A discrete districting plan - Saracco, A.; Saracco, G. (Netw. Heterog. Media, 2019)

Efficient reduction in shape parameter space dimension for ship propeller blade design - Andrea Mola; Marco Tezzele; Mahmoud Gadalla; Valdenazzi, Federica; Grassi, Davide; Padovan, Roberta; Gianluigi Rozza (8th International Conference on Computational Methods in Marine Engineering, MARINE 2019, 2019)
In this work, we present the results of a ship propeller design optimization campaign carried out in the framework of the research project PRELICA, funded by the Friuli Venezia Giulia regional government. The main idea of this work is to operate on a multidisciplinary level to identify propeller shapes that lead to reduced tip vortex-induced pressure and increased efficiency without altering the thrust. First, a specific tool for the bottom-up construction of parameterized propeller blade geometries has been developed. The algorithm proposed operates with a user defined number of arbitrary shaped or NACA airfoil sections, and employs arbitrary degree NURBS to represent the chord, pitch, skew and rake distribution as a function of the blade radial coordinate. The control points of such curves have been modified to generate, in a fully automated way, a family of blade geometries depending on as many as 20 shape parameters. Such geometries have then been used to carry out potential flow simulations with the Boundary Element Method based software PROCAL. Given the high number of parameters considered, such a preliminary stage allowed for a fast evaluation of the performance of several hundreds of shapes. In addition, the data obtained from the potential flow simulation allowed for the application of a parameter space reduction methodology based on active subspaces (AS) property, which suggested that the main propeller performance indices are, at a first but rather accurate approximation, only depending on a single parameter which is a linear combination of all the original geometric ones. AS analysis has also been used to carry out a constrained optimization exploiting response surface method in the reduced parameter space, and a sensitivity analysis based on such surrogate model. The few selected shapes were finally used to set up high fidelity RANS simulations and select an optimal shape.

An entropic interpolation proof of the HWI inequality - Ivan Gentil; Christian Léonard; Luigia Ripani; Luca Tamanini (Stochastic Processes and their Applications, 2019)
The HWI inequality is an “interpolation”inequality between the Entropy H, the Fisher information I and the Wasserstein distance W. We present a pathwise proof of the HWI inequality which is obtained through a zero noise limit of the Schrödinger problem. Our approach consists in making rigorous the Otto–Villani heuristics in Otto and Villani (2000) taking advantage of the entropic interpolations, which are regular both in space and time, rather than the displacement ones.
Keywords: Entropic interpolations; Fisher information; Relative entropy; Schrödinger problem; Wasserstein distance

Error estimates in weighted Sobolev norms for finite element immersed interface methods - Luca Heltai; Nella Rotundo (Computers & Mathematics with Applications, 2019)

On the existence of elastic minimizers for initially stressed materials - Davide Riccobelli; A. Agosti; Pasquale Ciarletta (Phil. Trans. R. Soc. A, 2019)

A Finite Volume approximation of the Navier-Stokes equations with nonlinear filtering stabilization - Michele Girfoglio; Annalisa Quaini; Gianluigi Rozza (Computers and Fluids, 2019)
We consider a Leray model with a nonlinear differential low-pass filter for the simulation of incompressible fluid flow at moderately large Reynolds number (in the range of a few thousands) with under-refined meshes. For the implementation of the model, we adopt the three-step algorithm Evolve-Filter-Relax (EFR). The Leray model has been extensively applied within a Finite Element (FE) framework. Here, we propose to combine the EFR algorithm with a computationally efficient Finite Volume (FV) method. Our approach is validated against numerical data available in the literature for the 2D flow past a cylinder and against experimental measurements for the 3D fluid flow in an idealized medical device, as recommended by the U.S. Food and Drug Administration. We will show that for similar levels of mesh refinement FV and FE methods provide significantly different results. Through our numerical experiments, we are able to provide practical directions to tune the parameters involved in EFR algorithm. Furthermore, we are able to investigate the impact of mesh features (element type, non-orthogonality, local refinement, and element aspect ratio) and the discretization method for the convective term on the agreement between numerical solutions and experimental data.

A Finite Volume approximation of the Navier-Stokes equations with nonlinear filtering stabilization - Michele Girfoglio; Annalisa Quaini; Gianluigi Rozza (Computers & Fluids, 2019)
We consider a Leray model with a nonlinear differential low-pass filter for the simulation of incompressible fluid flow at moderately large Reynolds number (in the range of a few thousands) with under-refined meshes. For the implementation of the model, we adopt the three-step algorithm Evolve-Filter-Relax (EFR). The Leray model has been extensively applied within a Finite Element (FE) framework. Here, we propose to combine the EFR algorithm with a computationally efficient Finite Volume (FV) method. Our approach is validated against numerical data available in the literature for the 2D flow past a cylinder and against experimental measurements for the 3D fluid flow in an idealized medical device, as recommended by the U.S. Food and Drug Administration. We will show that for similar levels of mesh refinement FV and FE methods provide significantly different results. Through our numerical experiments, we are able to provide practical directions to tune the parameters involved in the model. Furthermore, we are able to investigate the impact of mesh features (element type, non-orthogonality, local refinement, and element aspect ratio) and the discretization method for the convective term on the agreement between numerical solutions and experimental data.

Ground state energy of mixture of Bose gases - Alessandro Michelangeli; Phan Thanh Nam; Alessandro Olgiati (Reviews in Mathematical Physics, 2019)
We consider the asymptotic behavior of a system of multi-component trapped bosons, when the total particle number N becomes large. In the dilute regime, when the interaction potentials have the length scale of order O(N−1), we show that the leading order of the ground state energy is captured correctly by the Gross–Pitaevskii energy functional and that the many-body ground state fully condensates on the Gross–Pitaevskii minimizers. In the mean-field regime, when the interaction length scale is O(1), we are able to verify Bogoliubov’s approximation and obtain the second order expansion of the ground state energy. While such asymptotic results have several precursors in the literature on one-component condensates, the adaptation to the multi-component setting is non-trivial in various respects and the analysis will be presented in detail.

hp-adaptive discontinuous Galerkin methods for non-stationary convection–diffusion problems - Andrea Cangiani; E.H. Georgoulis; Stefano Giani; S. Metcalfe (Computers & Mathematics with Applications, 2019)
An a posteriori error estimator for the error in the (L2(H1)+L∞(L2))-type norm for an interior penalty discontinuous Galerkin (dG) spatial discretisation and backward Euler temporal discretisation of linear non-stationary convection–diffusion initial/boundary value problems is derived, allowing for anisotropic elements. The proposed error estimator is used to drive an hp-space–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–time adaptive algorithm is assessed via a number of standard test problems characterised by sharp and/or moving layers.
Applications of Partial Differential Equations in Science and Engineering
Keywords: A posteriori error estimation; Adaptive finite element methods; Anisotropic meshes; Discontinuous Galerkin; Unsteady convection–diffusion

Isomonodromy deformations at an irregular singularity with coalescing eigenvalues - Giordano Cotti; Boris Dubrovin; Davide Guzzetti (Duke Math. J., 2019)
We consider an n×n linear system of ODEs with an irregular singularity of Poincar\'e rank 1 at z=∞, holomorphically depending on parameter t within a polydisc in Cn centred at t=0. The eigenvalues of the leading matrix at z=∞ coalesce along a locus Δ contained in the polydisc, passing through t=0. Namely, z=∞ is a resonant irregular singularity for t∈Δ. We analyse the case when the leading matrix remains diagonalisable at Δ. We discuss the existence of fundamental matrix solutions, their asymptotics, Stokes phenomenon and monodromy data as t varies in the polydisc, and their limits for t tending to points of Δ. When the deformation is isomonodromic away from Δ, it is well known that a fundamental matrix solution has singularities at Δ. When the system also has a Fuchsian singularity at z=0, we show under minimal vanishing conditions on the residue matrix at z=0 that isomonodromic deformations can be extended to the whole polydisc, including Δ, in such a way that the fundamental matrix solutions and the constant monodromy data are well defined in the whole polydisc. These data can be computed just by considering the system at fixed t=0. Conversely, if the t-dependent system is isomonodromic in a small domain contained in the polydisc not intersecting Δ, if the entries of the Stokes matrices with indices corresponding to coalescing eigenvalues vanish, then we show that Δ is not a branching locus for the fundamental matrix solutions. The importance of these results for the analytic theory of Frobenius Manifolds is explained. An application to Painlev\'e equations is discussed.

Isoperimetric inequality under Measure-Contraction property - Fabio Cavalletti; Flavia Santarcangelo (Journal of Functional Analysis, 2019)
We prove that if (X,d,m) is an essentially non-branching metric measure space with m(X)=1, having Ricci curvature bounded from below by K and dimension bounded above by N∈(1,∞), understood as a synthetic condition called Measure-Contraction property, then a sharp isoperimetric inequality à la Lévy-Gromov holds true. Measure theoretic rigidity is also obtained.
Keywords: Isoperimetric inequality; Measure-Contraction property; Optimal transport; Ricci curvature

On Krylov solutions to infinite-dimensional inverse linear problems - Noe Caruso; Alessandro Michelangeli; Paolo Novati (Calcolo, 2019)
We discuss, in the context of inverse linear problems in Hilbert space, the notion of the associated infinite-dimensional Krylov subspace and we produce necessary and sufficient conditions for the Krylov-solvability of a given inverse problem, together with a series of model examples and numerical experiments.

Local well-posedness for quasi-linear NLS with large Cauchy data on the circle - Roberto Feola; Felice Iandoli (Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 2019)
We prove local in time well-posedness for a large class of quasilinear Hamiltonian, or parity preserving, Schrödinger equations on the circle. After a paralinearization of the equation, we perform several paradifferential changes of coordinates in order to transform the system into a paradifferential one with symbols which, at the positive order, are constant and purely imaginary. This allows to obtain a priori energy estimates on the Sobolev norms of the solutions.
Keywords: Dispersive equations; Energy method; Local wellposedness; NLS; Para-differential calculus; Quasi-linear PDEs

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

A localized reduced-order modeling approach for PDEs with bifurcating solutions - Martin W. Hess; Alla, Alessandro; Annalisa Quaini; Gianluigi Rozza; Max Gunzburger (Computer Methods in Applied Mechanics and Engineering, 2019)
Reduced-order modeling (ROM) commonly refers to the construction, based on a few solutions (referred to as snapshots) of an expensive discretized partial differential equation (PDE), and the subsequent application of low-dimensional discretizations of partial differential equations (PDEs) that can be used to more efficiently treat problems in control and optimization, uncertainty quantification, and other settings that require multiple approximate PDE solutions. Although ROMs have been successfully used in many settings, ROMs built specifically for the efficient treatment of PDEs having solutions that bifurcate as the values of input parameters change have not received much attention. In such cases, the parameter domain can be subdivided into subregions, each of which corresponds to a different branch of solutions. Popular ROM approaches such as proper orthogonal decomposition (POD), results in a global low-dimensional basis that does not respect the often large differences in the PDE solutions corresponding to different subregions. In this work, we develop and test a new ROM approach specifically aimed at bifurcation problems. In the new method, the k-means algorithm is used to cluster snapshots so that within cluster snapshots are similar to each other and are dissimilar to those in other clusters. This is followed by the construction of local POD bases, one for each cluster. The method also can detect which cluster a new parameter point belongs to, after which the local basis corresponding to that cluster is used to determine a ROM approximation. Numerical experiments show the effectiveness of the method both for problems for which bifurcation cause continuous and discontinuous changes in the solution of the PDE.

Multiscale modeling of vascularized tissues via non-matching immersed methods - Luca Heltai; Alfonso Caiazzo (International Journal for Numerical Methods in Biomedical Engineering, 2019)

N=2 gauge theories on unoriented/open four-manifolds and their AGT counterparts - Aditya Bawane; Benvenuti, Sergio; Giulio Bonelli; Muteeb, Nouman; Alessandro Tanzini (JHEP, 2019)

A neutrally stable shell in a Stokes flow: a rotational Taylor's sheet - Giovanni Corsi; Antonio DeSimone; C. Maurini; S. Vidoli (Proceedings of the Royal Society A: Mathematical, Physical and Engineering SciencesProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2019)

A non-intrusive approach for the reconstruction of POD modal coefficients through active subspaces - Nicola Demo; Marco Tezzele; Gianluigi Rozza (Comptes Rendus - Mecanique, 2019)
Reduced order modeling (ROM) provides an efficient framework to compute solutions of parametric problems. Basically, it exploits a set of precomputed high-fidelity solutions—computed for properly chosen parameters, using a full-order model—in order to find the low dimensional space that contains the solution manifold. Using this space, an approximation of the numerical solution for new parameters can be computed in real-time response scenario, thanks to the reduced dimensionality of the problem. In a ROM framework, the most expensive part from the computational viewpoint is the calculation of the numerical solutions using the full-order model. Of course, the number of collected solutions is strictly related to the accuracy of the reduced order model. In this work, we aim at increasing the precision of the model also for few input solutions by coupling the proper orthogonal decomposition with interpolation (PODI)—a data-driven reduced order method—with the active subspace (AS) property, an emerging tool for reduction in parameter space. The enhanced ROM results in a reduced number of input solutions to reach the desired accuracy. In this contribution, we present the numerical results obtained by applying this method to a structural problem and in a fluid dynamics one.

