle.net/1963/1855}, author = {Alberto Bressan} } @article {1997, title = {Homogeneous tangent vectors and high order necessary conditions for optimal controls}, journal = {J. Dynam. Control Systems 3 (1997), no. 2, 205--240}, number = {SISSA;113/95/M}, year = {1997}, publisher = {SISSA Library}, url = {http://hdl.handle.net/1963/1015}, author = {Fabio Ancona} } @article {10572, title = {Hilbert schemes of points on some K3 surfaces and Gieseker stable boundles}, journal = {MATH PROC CAMBRIDGE 120: 255-261 Part 2}, number = {SISSA;34/95/GEO-FM}, year = {1994}, publisher = {SISSA Library}, abstract = {By using a Fourier-Mukai transform for sheaves on K3 surfaces we show that for a wide class of K3 surfaces $X$ the punctual Hilbert schemes $\\\\Hilb^n(X)$ can be identified, for all $n\\\\geq 1$, with moduli spaces of Gieseker stable vector bundles on $X$ of rank $1+2n$. We also introduce a new Fourier-Mukai type transform for such surfaces.

}, url = {http://hdl.handle.net/1963/937}, author = {Ugo Bruzzo and Antony Maciocia} } @article {1992, title = {Hamiltonian formalism of Whitham-type hierarchies and topological Landau - Ginsburg models}, journal = {Communications in Mathematical Physics. Volume 145, Issue 1, March 1992, Pages 195-207}, year = {1992}, publisher = {SISSA}, abstract = {We show that the bi-hamiltonian structure of the averaged Gelfand-Dikii\\r\\nhierarchy is involved in the Landau-Ginsburg topological models (for An-Series):\\r\\nthe Casimirs for the first P.B. give the correct coupling parameters for the perturbed\\r\\ntopological minimal model; the correspondence {coupling parameters} ~ {primary\\r\\nfields} is determined by the second P.B. The partition function (at the tree level) and\\r\\nthe chiral algebra for LG models are calculated for any genus g.}, doi = {10.1007/BF02099286}, url = {http://hdl.handle.net/1963/6476}, author = {Boris Dubrovin} } @article {1989, title = {Hyperbolic equations as ordinary differential equations in Banach spaces}, number = {SISSA;11/89/M}, year = {1989}, publisher = {SISSA Library}, url = {http://hdl.handle.net/1963/773}, author = {Giovanni Vidossich} } @article {1988, title = {Holes and obstacles}, journal = {Ann. Inst. H. Poincare Anal. Non Lineaire 5 (1988), no. 4, 323-345}, number = {SISSA;20/87/M}, year = {1988}, publisher = {SISSA Library}, url = {http://hdl.handle.net/1963/501}, author = {Roberta Musina and Giovanni Mancini} } @article {1988, title = {H-surfaces with obstacles. (Italian)}, journal = {Ann. Univ. Ferrara Sez. VII (N.S.) 34 (1988), 1-14}, number = {SISSA;9/87/M}, year = {1988}, publisher = {SISSA Library}, url = {http://hdl.handle.net/1963/491}, author = {Roberta Musina} }