A Note About the Strong Maximum Principle on RCD Spaces - Nicola Gigli; Chiara Rigoni (Canadian Mathematical Bulletin, 2019)
We give a direct proof of the strong maximum principle on finite dimensional RCD spaces based on the Laplacian comparison of the squared distance.

On the Number of Flats Tangent to Convex Hypersurfaces in Random Position - Khazhgali Kozhasov; Antonio Lerario (Discrete & Computational Geometry, 2019)

Numerical approximation of the integral fractional Laplacian - Bonito, Andrea; Wenyu Lei; Joseph E Pasciak (Numerische Mathematik, 2019)
We propose a new nonconforming finite element algorithm to approximate the solution to the elliptic problem involving the fractional Laplacian. We first derive an integral representation of the bilinear form corresponding to the variational problem. The numerical approximation of the action of the corresponding stiffness matrix consists of three steps: (1) apply a sinc quadrature scheme to approximate the integral representation by a finite sum where each term involves the solution of an elliptic partial differential equation defined on the entire space, (2) truncate each elliptic problem to a bounded domain, (3) use the finite element method for the space approximation on each truncated domain. The consistency error analysis for the three steps is discussed together with the numerical implementation of the entire algorithm. The results of computations are given illustrating the error behavior in terms of the mesh size of the physical domain, the domain truncation parameter and the quadrature spacing parameter.

Nutations in growing plant shoots: The role of elastic deformations due to gravity loading - Daniele Agostinelli; Alessandro Lucantonio; Giovanni Noselli; Antonio DeSimone (Journal of the Mechanics and Physics of Solids, 2019)
The effect of elastic deformations induced by gravity loading on the active circumnutation movements of growing plant shoots is investigated. We consider first a discrete model (a gravitropic spring-pendulum system) and then a continuous rod model which is analyzed both analytically (under the assumption of small deformations) and numerically (in the large deformation regime). We find that, for a choice of material parameters consistent with values reported in the available literature on plant shoots, rods of sufficient length may exhibit lateral oscillations of increasing amplitude, which eventually converge to limit cycles. This behavior strongly suggests the occurrence of a Hopf bifurcation, just as for the gravitropic spring-pendulum system, for which this result is rigorously established. At least in this restricted set of material parameters, our analysis supports a view of Darwin’s circumnutations as a biological analogue to structural systems exhibiting flutter instabilities, i.e., spontaneous oscillations away from equilibrium configurations driven by non-conservative loads. Here, in the context of nutation movements of growing plant shoots, the energy needed to sustain oscillations is continuously supplied to the system by the internal biochemical machinery presiding the capability of plants to maintain a vertical pose.
Keywords: Circumnutations; Flutter instability; Gravitropism; Hopf bifurcation

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

POD-Galerkin Reduced Order Model of the Boussinesq Approximation for Buoyancy-Driven Enclosed Flows - Kelbij Star; Giovanni Stabile; Sokratia Georgaka; Francesco Belloni; Gianluigi Rozza; Joris Degroote (International Conference on Mathematics and Computational Methods Applied to Nuclear Science and Engineering, M and C 2019, 2019)

A POD-selective inverse distance weighting method for fast parametrized shape morphing - F. Ballarin; A. D'Amario; Simona Perotto; Gianluigi Rozza (International Journal for Numerical Methods in Engineering, 2019)
Efficient shape morphing techniques play a crucial role in the approximation of partial differential equations defined in parametrized domains, such as for fluid-structure interaction or shape optimization problems. In this paper, we focus on inverse distance weighting (IDW) interpolation techniques, where a reference domain is morphed into a deformed one via the displacement of a set of control points. We aim at reducing the computational burden characterizing a standard IDW approach without significantly compromising the accuracy. To this aim, first we propose an improvement of IDW based on a geometric criterion that automatically selects a subset of the original set of control points. Then, we combine this new approach with a dimensionality reduction technique based on a proper orthogonal decomposition of the set of admissible displacements. This choice further reduces computational costs. We verify the performances of the new IDW techniques on several tests by investigating the trade-off reached in terms of accuracy and efficiency.

Point-Like Perturbed Fractional Laplacians Through Shrinking Potentials of Finite Range - Alessandro Michelangeli; Raffaele Scandone (Complex Analysis and Operator Theory, 2019)
We construct the rank-one, singular (point-like) perturbations of the d-dimensional fractional Laplacian in the physically meaningful norm-resolvent limit of fractional Schrödinger operators with regular potentials centred around the perturbation point and shrinking to a delta-like shape. We analyse both possible regimes, the resonance-driven and the resonance-independent limit, depending on the power of the fractional Laplacian and the spatial dimension. To this aim, we also qualify the notion of zero-energy resonance for Schrödinger operators formed by a fractional Laplacian and a regular potential.

Quantum Systems at The Brink. Existence and Decay Rates of Bound States at Thresholds; Helium - Dirk Hundertmark; Michal Jex; Markus Lange (arXiv:1908.04883, 2019)

Quantum Systems at The Brink. Existence and Decay Rates of Bound States at Thresholds; Atoms - Dirk Hundertmark; Michal Jex; Markus Lange (arXiv:1908.05016, 2019)

Quasi-continuous vector fields on RCD spaces - Clément Debin; Nicola Gigli; Enrico Pasqualetto (2019)

The Real Polynomial Eigenvalue Problem is Well Conditioned on the Average - Carlos Beltrán; Khazhgali Kozhasov (Foundations of Computational Mathematics, 2019)
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.

A reduced basis approach for PDEs on parametrized geometries based on the shifted boundary finite element method and application to a Stokes flow - Efthymios N Karatzas; Giovanni Stabile; Leo Nouveau; Guglielmo Scovazzi; Gianluigi Rozza (Computer Methods in Applied Mechanics and Engineering, 2019)
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.

Reduced Basis Approaches for Parametrized Bifurcation Problems held by Non-linear Von Kármán Equations - Federico Pichi; Gianluigi Rozza (Journal of Scientific Computing, 2019)
This work focuses on the computationally efficient detection of the buckling phenomena and bifurcation analysis of the parametric Von Kármán plate equations based on reduced order methods and spectral analysis. The computational complexity—due to the fourth order derivative terms, the non-linearity and the parameter dependence—provides an interesting benchmark to test the importance of the reduction strategies, during the construction of the bifurcation diagram by varying the parameter(s). To this end, together the state equations, we carry out also an analysis of the linearized eigenvalue problem, that allows us to better understand the physical behaviour near the bifurcation points, where we lose the uniqueness of solution. We test this automatic methodology also in the two parameter case, understanding the evolution of the first buckling mode.

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

A reduced order variational multiscale approach for turbulent flows - Giovanni Stabile; F. Ballarin; G. Zuccarino; Gianluigi Rozza (Advances in Computational Mathematics, 2019)
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.

Reducibility for a fast-driven linear Klein–Gordon equation - Luca Franzoi; Alberto Maspero (Annali di Matematica Pura ed Applicata (1923 -), 2019)
We prove a reducibility result for a linear Klein–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.

Reducibility of first order linear operators on tori via Moser's theorem - Roberto Feola; Filippo Giuliani; Riccardo Montalto; Michela Procesi (Journal of Functional Analysis, 2019)
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'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–Moser; Reducibility

Regularity of minimizers for a model of charged droplets - Guido De Philippis; Jonas Hirsch; Giulia Vescovo (2019)

On the relaxed area of the graph of discontinuous maps from the plane to the plane taking three values with no symmetry assumptions - Giovanni Bellettini; Alaa Elshorbagy; Maurizio Paolini; Riccardo Scala (Annali di Matematica Pura ed Applicata (1923 -), 2019)
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 ``fill the hole'' 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.

The Serre–Swan theorem for normed modules - Danka Lučić; Enrico Pasqualetto (Rendiconti del Circolo Matematico di Palermo Series 2, 2019)

Shape optimization through proper orthogonal decomposition with interpolation and dynamic mode decomposition enhanced by active subspaces - Marco Tezzele; Nicola Demo; Gianluigi Rozza (8th International Conference on Computational Methods in Marine Engineering, MARINE 2019, 2019)
We propose a numerical pipeline for shape optimization in naval engineering involving two different non-intrusive reduced order method (ROM) techniques. Such methods are proper orthogonal decomposition with interpolation (PODI) and dynamic mode decomposition (DMD). The ROM proposed will be enhanced by active subspaces (AS) as a pre-processing tool that reduce the parameter space dimension and suggest better sampling of the input space. We will focus on geometrical parameters describing the perturbation of a reference bulbous bow through the free form deformation (FFD) technique. The ROM are based on a finite volume method (FV) to simulate the multi-phase incompressible flow around the deformed hulls. In previous works we studied the reduction of the parameter space in naval engineering through AS [38, 10] focusing on different parts of the hull. PODI and DMD have been employed for the study of fast and reliable shape optimization cycles on a bulbous bow in [9]. The novelty of this work is the simultaneous reduction of both the input parameter space and the output fields of interest. In particular AS will be trained computing the total drag resistance of a hull advancing in calm water and its gradients with respect to the input parameters. DMD will improve the performance of each simulation of the campaign using only few snapshots of the solution fields in order to predict the regime state of the system. Finally PODI will interpolate the coefficients of the POD decomposition of the output fields for a fast approximation of all the fields at new untried parameters given by the optimization algorithm. This will result in a non-intrusive data-driven numerical optimization pipeline completely independent with respect to the full order solver used and it can be easily incorporated into existing numerical pipelines, from the reference CAD to the optimal shape.

A Spectral Element Reduced Basis Method in Parametric CFD - Martin W. Hess; Gianluigi Rozza (Numerical Mathematics and Advanced Applications - ENUMATH 2017, 2019)
We consider the Navier-Stokes equations in a channel with varying Reynolds numbers. The model is discretized with high-order spectral element ansatz functions, resulting in 14 259 degrees of freedom. The steady-state snapshot solu- tions define a reduced order space, which allows to accurately evaluate the steady- state solutions for varying Reynolds number with a reduced order model within a fixed-point iteration. In particular, we compare different aspects of implementing the reduced order model with respect to the use of a spectral element discretization. It is shown, how a multilevel static condensation in the pressure and velocity boundary degrees of freedom can be combined with a reduced order modelling approach to enhance computational times in parametric many-query scenarios.

A spectral element reduced basis method in parametric CFD - Martin W. Hess; Gianluigi Rozza (Lecture Notes in Computational Science and Engineering, 2019)
We consider the Navier-Stokes equations in a channel with varying Reynolds numbers. The model is discretized with high-order spectral element ansatz functions, resulting in 14,259 degrees of freedom. The steady-state snapshot solutions define a reduced order space, which allows to accurately evaluate the steady-state solutions for varying Reynolds number with a reduced order model within a fixed-point iteration. In particular, we compare different aspects of implementing the reduced order model with respect to the use of a spectral element discretization. It is shown, how a multilevel static condensation (Karniadakis and Sherwin, Spectral/hp element methods for computational fluid dynamics, 2nd edn. Oxford University Press, Oxford, 2005) in the pressure and velocity boundary degrees of freedom can be combined with a reduced order modelling approach to enhance computational times in parametric many-query scenarios.

On the square distance function from a manifold with boundary - Giovanni Bellettini; Alaa Elshorbagy (2019)
We characterize arbitrary codimensional smooth manifolds $\mathcal{M}$ with boundary embedded in $\mathbb{R}^n$ using the square distance function and the signed distance function from $\mathcal{M}$ and from its boundary. The results are localized in an open set.

Strong Novikov conjecture for low degree cohomology and exotic group C*-algebras - Paolo Antonini; Buss, Alcides; Engel, Alexander; Siebenand , Timo (arXiv e-prints, 2019)
We strengthen a result of Hanke–Schick about the strong Novikov conjecture for low degree cohomology by showing that their non-vanishing result for the maximal group $C^*$-algebra even holds for the reduced group $C^*$-algebra. To achieve this we provide a Fell absorption principle for certain exotic crossed product functors.
Keywords: Mathematics - K-Theory and Homology; Mathematics - Operator Algebras

Swimming Euglena respond to confinement with a behavioural change enabling effective crawling - Giovanni Noselli; Alfred Beran; Marino Arroyo; Antonio DeSimone (Nature Physics, 2019)
Some euglenids, a family of aquatic unicellular organisms, can develop highly concerted, large-amplitude peristaltic body deformations. This remarkable behaviour has been known for centuries. Yet, its function remains controversial, and is even viewed as a functionless ancestral vestige. Here, by examining swimming Euglena gracilis in environments of controlled crowding and geometry, we show that this behaviour is triggered by confinement. Under these conditions, it allows cells to switch from unviable flagellar swimming to a new and highly robust mode of fast crawling, which can deal with extreme geometric confinement and turn both frictional and hydraulic resistance into propulsive forces. To understand how a single cell can control such an adaptable and robust mode of locomotion, we developed a computational model of the motile apparatus of Euglena cells consisting of an active striated cell envelope. Our modelling shows that gait adaptability does not require specific mechanosensitive feedback but instead can be explained by the mechanical self-regulation of an elastic and extended motor system. Our study thus identifies a locomotory function and the operating principles of the adaptable peristaltic body deformation of Euglena cells.

On the topological degree of planar maps avoiding normal cones - Alessandro Fonda; Giuliano Klun (TOPOLOGICAL METHODS IN NONLINEAR ANALYSIS, 2019)
The classical Poincaré-Bohl theorem provides the existence of a zero for a function avoiding external rays. When the domain is convex, the same holds true when avoiding normal cones. We consider here the possibility of dealing with nonconvex sets having inward corners or cusps, in which cases the normal cone vanishes. This allows us to deal with situations where the topological degree may be strictly greater than $1$.

Two-Dimensional Yang–Mills Theory on Surfaces with Corners in Batalin–Vilkovisky Formalism - Riccardo Iraso; Mnev, P. (Communications in Mathematical Physics, 2019)
In this paper we recover the non-perturbative partition function of 2D Yang–Mills theory from the perturbative path integral. To achieve this goal, we study the perturbative path integral quantization for 2D Yang–Mills theory on surfaces with boundaries and corners in the Batalin–Vilkovisky formalism (or, more precisely, in its adaptation to the setting with boundaries, compatible with gluing and cutting–-the BV-BFV formalism). We prove that cutting a surface (e.g. a closed one) into simple enough pieces–-building blocks–-and choosing a convenient gauge-fixing on the pieces, and assembling back the partition function on the surface, one recovers the known non-perturbative answers for 2D Yang–Mills theory.

Virtual element method for quasilinear elliptic problems - Andrea Cangiani; Chatzipantelidis, P; Diwan, G; E.H. Georgoulis (IMA Journal of Numerical Analysis, 2019)
A virtual element method for the quasilinear equation \\$-\\textrm\{div\} (\{\\boldsymbol \ąppa \}(u)\\operatorname\{grad\} u)=f\\$ using general polygonal and polyhedral meshes is presented and analysed. The nonlinear coefficient is evaluated with the piecewise polynomial projection of the virtual element ansatz. Well posedness of the discrete problem and optimal-order a priori error estimates in the \\$H^1\\$- and \\$L^2\\$-norm are proven. In addition, the convergence of fixed-point iterations for the resulting nonlinear system is established. Numerical tests confirm the optimal convergence properties of the method on general meshes.

A Weighted POD Method for Elliptic PDEs with Random Inputs - L.Venturi; F. Ballarin; Gianluigi Rozza (Journal of Scientific Computing, 2019)
In this work we propose and analyze a weighted proper orthogonal decomposition method to solve elliptic partial differential equations depending on random input data, for stochastic problems that can be transformed into parametric systems. The algorithm is introduced alongside the weighted greedy method. Our proposed method aims to minimize the error in a L2 norm and, in contrast to the weighted greedy approach, it does not require the availability of an error bound. Moreover, we consider sparse discretization of the input space in the construction of the reduced model; for high-dimensional problems, provided the sampling is done accordingly to the parameters distribution, this enables a sensible reduction of computational costs, while keeping a very good accuracy with respect to high fidelity solutions. We provide many numerical tests to assess the performance of the proposed method compared to an equivalent reduced order model without weighting, as well as to the weighted greedy approach, in both low and high dimensional problems.

Weighted Reduced Order Methods for Parametrized Partial Differential Equations with Random Inputs - L. Venturi; D. Torlo; F. Ballarin; Gianluigi Rozza (PoliTO Springer Series, 2019)
In this manuscript we discuss weighted reduced order methods for stochastic partial differential equations. Random inputs (such as forcing terms, equation coefficients, boundary conditions) are considered as parameters of the equations. We take advantage of the resulting parametrized formulation to propose an efficient reduced order model; we also profit by the underlying stochastic assumption in the definition of suitable weights to drive to reduction process. Two viable strategies are discussed, namely the weighted reduced basis method and the weighted proper orthogonal decomposition method. A numerical example on a parametrized elasticity problem is shown.

Zero modes and low-energy resolvent expansion for three dimensional Schrodinger operators with point interactions - Raffaele Scandone (2019)

Accelerating the iterative solution of convection-diffusion problems using singular value decomposition - Giuseppe Pitton; Luca Heltai (NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, 2018)

Adaptive discontinuous Galerkin methods for elliptic interface problems - Andrea Cangiani; E.H. Georgoulis; Sabawi, Younis A. (Math. Comp., 2018)

Analysis of a Dynamic Peeling Test with Speed-Dependent Toughness - Giuliano Lazzaroni; Lorenzo Nardini (SIAM Journal on Applied Mathematics, 2018)
We analyse a one-dimensional model of dynamic debonding for a thin film, where the local toughness of the glue between the film and the substrate also depends on the debonding speed. The wave equation on the debonded region is strongly coupled with Griffith's criterion for the evolution of the debonding front. We provide an existence and uniqueness result and find explicitly the solution in some concrete examples. We study the limit of solutions as inertia tends to zero, observing phases of unstable propagation, as well as time discontinuities, even though the toughness diverges at a limiting debonding speed.

On asymptotic expansions in spin-boson models - Bräunlich, Gerhard; Hasler, David; Markus Lange (Ann. Henri Poincaré, 2018)

An authenticated theoretical modeling of electrified fluid jet in core–shell nanofibers production - Rafiei, S.; Noroozi, B.; Luca Heltai; Haghi, A. K. (JOURNAL OF INDUSTRIAL TEXTILES, 2018)

The Baum–Connes conjecture localised at the unit element of a discrete group - Paolo Antonini; Azzali, S.; Skandalis, G. (ArXiv e-prints, 2018)
Keywords: 19K35; 46L80; 46L85; 58J22; Mathematics - K-Theory and Homology; Mathematics - Operator Algebras

Canonical Surfaces and Hypersurfaces in Abelian Varieties - Luca Cesarano (2018)

On the Cauchy problem for the wave equation on time-dependent domains - Gianni Dal Maso; Rodica Toader (SISSA, 2018)
We introduce a notion of solution to the wave equation on a suitable class of time-dependent domains and compare it with a previous de nition. We prove an existence result for the solution of the Cauchy problem and present some additional conditions which imply uniqueness.

Certified Reduced Basis Approximation for the Coupling of Viscous and Inviscid Parametrized Flow Models - Immanuel Martini; Bernard Haasdonk; Gianluigi Rozza (Journal of Scientific Computing, 2018)

Characteristic boundary layers for mixed hyperbolic systems in one space dimension and applications to the Navier-Stokes and MHD equations - Stefano Bianchini; Laura Spinolo (SISSA, 2018)
We provide a detailed analysis of the boundary layers for mixed hyperbolic-parabolic systems in one space dimension and small amplitude regimes. As an application of our results, we describe the solution of the so-called boundary Riemann problem recovered as the zero viscosity limit of the physical viscous approximation. In particular, we tackle the so called doubly characteristic case, which is considerably more demanding from the technical viewpoint and occurs when the boundary is characteristic for both the mixed hyperbolic-parabolic system and for the hyperbolic system obtained by neglecting the second order terms. Our analysis applies in particular to the compressible Navier-Stokes and MHD equations in Eulerian coordinates, with both positive and null conductivity. In these cases, the doubly characteristic case occurs when the velocity is close to 0. The analysis extends to non-conservative systems.

Cohesive fracture with irreversibility: Quasistatic evolution for a model subject to fatigue - Vito Crismale; Giuliano Lazzaroni; Gianluca Orlando (Mathematical Models and Methods in Applied Sciences, 2018)
In this paper we prove the existence of quasistatic evolutions for a cohesive fracture on a prescribed crack surface, in small-strain antiplane elasticity. The main feature of the model is that the density of the energy dissipated in the fracture process depends on the total variation of the amplitude of the jump. Thus, any change in the crack opening entails a loss of energy, until the crack is complete. In particular this implies a fatigue phenomenon, i.e. a complete fracture may be produced by oscillation of small jumps. The first step of the existence proof is the construction of approximate evolutions obtained by solving discrete-time incremental minimum problems. The main difficulty in the passage to the continuous-time limit is that we lack of controls on the variations of the jump of the approximate evolutions. Therefore we resort to a weak formulation where the variation of the jump is replaced by a Young measure. Eventually, after proving the existence in this weak formulation, we improve the result by showing that the Young measure is concentrated on a function and coincides with the variation of the jump of the displacement.

Combined parameter and model reduction of cardiovascular problems by means of active subspaces and POD-Galerkin methods - Marco Tezzele; F. Ballarin; Gianluigi Rozza (Mathematical and Numerical Modeling of the Cardiovascular System and Applications, 2018)

A Comparison Between Active Strain and Active Stress in Transversely Isotropic Hyperelastic Materials - Giulia Giantesio; Alessandro Musesti; Davide Riccobelli (J. Elast., 2018)

Computational methods in cardiovascular mechanics - Auricchio, Ferdinando; Conti, Michele; Lefieux, Adrian; Morganti, Simone; Alessandro Reali; Gianluigi Rozza; Veneziani, Alessandro (Cardiovascular Mechanics, 2018)
The introduction of computational models in cardiovascular sciences has been progressively bringing new and unique tools for the investigation of the physiopathology. Together with the dramatic improvement of imaging and measuring devices on one side, and of computational architectures on the other one, mathematical and numerical models have provided a new, clearly noninvasive, approach for understanding not only basic mechanisms but also patient-specific conditions, and for supporting the design and the development of new therapeutic options. The terminology in silico is, nowadays, commonly accepted for indicating this new source of knowledge added to traditional in vitro and in vivo investigations. The advantages of in silico methodologies are basically the low cost in terms of infrastructures and facilities, the reduced invasiveness and, in general, the intrinsic predictive capabilities based on the use of mathematical models. The disadvantages are generally identified in the distance between the real cases and their virtual counterpart required by the conceptual modeling that can be detrimental for the reliability of numerical simulations.

deal2lkit: A toolkit library for high performance programming in deal.II - Alberto Sartori; Nicola Giuliani; Mauro Bardelloni; Luca Heltai (SOFTWAREX, 2018)

The deal.II Library, Version 9.0 - Giovanni Alzetta; Daniel Arndt; Wolfgang Bangerth; Boddu, Vishal; Brands, Benjamin; Denis Davydov; Rene Gassmöller; Timo Heister; Luca Heltai; Kormann, Katharina; Martin Kronbichler; Matthias Maier; et al. (JOURNAL OF NUMERICAL MATHEMATICS, 2018)

Differential of metric valued Sobolev maps - Nicola Gigli; Enrico Pasqualetto; Elefterios Soultanis (2018)

Dimension reduction for thin films with transversally varying prestrain: the oscillatory and the non-oscillatory case - Marta Lewicka; Danka Lučić (2018)

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

Discriminant circle bundles over local models of Strebel graphs and Boutroux curves - Marco Bertola; Korotkin, D. A. (Teoret. Mat. Fiz., 2018)

A distributed lagrange formulation of the finite element immersed boundary method for fluids interacting with compressible solids - Boffi, Daniele; Gastaldi, Lucia; Luca Heltai (Mathematical and Numerical Modeling of the Cardiovascular System and Applications, 2018)

Effective non-linear spinor dynamics in a spin-1 Bose–Einstein condensate - Alessandro Michelangeli; Alessandro Olgiati (Journal of Physics A: Mathematical and Theoretical, 2018)
We derive from first principles the experimentally observed effective dynamics of a spinor Bose gas initially prepared as a Bose–Einstein condensate and then left free to expand ballistically. In spinor condensates, which represent one of the recent frontiers in the manipulation of ultra-cold atoms, particles interact with a two-body spatial interaction and a spin–spin interaction. The effective dynamics is well-known to be governed by a system of coupled semi-linear Schrödinger equations: we recover this system, in the sense of marginals in the limit of infinitely many particles, with a mean-field re-scaling of the many-body Hamiltonian. When the resulting control of the dynamical persistence of condensation is quantified with the parameters of modern observations, we obtain a bound that remains quite accurate for the whole typical duration of the experiment.

An efficient shape parametrisation by free-form deformation enhanced by active subspace for hull hydrodynamic ship design problems in open source environment - Nicola Demo; Marco Tezzele; Andrea Mola; Gianluigi Rozza (The 28th International Ocean and Polar Engineering Conference, 2018)
In this contribution, we present the results of the application of a parameter space reduction methodology based on active subspaces to the hull hydrodynamic design problem. Several parametric deformations of an initial hull shape are considered to assess the influence of the shape parameters considered on the hull total drag. The hull resistance is typically computed by means of numerical simulations of the hydrodynamic flow past the ship. Given the high number of parameters involved - which might result in a high number of time consuming hydrodynamic simulations - assessing whether the parameters space can be reduced would lead to considerable computational cost reduction. Thus, the main idea of this work is to employ the active subspaces to identify possible lower dimensional structures in the parameter space, or to verify the parameter distribution in the position of the control points. To this end, a fully automated procedure has been implemented to produce several small shape perturbations of an original hull CAD geometry which are then used to carry out high-fidelity flow simulations and collect data for the active subspaces analysis. To achieve full automation of the open source pipeline described, both the free form deformation methodology employed for the hull perturbations and the solver based on unsteady potential flow theory, with fully nonlinear free surface treatment, are directly interfaced with CAD data structures and operate using IGES vendor-neutral file formats as input files. The computational cost of the fluid dynamic simulations is further reduced through the application of dynamic mode decomposition to reconstruct the steady state total drag value given only few initial snapshots of the simulation. The active subspaces analysis is here applied to the geometry of the DTMB-5415 naval combatant hull, which is which is a common benchmark in ship hydrodynamics simulations.
Keywords: Active subspaces; Boundary element method; Dynamic mode decomposition; Fluid structure interaction; Free form deformation; Fully nonlinear potential; Numerical towing tank

Existence and uniqueness of dynamic evolutions for a one dimensional debonding model with damping - Lorenzo Nardini; Filippo Riva (2018)
In this paper we analyse a one-dimensional debonding model for a thin film peeled from a substrate when friction is taken into account. It is described by the weakly damped wave equation whose domain, the debonded region, grows according to a Griffth's criterion. Firstly we prove that the equation admits a unique solution when the evolution of the debonding front is assigned. Finally we provide an existence and uniqueness result for the coupled problem given by the wave equation together with Griffth's criterion.

Existence for elastodynamic Griffith fracture with a weak maximal dissipation condition - Gianni Dal Maso; Cristopher J. Larsen; Rodica Toader (2018)
We consider a model of elastodynamics with fracture evolution, based on energy-dissipation balance and a maximal dissipation condition. We prove an existence result in the case of planar elasticity with a free crack path, where the maximal dissipation condition is satisfied among suitably regular competitor cracks.

EZyRB: Easy Reduced Basis method - Nicola Demo; Marco Tezzele; Gianluigi Rozza (The Journal of Open Source Software, 2018)

Failure of the Chain Rule in the Non Steady Two-Dimensional Setting - Stefano Bianchini; Paolo Bonicatto (Current Research in Nonlinear Analysis: In Honor of Haim Brezis and Louis Nirenberg, 2018)

Finite volume POD-Galerkin stabilised reduced order methods for the parametrised incompressible Navier–Stokes equations - Giovanni Stabile; Gianluigi Rozza (Computers and Fluids, 2018)
In this work a stabilised and reduced Galerkin projection of the incompressible unsteady Navier–Stokes equations for moderate Reynolds number is presented. The full-order model, on which the Galerkin projection is applied, is based on a finite volumes approximation. The reduced basis spaces are constructed with a POD approach. Two different pressure stabilisation strategies are proposed and compared: the former one is based on the supremizer enrichment of the velocity space, and the latter one is based on a pressure Poisson equation approach.

Foldable structures made of hydrogel bilayers - Virginia Agostiniani; Antonio DeSimone; Alessandro Lucantonio; Danka Lučić (2018)

Fractional powers and singular perturbations of quantum differential Hamiltonians - Alessandro Michelangeli; Andrea Ottolini; Raffaele Scandone (Journal of Mathematical Physics, 2018)
We consider the fractional powers of singular (point-like) perturbations of the Laplacian and the singular perturbations of fractional powers of the Laplacian, and we compare two such constructions focusing on their perturbative structure for resolvents and on the local singularity structure of their domains. In application to the linear and non-linear Schrödinger equations for the corresponding operators, we outline a programme of relevant questions that deserve being investigated.

On fractional powers of singular perturbations of the Laplacian - Vladimir Georgiev; Alessandro Michelangeli; Raffaele Scandone (Journal of Functional Analysis, 2018)
We qualify a relevant range of fractional powers of the so-called Hamiltonian of point interaction in three dimensions, namely the singular perturbation of the negative Laplacian with a contact interaction supported at the origin. In particular we provide an explicit control of the domain of such a fractional operator and of its decomposition into regular and singular parts. We also qualify the norms of the resulting singular fractional Sobolev spaces and their mutual control with the corresponding classical Sobolev norms.
Keywords: Point interactions; Regular and singular component of a point-interaction operator; Singular perturbations of the Laplacian

Framed symplectic sheaves on surfaces - Jacopo Vittorio Scalise (International Journal of Mathematics, 2018)
A framed symplectic sheaf on a smooth projective surface $X$ is a torsion-free sheaf $E$ together with a trivialization on a divisor $D \subset X$ and a morphism $Λ^2 E → \mathcal{O}_X$ satisfying some additional conditions. We construct a moduli space for framed symplectic sheaves on a surface, and present a detailed study for $X =\mathbb{P}_\mathbb{C}^2$. In this case, the moduli space is irreducible and admits an ADHM-type description and a birational proper map onto the space of framed symplectic ideal instantons.

Free-form deformation, mesh morphing and reduced-order methods: enablers for efficient aerodynamic shape optimisation - Filippo Salmoiraghi; Scardigli, Angela; Telib, Haysam; Gianluigi Rozza (International Journal of Computational Fluid Dynamics, 2018)
In this work, we provide an integrated pipeline for the model-order reduction of turbulent flows around parametrised geometries in aerodynamics. In particular, free-form deformation is applied for geometry parametrisation, whereas two different reduced-order models based on proper orthogonal decomposition (POD) are employed in order to speed-up the full-order simulations: the first method exploits POD with interpolation, while the second one is based on domain decomposition. For the sampling of the parameter space, we adopt a Greedy strategy coupled with Constrained Centroidal Voronoi Tessellations, in order to guarantee a good compromise between space exploration and exploitation. The proposed framework is tested on an industrially relevant application, i.e. the front-bumper morphing of the DrivAer car model, using the finite-volume method for the full-order resolution of the Reynolds-Averaged Navier–Stokes equations.

On fully real eigenconfigurations of tensors - Khazhgali Kozhasov (SIAM Journal on Applied Algebra and Geometry, 2018)
We construct generic real symmetric tensors with only real eigenvectors or, equivalently, real homogeneous polynomials with the maximum possible finite number of critical points on the sphere.

On Geometric Quantum Confinement in Grushin-Like Manifolds - Matteo Gallone; Alessandro Michelangeli; Eugenio Pozzoli (2018)
We study the problem of so-called geometric quantum confinement in a class of two-dimensional incomplete Riemannian manifold with metric of Grushin type. We employ a constant-fibre direct integral scheme, in combination with Weyl's analysis in each fibre, thus fully characterising the regimes of presence and absence of essential self-adjointness of the associated Laplace-Beltrami operator.
16 pages

Global, finite energy, weak solutions for the NLS with rough, time-dependent magnetic potentials - Paolo Antonelli; Alessandro Michelangeli; Raffaele Scandone (Zeitschrift für angewandte Mathematik und Physik, 2018)
We prove the existence of weak solutions in the space of energy for a class of nonlinear Schrödinger equations in the presence of a external, rough, time-dependent magnetic potential. Under our assumptions, it is not possible to study the problem by means of usual arguments like resolvent techniques or Fourier integral operators, for example. We use a parabolic regularisation, and we solve the approximating Cauchy problem. This is achieved by obtaining suitable smoothing estimates for the dissipative evolution. The total mass and energy bounds allow to extend the solution globally in time. We then infer sufficient compactness properties in order to produce a global-in-time finite energy weak solution to our original problem.

Ground states and spectral properties in quantum field theories - Markus Lange (Friedrich-Schiller-University Jena, 2018)

Heterogeneous elastic plates with in-plane modulation of the target curvature and applications to thin gel sheets - Virginia Agostiniani; Alessandro Lucantonio; Danka Lučić (ESAIM: Control, Optimisation and Calculus of Variations, 2018)

Hydrogenoid Spectra with Central Perturbations - Matteo Gallone; Alessandro Michelangeli (2018)
Through the Kreĭn-Višik-Birman extension scheme, unlike the previous classical analysis based on von Neumann'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 Schördinger 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 Kreĭn-Višik-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.
Mathematics Subject Classification (2010) 34L10 . 34L15 . 34L16 . 47B15 . 47B25 . 47N20 . 81Q10 . 81Q80

On the isoperimetric problem with double density - Pratelli, A.; Saracco, G. (Nonlinear Anal., 2018)

Iterative map-making with two-level preconditioning for polarized cosmic microwave background data sets. A worked example for ground-based experiments - Puglisi, Giuseppe; Poletti, Davide; Fabbian, Giulio; Baccigalupi, Carlo; Luca Heltai; Stompor, Radek (ASTRONOMY & ASTROPHYSICS, 2018)

Local moduli of semisimple Frobenius coalescent structures - Giordano Cotti; Boris Dubrovin; Davide Guzzetti (SISSA, 2018)
There is a conjectural relation, formulated by the second author, between the enumerative geometry of a wide class of smooth projective varieties and their derived category of coherent sheaves. In particular, there is an increasing interest for an explicit description of certain local invariants, called monodromy data, of semisimple quantum cohomologies in terms of characteristic classes of exceptional collections in the derived categories. Being intentioned to address this problem, which, to our opinion, is still not well understood, we have realized that some issues in the theory of Frobenius manifolds need to be preliminarily clarified, and that an extension of the theory itself is necessary, in view of the fact that quantum cohomologies of certain classes of homogeneous spaces may show a coalescence phenomenon.

Long time existence for fully nonlinear NLS with small Cauchy data on the circle - Feola Roberto; Felice Iandoli (2018)

Lp-Boundedness of Wave Operators for the Three-Dimensional Multi-Centre Point Interaction - Gianfausto Dell'Antonio; Alessandro Michelangeli; Raffaele Scandone; Kenji Yajima (Annales Henri Poincaré, 2018)
We prove that, for arbitrary centres and strengths, the wave operators for three-dimensional Schrödinger operators with multi-centre local point interactions are bounded in Lp(R3)for 1<p<3 and unbounded otherwise.

A minimization approach to the wave equation on time-dependent domains - Gianni Dal Maso; Lucia De Luca (SISSA, 2018)
We prove the existence of weak solutions to the homogeneous wave equation on a suitable class of time-dependent domains. Using the approach suggested by De Giorgi and developed by Serra and Tilli, such solutions are approximated by minimizers of suitable functionals in space-time.

Minimizing movements for mean curvature flow of droplets with prescribed contact angle - Giovanni Bellettini; Matteo Novaga; Shokhrukh Kholmatov (Journal de Mathématiques Pures et Appliquées, 2018)
We study the mean curvature motion of a droplet flowing by mean curvature on a horizontal hyperplane with a possibly nonconstant prescribed contact angle. Using the solutions constructed as a limit of an approximation algorithm of Almgren–Taylor–Wang and Luckhaus–Sturzenhecker, we show the existence of a weak evolution, and its compatibility with a distributional solution. We also prove various comparison results. Résumé Nous étudions le mouvement par courbure moyenne d'une goutte qui glisse par courbure moyenne sur un hyperplan horizontal avec un angle de contact prescrit éventuellement non constant. En utilisant les solutions construites comme limites d'un algorithme d'approximation dû à Almgren, Taylor et Wang et Luckhaus et Sturzenhecker, nous montrons l'existence d'une évolution faible, et sa compatibilité avec une solution au sens des distributions. Nous démontrons également plusieurs résultats de comparaison.
Keywords: Capillary functional; Mean curvature flow with prescribed contact angle; Minimizing movements; Sets of finite perimeter

Minimizing Movements for Mean Curvature Flow of Partitions - Giovanni Bellettini; Shokhrukh Kholmatov (SIAM Journal on Mathematical Analysis, 2018)
We prove the existence of a weak global in time mean curvature flow of a bounded partition of space using the method of minimizing movements. The result is extended to the case when suitable driving forces are present. We also prove some consistency results for a minimizing movement solution with smooth and viscosity solutions when the evolution starts from a partition made by a union of bounded sets at a positive distance. In addition, the motion starting from the union of convex sets at a positive distance agrees with the classical mean curvature flow and is stable with respect to the Hausdorff convergence of the initial partitions.

Model Order Reduction by means of Active Subspaces and Dynamic Mode Decomposition for Parametric Hull Shape Design Hydrodynamics - Marco Tezzele; Nicola Demo; Mahmoud Gadalla; Andrea Mola; Gianluigi Rozza (Technology and Science for the Ships of the Future: Proceedings of NAV 2018: 19th International Conference on Ship & Maritime Research, 2018)
We present the results of the application of a parameter space reduction methodology based on active subspaces (AS) to the hull hydrodynamic design problem. Several parametric deformations of an initial hull shape are considered to assess the influence of the shape parameters on the hull wave resistance. Such problem is relevant at the preliminary stages of the ship design, when several flow simulations are carried out by the engineers to establish a certain sensibility with respect to the parameters, which might result in a high number of time consuming hydrodynamic simulations. The main idea of this work is to employ the AS to identify possible lower dimensional structures in the parameter space. The complete pipeline involves the use of free form deformation to parametrize and deform the hull shape, the full order solver based on unsteady potential flow theory with fully nonlinear free surface treatment directly interfaced with CAD, the use of dynamic mode decomposition to reconstruct the final steady state given only few snapshots of the simulation, and the reduction of the parameter space by AS, and shared subspace. Response surface method is used to minimize the total drag.

Model Reduction for Parametrized Optimal Control Problems in Environmental Marine Sciences and Engineering - Maria Strazzullo; F. Ballarin; Mosetti, R.; Gianluigi Rozza (SIAM Journal on Scientific Computing, 2018)

Morpho-elastic model of the tortuous tumour vessels - Davide Riccobelli; Pasquale Ciarletta (Int. J. Non-Linear Mech., 2018)

Noncommutative Painlevé Equations and Systems of Calogero Type - Marco Bertola; Mattia Cafasso; V. Rubtsov (Comm. Math. Phys, 2018)

Non-linear Gross-Pitaevskii dynamics of a 2D binary condensate: a numerical analysis - Alessandro Michelangeli; Giuseppe Pitton (2018)
We present a numerical study of the two-dimensional Gross-Pitaevskii systems in a wide range of relevant regimes of population ratios and intra-species and inter-species interactions. Our numerical method is based on a Fourier collocation scheme in space combined with a fourth order integrating factor scheme in time.

On the notion of parallel transport on RCD spaces - Nicola Gigli; Enrico Pasqualetto (2018)

A novel reduced order model for vortex induced vibrations of long flexible cylinders - Giovanni Stabile; Hermann G. Matthies; Claudio Borri (Elsevier BV, 2018)

Numerical study of the Kadomtsev-Petviashvili equation and dispersive shock waves - Tamara Grava; Christian Klein; Giuseppe Pitton (Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2018)
A detailed numerical study of the long time behaviour of dispersive shock waves in solutions to the Kadomtsev–Petviashvili (KP) I equation is presented. It is shown that modulated lump solutions emerge from the dispersive shock waves. For the description of dispersive shock waves, Whitham modulation equations for KP are obtained. It is shown that the modulation equations near the soliton line are hyperbolic for the KPII equation while they are elliptic for the KPI equation leading to a focusing effect and the formation of lumps. Such a behaviour is similar to the appearance of breathers for the focusing nonlinear Schrödinger equation in the semiclassical limit.

NURBS-SEM: A hybrid spectral element method on NURBS maps for the solution of elliptic PDEs on surfaces - Giuseppe Pitton; Luca Heltai (COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING, 2018)

Observables in the equivariant A-model - Bonechi, F.; Cattaneo, A.S.; Riccardo Iraso; Maxim Zabzine (2018)

Painlevé IV Critical Asymptotics for Orthogonal Polynomials in the Complex Plane - Marco Bertola; José Gustavo Elias Rebelo; Tamara Grava (Symmetry, Integrability and Geometry. Methods and Applications, 2018)
We study the asymptotic behaviour of orthogonal polynomials in the complex plane that are associated to a certain normal matrix model. The model depends on a parameter and the asymptotic distribution of the eigenvalues undergoes a transition for a special value of the parameter, where it develops a corner-type singularity. In the double scaling limit near the transition we determine the asymptotic behaviour of the orthogonal polynomials in terms of a solution of the Painlev´e IV equation. We determine the Fredholm determinant associated to such solution and we compute it numerically on the real line, showing also that the corresponding Painlev´e transcendent is pole-free on a semiaxis.

Peristaltic Waves as Optimal Gaits in Metameric Bio-Inspired Robots - Daniele Agostinelli; François Alouges; Antonio DeSimone (Frontiers in Robotics and AI, 2018)
Peristalsis, i.e., a motion pattern arising from the propagation of muscle contraction and expansion waves along the body, is a common locomotion strategy for limbless animals. Mimicking peristalsis in bio-inspired robots has attracted considerable attention in the literature. It has recently been observed that maximal velocity in a metameric earthworm-like robot is achieved by actuating the segments using a “phase coordination” principle. This paper shows that, in fact, peristalsis (which requires not only phase coordination, but also that all segments oscillate at same frequency and amplitude) emerges from optimization principles. More precisely, basing our analysis on the assumption of small deformations, we show that peristaltic waves provide the optimal actuation solution in the ideal case of a periodic infinite system, and that this is approximately true, modulo edge effects, for the real, finite length system. Therefore, this paper confirms the effectiveness of mimicking peristalsis in bio-inspired robots, at least in the small-deformation regime. Further research will be required to test the effectiveness of this strategy if large deformations are allowed.
Keywords: Biomimetic robots; Crawling motility; Lumbricus terrestris; Metameric robots; Optimization; Peristalsis; Self-propulsion; Soft robotics

Positive solutions for super-sublinear indefinite problems: high multiplicity results via coincidence degree - Alberto Boscaggin; Guglielmo Feltrin; Fabio Zanolin (Trans. Amer. Math. Soc., 2018)
We study the periodic boundary value problem associated with the second order nonlinear equation u''+(λa+(t)−μa−(t))g(u)=0, where g(u) has superlinear growth at zero and sublinear growth at infinity. For λ,μ positive and large, we prove the existence of 3^m−1 positive T-periodic solutions when the weight function a(t) has m positive humps separated by m negative ones (in a T-periodicity interval). As a byproduct of our approach we also provide abundance of positive subharmonic solutions and symbolic dynamics. The proof is based on coincidence degree theory for locally compact operators on open unbounded sets and also applies to Neumann and Dirichlet boundary conditions. Finally, we deal with radially symmetric positive solutions for the Neumann and the Dirichlet problems associated with elliptic PDEs.
AMS Subject Classification: 34B15, 34B18, 34C25, 34C28, 47H11.

Positive subharmonic solutions to nonlinear ODEs with indefinite weight - Alberto Boscaggin; Guglielmo Feltrin (Communications in Contemporary Mathematics, 2018)
We prove that the superlinear indefinite equation u″ + a(t)up = 0, where p > 1 and a(t) is a T-periodic sign-changing function satisfying the (sharp) mean value condition ∫0Ta(t)dt < 0, has positive subharmonic solutions of order k for any large integer k, thus providing a further contribution to a problem raised by Butler in its pioneering paper [Rapid oscillation, nonextendability, and the existence of periodic solutions to second order nonlinear ordinary differential equations, J. Differential Equations 22 (1976) 467–477]. The proof, which applies to a larger class of indefinite equations, combines coincidence degree theory (yielding a positive harmonic solution) with the Poincaré–Birkhoff fixed point theorem (giving subharmonic solutions oscillating around it).

Predicting and Optimizing Microswimmer Performance from the Hydrodynamics of Its Components: The Relevance of Interactions - Nicola Giuliani; Luca Heltai; Antonio DeSimone (SOFT ROBOTICS, 2018)

The prescribed mean curvature equation in weakly regular domains - Leonardi, G. P.; Saracco, G. (NoDEA Nonlinear Differ. Equ. Appl., 2018)

Principal fibrations over noncommutative spheres - Michel Dubois-Violette; Xiao Han; Giovanni Landi (Reviews in Mathematical Physics, 2018)
We present examples of noncommutative four-spheres that are base spaces of $SU(2)$-principal bundles with noncommutative seven-spheres as total spaces. The noncommutative coordinate algebras of the four-spheres are generated by the entries of a projection which is invariant under the action of $SU(2)$. We give conditions for the components of the Connes–Chern character of the projection to vanish but the second (the top) one. The latter is then a non-zero Hochschild cycle that plays the role of the volume form for the noncommutative four-spheres.

PyDMD: Python Dynamic Mode Decomposition - Nicola Demo; Marco Tezzele; Gianluigi Rozza (The Journal of Open Source Software, 2018)

On the Quasistatic Limit of Dynamic Evolutions for a Peeling Test in Dimension One - Giuliano Lazzaroni; Lorenzo Nardini (Journal of Nonlinear Science, 2018)
The aim of this paper is to study the quasistatic limit of a one-dimensional model of dynamic debonding. We start from a dynamic problem that strongly couples the wave equation in a time-dependent domain with Griffith's criterion for the evolution of the domain. Passing to the limit as inertia tends to zero, we find that the limit evolution satisfies a stability condition; however, the activation rule in Griffith's (quasistatic) criterion does not hold in general, thus the limit evolution is not rate-independent.

On real resonances for the three-dimensional, multi-centre point interaction - Alessandro Michelangeli; Raffaele Scandone (2018)

Recognizing the flat torus among RCD*(0,N) spaces via the study of the first cohomology group - Nicola Gigli; Chiara Rigoni (Calculus of Variations and Partial Differential Equations, 2018)
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.

Reduced Basis Approximation and A Posteriori Error Estimation: Applications to Elasticity Problems in Several Parametric Settings - Huynh, D. B. P.; Federico Pichi; Gianluigi Rozza (Numerical Methods for PDEs, 2018)

Reduced Basis Approximation and A Posteriori Error Estimation: Applications to Elasticity Problems in Several Parametric Settings - D.B.P. Huynh; Federico Pichi; Gianluigi Rozza (SEMA SIMAI Springer Series, 2018)
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 “parametric manifold”—dimension reduction; an efficient and effective greedy sampling methods for identification of optimal and numerically stable approximations—rapid convergence; an a posteriori error estimation procedures—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—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—to model the materials and loads—and geometrical parameters—to model different geometrical configurations—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.

Reducibility for a class of weakly dispersive linear operators arising from the Degasperis Procesi equation - Roberto Feola; Filippo Giuliani; Michela Procesi (2018)

Regularity estimates for scalar conservation laws in one space dimension - Elio Marconi (Journal of Hyperbolic Differential Equations, 2018)
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′∘ 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.

Renormalization analysis for degenerate ground states - Hasler, David; Markus Lange (J. Funct. Anal., 2018)

Revealing new dynamical patterns in a reaction–diffusion model with cyclic competition via a novel computational framework - Andrea Cangiani; E.H. Georgoulis; Morozov, A. Yu.; Sutton, O. J. (Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2018)
Understanding how patterns and travelling waves form in chemical and biological reaction–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–diffusion models, such as Lotka–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–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.

Second order differentiation formula on RCD(K, N) spaces - Nicola Gigli; Luca Tamanini (Rendiconti Lincei-Matematica e Applicazioni, 2018)

Second order differentiation formula on RCD*(K,N) spaces - Nicola Gigli; Luca Tamanini (2018)

Shape Optimization by means of Proper Orthogonal Decomposition and Dynamic Mode Decomposition - Nicola Demo; Marco Tezzele; Gianluca Gustin; Gianpiero Lavini; Gianluigi Rozza (Technology and Science for the Ships of the Future: Proceedings of NAV 2018: 19th International Conference on Ship & Maritime Research, 2018)
Shape optimization is a challenging task in many engineering fields, since the numerical solutions of parametric system may be computationally expensive. This work presents a novel optimization procedure based on reduced order modeling, applied to a naval hull design problem. The advantage introduced by this method is that the solution for a specific parameter can be expressed as the combination of few numerical solutions computed at properly chosen parametric points. The reduced model is built using the proper orthogonal decomposition with interpolation (PODI) method. We use the free form deformation (FFD) for an automated perturbation of the shape, and the finite volume method to simulate the multiphase incompressible flow around the deformed hulls. Further computational reduction is done by the dynamic mode decomposition (DMD) technique: from few high dimensional snapshots, the system evolution is reconstructed and the final state of the simulation is faithfully approximated. Finally the global optimization algorithm iterates over the reduced space: the approximated drag and lift coefficients are projected to the hull surface, hence the resistance is evaluated for the new hulls until the convergence to the optimal shape is achieved. We will present the results obtained applying the described procedure to a typical Fincantieri cruise ship.

Shape transitions in a soft incompressible sphere with residual stresses - Davide Riccobelli; Pasquale Ciarletta (Math. Mech. Solids, 2018)

On sinc quadrature approximations of fractional powers of regularly accretive operators - Bonito, Andrea; Wenyu Lei; Joseph E Pasciak (Journal of Numerical Mathematics, 2018)

Singular Hartree equation in fractional perturbed Sobolev spaces - Alessandro Michelangeli; Alessandro Olgiati; Raffaele Scandone (Journal of Nonlinear Mathematical Physics, 2018)
We establish the local and global theory for the Cauchy problem of the singular Hartree equation in three dimensions, that is, the modification of the non-linear Schrödinger equation with Hartree non-linearity, where the linear part is now given by the Hamiltonian of point interaction. The latter is a singular, self-adjoint perturbation of the free Laplacian, modelling a contact interaction at a fixed point. The resulting non-linear equation is the typical effective equation for the dynamics of condensed Bose gases with fixed point-like impurities. We control the local solution theory in the perturbed Sobolev spaces of fractional order between the mass space and the operator domain. We then control the global solution theory both in the mass and in the energy space.

On some rigorous aspects of fragmented condensation - Daniele Dimonte; Marco Falconi; Alessandro Olgiati (2018)

Spectral triples on the Jiang-Su algebra - Jacopo Bassi; Ludwik Dabrowski (Journal of Mathematical Physics, 2018)

Spontaneous morphing of equibiaxially pre-stretched elastic bilayers: The role of sample geometry - Noe Caruso; Aleksandar Cvetković; Alessandro Lucantonio; Giovanni Noselli; Antonio DeSimone (International Journal of Mechanical Sciences, 2018)
An elastic bilayer, consisting of an equibiaxially pre-stretched sheet bonded to a stress-free one, spontaneously morphs into curved shapes in the absence of external loads or constraints. Using experiments and numerical simulations, we explore the role of geometry for square and rectangular samples in determining the equilibrium shape of the system, for a fixed pre-stretch. We classify the observed shapes over a wide range of aspect ratios according to their curvatures and compare measured and computed values, which show good agreement. In particular, as the bilayer becomes thinner, a bifurcation of the principal curvatures occurs, which separates two scaling regimes for the energy of the system. We characterize the transition between these two regimes and show the peculiar features that distinguish square from rectangular samples. The results for our model bilayer system may help explaining morphing in more complex systems made of active materials.
Keywords: Bifurcation; Elastic bilayer; Pre-stretch; Shape programming

SRTP 2.0 - The evolution of the safe return to port concept - D. Cangelosi; A. Bonvicini; M. Nardo; Andrea Mola; A. Marchese; Marco Tezzele; Gianluigi Rozza (Technology and Science for the Ships of the Future - Proceedings of NAV 2018: 19th International Conference on Ship and Maritime Research, 2018)
In 2010 IMO (International Maritime Organisation) introduced new rules in SOLAS with the aim of intrinsically increase the safety of passenger ships. This requirement is achieved by providing safe areas for passengers and essential services for allowing ship to Safely Return to Port (SRtP). The entry into force of these rules has changed the way to design passenger ships. In this respect big effort in the research has been done by industry to address design issues related to the impact on failure analysis of the complex interactions among systems. Today the research activity is working to bring operational matters in the design stage. This change of research focus was necessary because human factor and the way to operate the ship itself after a casualty on board may have a big impact in the design of the ship/systems. Also the management of the passengers after a casualty is becoming a major topic for safety. This paper presents the state of the art of Italian knowledge in the field of system engineering applied to passenger ship address to safety improvement and design reliability. An overview of present tools and methodologies will be offered together with future focuses in the research activity.

Stabilized weighted reduced basis methods for parametrized advection dominated problems with random inputs - D. Torlo; F. Ballarin; Gianluigi Rozza (SIAM-ASA Journal on Uncertainty Quantification, 2018)
In this work, we propose viable and eficient strategies for stabilized parametrized advection dominated problems, with random inputs. In particular, we investigate the combination of the wRB (weighted reduced basis) method for stochastic parametrized problems with the stabilized RB (reduced basis) method, which is the integration of classical stabilization methods (streamline/upwind Petrov-Galerkin (SUPG) in our case) in the ofine-online structure of the RB method. Moreover, we introduce a reduction method that selectively enables online stabilization; this leads to a sensible reduction of computational costs, while keeping a very good accuracy with respect to high-fdelity solutions. We present numerical test cases to assess the performance of the proposed methods in steady and unsteady problems related to heat transfer phenomena.

Stochastic homogenisation of free-discontinuity problems - Filippo Cagnetti; Gianni Dal Maso; Lucia Scardia; Caterina Ida Zeppieri (2018)
In this paper we study the stochastic homogenisation of free-discontinuity functionals. Assuming stationarity for the random volume and surface integrands, we prove the existence of a homogenised random free-discontinuity functional, which is deterministic in the ergodic case. Moreover, by establishing a connection between the deterministic convergence of the functionals at any fixed realisation and the pointwise Subadditive Ergodic Theorem by Akcoglou and Krengel, we characterise the limit volume and surface integrands in terms of asymptotic cell formulas.

Symplectic invariants for parabolic orbits and cusp singularities of integrable systems - Alexey Bolsinov; Lorenzo Guglielmi; Elena Kudryavtseva (Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2018)
We discuss normal forms and symplectic invariants of parabolic orbits and cuspidal tori in integrable Hamiltonian systems with two degrees of freedom. Such singularities appear in many integrable systems in geometry and mathematical physics and can be considered as the simplest example of degenerate singularities. We also suggest some new techniques which apparently can be used for studying symplectic invariants of degenerate singularities of more general type. This article is part of the theme issue ‘Finite dimensional integrable systems: new trends and methods’.

Transmission conditions obtained by homogenisation - Gianni Dal Maso; Giovanni Franzina; Davide Zucco (2018)
We study the asymptotic behaviour of solutions to variational problems in perforated domains with Neumann boundary conditions. We consider perforations that in the limit concentrate on a smooth manifold. We characterise the limits of the solutions and show that they solve a variational problem with a transmission condition across the manifold. This is expressed through a measure on the manifold, vanishing on sets of capacity zero. Then, we prove that every such measure can be obtained by homogenising suitable perforations. Eventually, we provide an asymptotic formula for this measure by using some auxiliary minimum problems.

Truncation and convergence issues for bounded linear inverse problems in Hilbert space - Noe Caruso; Alessandro Michelangeli; Paolo Novati (SISSA, 2018)
We present a general discussion of the main features and issues that (bounded) inverse linear problems in Hilbert space exhibit when the dimension of the space is infinite. This includes the set-up of a consistent notation for inverse problems that are genuinely infinite-dimensional, the analysis of the finite-dimensional truncations, a discussion of the mechanisms why the error or the residual generically fail to vanish in norm, and the identification of practically plausible sufficient conditions for such indicators to be small in some weaker sense. The presentation is based on theoretical results together with a series of model examples and numerical tests.

Two examples of minimal Cheeger sets in the plane - Leonardi, G. P.; Saracco, G. (Ann. Mat. Pura Appl. (4), 2018)

On Uniqueness of Weak Solutions to Transport Equation with Non-smooth Velocity Field - Paolo Bonicatto (Theory, Numerics and Applications of Hyperbolic Problems I, 2018)

Virtual element methods for elliptic problems on polygonal meshes - Andrea Cangiani; Sutton, Oliver J.; Gyrya, Vitaliy; Manzini, Gianmarco (Generalized barycentric coordinates in computer graphics and computational mechanics, 2018)

Weighted Cheeger sets are domains of isoperimetry - Saracco, G. (Manuscripta Math., 2018)

Wilson loop and its correlators in the limit of large coupling constant - Sysoeva, Ekaterina (Nucl. Phys. B, 2018)

Wilson loops and its correlators with chiral operators in $\mathcalN=2, 4$ SCFT at large $N$ - Sysoeva, Ekaterina (JHEP, 2018)

π-BEM : A flexible parallel implementation for adaptive , geometry aware , and high order boundary element methods - Nicola Giuliani; Andrea Mola; Luca Heltai (Advances in Engineering Software, 2018)

On the 1D wave equation in time-dependent domains and the problem of debond initiation - Giuliano Lazzaroni; Lorenzo Nardini (SISSA, 2017)
Motivated by a debonding model for a thin film peeled from a substrate, we analyse the one-dimensional wave equation, in a time-dependent domain which is degenerate at the initial time. In the first part of the paper we prove existence for the wave equation when the evolution of the domain is given; in the second part of the paper, the evolution of the domain is unknown and is governed by an energy criterion coupled with the wave equation. Our existence result for such coupled problem is a contribution to the study of crack initiation in dynamic fracture.

Almost global existence of solutions for capillarity-gravity water waves equations with periodic spatial boundary conditions - Massimiliano Berti; Jean-Marc Delort (2017)
The goal of this monograph is to prove that any solution of the Cauchy problem for the capillarity-gravity water waves equations, in one space dimension, with periodic, even in space, initial data of small size ϵ, is almost globally defined in time on Sobolev spaces, i.e. it exists on a time interval of length of magnitude ϵ−N for any N, as soon as the initial data are smooth enough, and the gravity-capillarity parameters are taken outside an exceptional subset of zero measure. In contrast to the many results known for these equations on the real line, with decaying Cauchy data, one cannot make use of dispersive properties of the linear flow. Instead, our method is based on a normal forms procedure, in order to eliminate those contributions to the Sobolev energy that are of lower degree of homogeneity in the solution. Since the water waves equations are a quasi-linear system, usual normal forms approaches would face the well known problem of losses of derivatives in the unbounded transformations. In this monograph, to overcome such a difficulty, after a paralinearization of the capillarity-gravity water waves equations, necessary to obtain energy estimates, and thus local existence of the solutions, we first perform several paradifferential reductions of the equations to obtain a diagonal system with constant coefficients symbols, up to smoothing remainders. Then we may start with a normal form procedure where the small divisors are compensated by the previous paradifferential regularization.The reversible structure of the water waves equations, and the fact that we look for solutions even in x, guarantees a key cancellation which prevents the growth of the Sobolev norms of the solutions.

Analytic geometry of semisimple coalescent Frobenius structures - Giordano Cotti; Davide Guzzetti (Random Matrices: Theory and Applications, 2017)
We present some results of a joint paper with Dubrovin (see references), as exposed at the Workshop “Asymptotic and Computational Aspects of Complex Differential Equations” at the CRM in Pisa, in February 2017. The analytical description of semisimple Frobenius manifolds is extended at semisimple coalescence points, namely points with some coalescing canonical coordinates although the corresponding Frobenius algebra is semisimple. After summarizing and revisiting the theory of the monodromy local invariants of semisimple Frobenius manifolds, as introduced by Dubrovin, it is shown how the definition of monodromy data can be extended also at semisimple coalescence points. Furthermore, a local Isomonodromy theorem at semisimple coalescence points is presented. Some examples of computation are taken from the quantum cohomologies of complex Grassmannians.

An application of coincidence degree theory to cyclic feedback type systems associated with nonlinear differential operators - Guglielmo Feltrin; Fabio Zanolin (Topol. Methods Nonlinear Anal., 2017)

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

The approximation of parabolic equations involving fractional powers of elliptic operators - Bonito, Andrea; Wenyu Lei; Joseph E Pasciak (J. Comput. Appl. Math., 2017)

An avoiding cones condition for the Poincaré–Birkhoff Theorem - Alessandro Fonda; Paolo Gidoni (Journal of Differential Equations, 2017)
We provide a geometric assumption which unifies and generalizes the conditions proposed in [11], [12], so to obtain a higher dimensional version of the Poincaré–Birkhoff fixed point Theorem for Poincaré maps of Hamiltonian systems.
Keywords: Avoiding cones condition; Hamiltonian systems; Periodic solutions; Poincaré–Birkhoff theorem

Certi fied Reduced Basis Method for Affinely Parametric Isogeometric Analysis NURBS Approximation - Denis Devaud; Gianluigi Rozza (Spectral and High Order Methods for Partial Differential Equations, 2017)
In this work we apply reduced basis methods for parametric PDEs to an isogeometric formulation based on NURBS. The motivation for this work is an integrated and complete work pipeline from CAD to parametrization of domain geometry, then from full order to certified reduced basis solution. IsoGeometric Analysis  (IGA) is a growing research theme in scientic computing and computational mechanics, as well as reduced basis methods for parametric PDEs. Their combination enhances the solution of some class of problems, especially the ones characterized by parametrized geometries we introduced in this work. For a general overview on Reduced Basis (RB) methods we recall [7, 15] and on IGA [3]. This work wants to demonstrate that it is also possible for some class of problems to deal with ane geometrical parametrization combined with a NURBS IGA formulation. This is what this work brings as original ingredients with respect to other works dealing with reduced order methods and IGA (set in a non-affine formulation, and using a POD [2]  sampling without certication: see for example for potential flows [12] and for Stokes flows [17]). In this work we show a certication of accuracy and a complete integration between IGA formulation and parametric certified greedy RB formulation. Section 2 recalls the abstract setting for parametrized PDEs, Section 3 recalls IGA setting, Section 4 deals with RB formulation, and Section 5 illustrates two numerical examples in heat transfer with different parametrization.

On a certified smagorinsky reduced basis turbulence model - Rebollo, T.C.; E.D. Ávila; Marmol, M.G.; F. Ballarin; Gianluigi Rozza (SIAM Journal on Numerical Analysis, 2017)

The Cheeger constant of a Jordan domain without necks - Leonardi, G. P.; Neumayer, R.; Saracco, G. (Calc. Var. Partial Differential Equations, 2017)

Clifford Tori and the singularly perturbed Cahn–Hilliard equation - Matteo Rizzi (Journal of Differential Equations, 2017)
In this paper we construct entire solutions uε to the Cahn–Hilliard equation −ε2Δ(−ε2Δu+W′(u))+W″(u)(−ε2Δu+W′(u))=ε4λε(1−uε), under the volume constraint ∫R3(1−uε)2dx=82π2cε, with cε→1 as ε→0, whose nodal set approaches the Clifford Torus, that is the Torus with radii of ratio 1/2 embedded in R3, as ε→0. It is crucial that the Clifford Torus is a Willmore hypersurface and it is non-degenerate, up to conformal transformations. The proof is based on the Lyapunov–Schmidt reduction and on careful geometric expansions of the Laplacian.
Keywords: Cahn–Hilliard equation; Clifford Torus; Lyapunov–Schmidt reduction; Willmore surface

Complex Friedrichs systems and applications - Nenad Antonić; Krešimir Burazin; Ivana Crnjac; Marko Erceg (2017)
We provide a suitable extension of the theory of abstract Friedrichs systems from real Hilbert spaces to the complex Hilbert space setting, which allows for applications to partial differential equations with complex coeffcients. We also provide examples where the involved Hilbert space is not the space of square integrable functions, as it was the case in previous works, but rather its closed subspace or the space Hs(Rd;Cr), for real s. This setting appears to be suitable for particular systems of partial differential equations, such as the Dirac system, the Dirac-Klein-Gordon system, the Dirac-Maxwell system, and the time-harmonic Maxwell system, which are all addressed in the paper. Moreover, for the time-harmonic Maxwell system we also applied a suitable version of the two-field theory with partial coercivity assumption which is developed in the paper.

Computational reduction strategies for the detection of steady bifurcations in incompressible fluid-dynamics: Applications to Coanda effect in cardiology - Giuseppe Pitton; Annalisa Quaini; Gianluigi Rozza (Journal of Computational Physics, 2017)
<div class="abstract svAbstract " style="border: 0px; font-size: 16px; margin: 0px; padding: 0px; vertical-align: baseline; font-family: Arial, Helvetica, 'Lucida Sans Unicode', 'Microsoft Sans Serif', 'Segoe UI Symbol', STIXGeneral, 'Cambria Math', 'Arial Unicode MS', sans-serif; color: #2e2e2e;" data-etype="ab"> We focus on reducing the computational costs associated with the hydrodynamic stability of solutions of the incompressible Navier–Stokes equations for a Newtonian and viscous fluid in contraction–expansion channels. In particular, we are interested in studying steady bifurcations, occurring when non-unique stable solutions appear as physical and/or geometric control parameters are varied. The formulation of the stability problem requires solving an eigenvalue problem for a partial differential operator. An alternative to this approach is the direct simulation of the flow to characterize the asymptotic behavior of the solution. Both approaches can be extremely expensive in terms of computational time. We propose to apply Reduced Order Modeling (ROM) techniques to reduce the demanding computational costs associated with the detection of a type of steady bifurcations in fluid dynamics. The application that motivated the present study is the onset of asymmetries (i.e., symmetry breaking bifurcation) in blood flow through a regurgitant mitral valve, depending on the Reynolds number and the regurgitant mitral valve orifice shape.
Keywords: Parametrized Navier–Stokes equations; Reduced basis method; Stability of flows; Symmetry breaking bifurcation

Computer simulations of phase field drops on super-hydrophobic surfaces - Livio Fedeli (Journal of Computational Physics, 2017)
We present a novel quasi-Newton continuation procedure that efficiently solves the system of nonlinear equations arising from the discretization of a phase field model for wetting phenomena. We perform a comparative numerical analysis that shows the improved speed of convergence gained with respect to other numerical schemes. Moreover, we discuss the conditions that, on a theoretical level, guarantee the convergence of this method. At each iterative step, a suitable continuation procedure develops and passes to the nonlinear solver an accurate initial guess. Discretization performs through cell-centered finite differences. The resulting system of equations is solved on a composite grid that uses dynamic mesh refinement and multi-grid techniques. The final code achieves three-dimensional, realistic computer experiments comparable to those produced in laboratory settings. This code offers not only new insights into the phenomenology of super-hydrophobicity, but also serves as a reliable predictive tool for the study of hydrophobic surfaces.
Keywords: Multigrid; Phase field; Quasi-Newton; Super-hydrophobicity

Conforming and nonconforming virtual element methods for elliptic problems - Andrea Cangiani; Manzini, Gianmarco; Sutton, Oliver J. (IMA J. Numer. Anal., 2017)

On contact interactions realised as Friedrichs systems - Marko Erceg; Alessandro Michelangeli (2017)
We realise the Hamiltonians of contact interactions in quantum mechanics within the framework of abstract Friedrichs systems. In particular, we show that the construction of the self-adjoint (or even only closed) operators of contact interaction supported at a fixed point can be associated with the construction of the bijective realisations of a suitable pair of abstract Friedrich operators. In this respect, the Hamiltonians of contact interaction provide novel examples of abstract Friedrich systems.

Coupling effects on the dynamic response of moored floating platforms for offshore wind energy plants - Giusti, Alessandro; Giovanni Stabile; Marino, Enzo; Claudio Borri (Procedia engineering, 2017)

Curvature terms in small time heat kernel expansion for a model class of hypoelliptic Hörmander operators - Davide Barilari; Elisa Paoli (Nonlinear Analysis, 2017)
We consider the heat equation associated with a class of second order hypoelliptic Hörmander operators with constant second order term and linear drift. We completely describe the small time heat kernel expansions on the diagonal giving a geometric characterization of the coefficients in terms of the divergence of the drift field and the curvature-like invariants of the optimal control problem associated with the diffusion operator.
Keywords: Curvature; Hypoelliptic heat equation; Small time asymptotics

Curvature-adapted remeshing of CAD surfaces - Franco Dassi; Andrea Mola; Hang Si (Engineering with Computers, 2017)

The deal.II Library, Version 8.5 - Daniel Arndt; Wolfgang Bangerth; Denis Davydov; Timo Heister; Luca Heltai; Martin Kronbichler; Matthias Maier; Jean-Paul Pelteret; Bruno Turcksin; David Wells (JOURNAL OF NUMERICAL MATHEMATICS, 2017)

Derivation of a rod theory from lattice systems with interactions beyond nearest neighbours - Roberto Alicandro; Giuliano Lazzaroni; Mariapia Palombaro (2017)
We study continuum limits of discrete models for (possibly heterogeneous) nanowires. The lattice energy includes at least nearest and next-to-nearest neighbour interactions: the latter have the role of penalising changes of orientation. In the heterogeneous case, we obtain an estimate on the minimal energy spent to match different equilibria. This gives insight into the nucleation of dislocations in epitaxially grown heterostructured nanowires.

Discrete spectra for critical Dirac-Coulomb Hamiltonians - Matteo Gallone; Alessandro Michelangeli (2017)
The one-particle Dirac Hamiltonian with Coulomb interaction is known to be realised, in a regime of large (critical) couplings, by an infinite multiplicity of distinct self-adjoint operators, including a distinguished physically most natural one. For the latter, Sommerfeld’s celebrated fine structure formula provides the well-known expression for the eigenvalues in the gap of the continuum spectrum. Exploiting our recent general classification of all other self-adjoint realisations, we generalise Sommerfeld’s formula so as to determine the discrete spectrum of all other self-adjoint versions of the Dirac-Coulomb Hamiltonian. Such discrete spectra display naturally a fibred structure, whose bundle covers the whole gap of the continuum spectrum.

Dispersive Estimates for Schrödinger Operators with Point Interactions in ℝ3 - Felice Iandoli; Raffaele Scandone (Advances in Quantum Mechanics: Contemporary Trends and Open Problems, 2017)
The study of dispersive properties of Schrödinger operators with point interactions is a fundamental tool for understanding the behavior of many body quantum systems interacting with very short range potential, whose dynamics can be approximated by non linear Schrödinger equations with singular interactions. In this work we proved that, in the case of one point interaction in $\mathbb{R}^3$, the perturbed Laplacian satisfies the same $L^p$−$L^q$ estimates of the free Laplacian in the smaller regime $q \in [2,3)$. These estimates are implied by a recent result concerning the Lpboundedness of the wave operators for the perturbed Laplacian. Our approach, however, is more direct and relatively simple, and could potentially be useful to prove optimal weighted estimates also in the regime $q ≥ 3$.

On the effect of interactions beyond nearest neighbours on non-convex lattice systems - Roberto Alicandro; Giuliano Lazzaroni; Mariapia Palombaro (2017)
We analyse the rigidity of non-convex discrete energies where at least nearest and next-to-nearest neighbour interactions are taken into account. Our purpose is to show that interactions beyond nearest neighbours have the role of penalising changes of orientation and, to some extent, they may replace the positive-determinant constraint that is usually required when only nearest neighbours are accounted for. In a discrete to continuum setting, we prove a compactness result for a family of surface-scaled energies and we give bounds on its possible Gamma-limit in terms of interfacial energies that penalise changes of orientation.

Effective Non-linear Dynamics of Binary Condensates and Open Problems - Alessandro Olgiati (Advances in Quantum Mechanics: Contemporary Trends and Open Problems, 2017)
We report on a recent result concerning the effective dynamics for a mixture of Bose-Einstein condensates, a class of systems much studied in physics and receiving a large amount of attention in the recent literature in mathematical physics; for such models, the effective dynamics is described by a coupled system of non-linear Schödinger equations. After reviewing and commenting our proof in the mean-field regime from a previous paper, we collect the main details needed to obtain the rigorous derivation of the effective dynamics in the Gross-Pitaevskii scaling limit.

Elliptic diffeomorphisms of symplectic 4-manifolds - Vsevolod Shevchishin; Gleb Smirnov (2017)

Energy release rate and quasi-static evolution via vanishing viscosity in a fracture model depending on the crack opening - Stefano Almi (ESAIM: Control, Optimisation and Calculus of Variations, 2017)
In the setting of planar linearized elasticity, we study a fracture model depending on the crack opening. Assuming that the crack path is known a priori and sufficiently smooth, we prove that the energy release rate is well defined. Then, we consider the problem of quasi-static evolution for our model. Thanks to a vanishing viscosity approach, we show the existence of such an evolution satisfying a weak Griffith’s criterion.

Friedrichs systems in a Hilbert space framework: solvability and multiplicity - Nenad Antonić; Marko Erceg; Alessandro Michelangeli (2017)
The Friedrichs (1958) theory of positive symmetric systems of first order partial differential equations encompasses many standard equations of mathematical physics, irrespective of their type. This theory was recast in an abstract Hilbert space setting by Ern, Guermond and Caplain (2007), and by Antonić and Burazin (2010). In this work we make a further step, presenting a purely operator-theoretic description of abstract Friedrichs systems, and proving that any pair of abstract Friedrichs operators admits bijective extensions with a signed boundary map. Moreover, we provide suffcient and necessary conditions for existence of infinitely many such pairs of spaces, and by the universal operator extension theory (Grubb, 1968) we get a complete identification of all such pairs, which we illustrate on two concrete one-dimensional examples.

Gamma-Convergence of Free-discontinuity problems - Filippo Cagnetti; Gianni Dal Maso; Lucia Scardia; Caterina Ida Zeppieri (SISSA, 2017)
We study the Gamma-convergence of sequences of free-discontinuity functionals depending on vector-valued functions u which can be discontinuous across hypersurfaces whose shape and location are not known a priori. The main novelty of our result is that we work under very general assumptions on the integrands which, in particular, are not required to be periodic in the space variable. Further, we consider the case of surface integrands which are not bounded from below by the amplitude of the jump of u. We obtain three main results: compactness with respect to Gamma-convergence, representation of the Gamma-limit in an integral form and identification of its integrands, and homogenisation formulas without periodicity assumptions. In particular, the classical case of periodic homogenisation follows as a by-product of our analysis. Moreover, our result covers also the case of stochastic homogenisation, as we will show in a forthcoming paper.

Gauge theories on compact toric surfaces, conformal field theories and equivariant Donaldson invariants - Mikhail Bershtein; Giulio Bonelli; Massimiliano Ronzani; Alessandro Tanzini (Journal of Geometry and Physics, 2017)
We show that equivariant Donaldson polynomials of compact toric surfaces can be calculated as residues of suitable combinations of Virasoro conformal blocks, by building on AGT correspondence between $\mathcal{N}=2$ supersymmetric gauge theories and two-dimensional conformal field theory. Talk presented by A.T. at the conference Interactions between Geometry and Physics — in honor of Ugo Bruzzo’s 60th birthday 17–22 August 2015, Guarujá, São Paulo, Brazil, mostly based on Bawane et al. (0000) and Bershtein et al. (0000).
Interactions between Geometry and Physics. A Special Issue in Honor of Ugo Bruzzo’s 60th Birthday
Keywords: AGT; Donaldson invariants; Equivariant localization; Exact partition function; Supersymmetry; Virasoro conformal blocks

On the generalized Cheeger problem and an application to 2d strips - Pratelli, A.; Saracco, G. (Rev. Mat. Iberoam., 2017)

On the genesis of directional friction through bristle-like mediating elements - Paolo Gidoni; Antonio DeSimone (ESAIM: COCV, 2017)
We propose an explanation of the genesis of directional dry friction, as emergent property of the oscillations produced in a bristle-like mediating element by the interaction with microscale fluctuations on the surface. Mathematically, we extend a convergence result by Mielke, for Prandtl–Tomlinson-like systems, considering also non-homothetic scalings of a wiggly potential. This allows us to apply the result to some simple mechanical models, that exemplify the interaction of a bristle with a surface having small fluctuations. We find that the resulting friction is the product of two factors: a geometric one, depending on the bristle angle and on the fluctuation profile, and a energetic one, proportional to the normal force exchanged between the bristle-like element and the surface. Finally, we apply our result to discuss the with the nap/against the nap asymmetry.

Globally stable quasistatic evolution for strain gradient plasticity coupled with damage - Vito Crismale (Annali di Matematica Pura ed Applicata (1923 -), 2017)
We consider evolutions for a material model which couples scalar damage with strain gradient plasticity, in small strain assumptions. For strain gradient plasticity, we follow the Gurtin–Anand formulation (J Mech Phys Solids 53:1624–1649, 2005). The aim of the present model is to account for different phenomena: On the one hand, the elastic stiffness reduces and the plastic yield surface shrinks due to material's degradation, on the other hand the dislocation density affects the damage growth. The main result of this paper is the existence of a globally stable quasistatic evolution (in the so-called energetic formulation). Furthermore, we study the limit model as the strain gradient terms tend to zero. Under stronger regularity assumptions, we show that the evolutions converge to the ones for the coupled elastoplastic damage model studied in Crismale (ESAIM Control Optim Calc Var 22:883-912, 2016).

Gross-Pitaevskii non-linear dynamics for pseudo-spinor condensates - Alessandro Michelangeli; Alessandro Olgiati (Journal of Nonlinear Mathematical Physics, 2017)
We derive the equations for the non-linear effective dynamics of a so called pseudo-spinor Bose-Einstein condensate, which emerges from the linear many-body Schrödinger equation at the leading order in the number of particles. The considered system is a three-dimensional diluted gas of identical bosons with spin, possibly confined in space, and coupled with an external time-dependent magnetic field; particles also interact among themselves through a short-scale repulsive interaction. The limit of infinitely many particles is monitored in the physically relevant Gross-Pitaevskii scaling. In our main theorem, if at time zero the system is in a phase of complete condensation (at the level of the reduced one-body marginal) and with energy per particle fixed by the Gross-Pitaevskii functional, then such conditions persist also at later times, with the one-body orbital of the condensate evolving according to a system of non-linear cubic Schrödinger equations coupled among themselves through linear (Rabi) terms. The proof relies on an adaptation to the spinor setting of Pickl’s projection counting method developed for the scalar case. Quantitative rates of convergence are available, but not made explicit because evidently non-optimal. In order to substantiate the formalism and the assumptions made in the main theorem, in an introductory section we review the mathematical formalisation of modern typical experiments with pseudo-spinor condensates.

Homotopically invisible singular curves - Andrei A. Agrachev; Francesco Boarotto; Antonio Lerario (Calculus of Variations and Partial Differential Equations, 2017)

Homotopy properties of horizontal path spaces and a theorem of Serre in subriemannian geometry - Francesco Boarotto; Antonio Lerario (Communications in Analysis and Geometry, 2017)

$hp$-version discontinuous Galerkin methods on polygonal and polyhedral meshes - Andrea Cangiani; Dong, Zhaonan; E.H. Georgoulis; Houston, Paul (SpringerBriefs in Mathematics, 2017)

$hp$-version space-time discontinuous Galerkin methods for parabolic problems on prismatic meshes - Andrea Cangiani; Dong, Zhaonan; E.H. Georgoulis (SIAM J. Sci. Comput., 2017)

The injectivity radius of Lie manifolds - Paolo Antonini; Guido De Philippis; Nicola Gigli (ArXiv e-prints, 2017)
We prove in a direct, geometric way that for any compatible Riemannian metric on a Lie manifold the injectivity radius is positive
Keywords: (58J40); 53C21; Mathematics - Differential Geometry

Integrability of dominated decompositions on three-dimensional manifolds - Stefano Luzzatto; Sina Türeli; Khadim Mbacke War (Ergodic Theory and Dynamical Systems, 2017)
We investigate the integrability of two-dimensional invariant distributions (tangent sub-bundles) which arise naturally in the context of dynamical systems on 3-manifolds. In particular, we prove unique integrability of dynamically dominated and volume-dominated Lipschitz continuous invariant decompositions as well as distributions with some other regularity conditions.

Integrable lifts for transitive Lie algebroids - Androulidakis, I.; Paolo Antonini (ArXiv e-prints, 2017)
Inspired by the work of Molino, we show that the integrability obstruction for transitive Lie algebroids can be made to vanish by adding extra dimensions. In particular, we prove that the Weinstein groupoid of a non-integrable transitive and abelian Lie algebroid, is the quotient of a finite dimensional Lie groupoid. Two constructions as such are given: First, explaining the counterexample to integrability given by Almeida and Molino, we see that it can be generalized to the construction of an "Almeida-Molino" integrable lift when the base manifold is simply connected. On the other hand, we notice that the classical de Rham isomorphism provides a universal integrable algebroid. Using it we construct a "de Rham" integrable lift for any given transitive Abelian Lie algebroid.
Keywords: 14F40; 58H05; Mathematics - Differential Geometry

Kinematics of flagellar swimming in Euglena gracilis: Helical trajectories and flagellar shapes - Massimiliano Rossi; Giancarlo Cicconofri; Alfred Beran; Giovanni Noselli; Antonio DeSimone (Proceedings of the National Academy of Sciences, 2017)
Active flagella provide the propulsion mechanism for a large variety of swimming eukaryotic microorganisms, from protists to sperm cells. Planar and helical beating patterns of these structures are recurrent and widely studied. The fast spinning motion of the locomotory flagellum of the alga Euglena gracilis constitutes a remarkable exception to these patterns. We report a quantitative description of the 3D flagellar beating in swimming E. gracilis. Given their complexity, these shapes cannot be directly imaged with current microscopy techniques. We show how to overcome these limitations by developing a method to reconstruct in full the 3D kinematics of the cell from conventional 2D microscopy images, based on the exact characterization of the helical motion of the cell body.The flagellar swimming of euglenids, which are propelled by a single anterior flagellum, is characterized by a generalized helical motion. The 3D nature of this swimming motion, which lacks some of the symmetries enjoyed by more common model systems, and the complex flagellar beating shapes that power it make its quantitative description challenging. In this work, we provide a quantitative, 3D, highly resolved reconstruction of the swimming trajectories and flagellar shapes of specimens of Euglena gracilis. We achieved this task by using high-speed 2D image recordings taken with a conventional inverted microscope combined with a precise characterization of the helical motion of the cell body to lift the 2D data to 3D trajectories. The propulsion mechanism is discussed. Our results constitute a basis for future biophysical research on a relatively unexplored type of eukaryotic flagellar movement.

The Kontsevich matrix integral: convergence to the Painlevé hierarchy and Stokes' phenomenon - Marco Bertola; Mattia Cafasso (Comm. Math. Phys, 2017)

Krein-Visik-Birman self-adjoint extension theory revisited - Matteo Gallone; Alessandro Michelangeli; Andrea Ottolini (2017)
The core results of the so-called KreIn-Visik-Birman theory of self-adjoint extensions of semi-bounded symmetric operators are reproduced, both in their original and in a more modern formulation, within a comprehensive discussion that includes missing details, elucidative steps, and intermediate results of independent interest.

A Lagrangian approach for scalar multi-d conservation laws - Stefano Bianchini; Paolo Bonicatto; Elio Marconi (2017)

Lagrangian representations for linear and nonlinear transport - Stefano Bianchini; Paolo Bonicatto; Elio Marconi (Contemporary Mathematics. Fundamental Directions, 2017)
In this note we present a unifying approach for two classes of first order partial differential equations: we introduce the notion of Lagrangian representation in the settings of continuity equation and scalar conservation laws. This yields, on the one hand, the uniqueness of weak solutions to transport equation driven by a two dimensional BV nearly incompressible vector field. On the other hand, it is proved that the entropy dissipation measure for scalar conservation laws in one space dimension is concentrated on countably many Lipschitz curves.

Limit of viscous dynamic processes in delamination as the viscosity and inertia vanish - Riccardo Scala (ESAIM: COCV, 2017)
We introduce a model of dynamic evolution of a delaminated visco-elastic body with viscous adhesive. We prove the existence of solutions of the corresponding system of PDEs and then study the behavior of such solutions when the data of the problem vary slowly. We prove that a rescaled version of the dynamic evolutions converge to a “local” quasistatic evolution, which is an evolution satisfying an energy inequality and a momentum balance at all times. In the one-dimensional case we give a more detailed description of the limit evolution and we show that it behaves in a very similar way to the limit of the solutions of the dynamic model in [T. Roubicek, SIAM J. Math. Anal. 45 (2013) 101–126], where no viscosity in the adhesive is taken into account.

Linear Hyperbolic Systems in Domains with Growing Cracks - Maicol Caponi (Milan Journal of Mathematics, 2017)
We consider the hyperbolic system ü$${ - {\rm div} (\mathbb{A} \nabla u) = f}$$in the time varying cracked domain $${Ω \backslash Γ_t}$$, where the set $${Ω \subset \mathbb{R}^d}$$is open, bounded, and with Lipschitz boundary, the cracks $${Γ_t, t \in [0, T]}$$, are closed subsets of $${\bar{Ω}}$$, increasing with respect to inclusion, and $${u(t) : Ω \backslash Γ_t → \mathbb{R}^d}$$for every $${t \in [0, T]}$$. We assume the existence of suitable regular changes of variables, which reduce our problem to the transformed system v̈$${ - {\rm div} (\mathbb{B}\nabla v) + a\nabla v - 2 \nabla \dot{v}b = g}$$on the fixed domain $${Ω \backslash Γ_0}$$. Under these assumptions, we obtain existence and uniqueness of weak solutions for these two problems. Moreover, we show an energy equality for the functions v, which allows us to prove a continuous dependence result for both systems. The same study has already been carried out in [3, 7] in the scalar case.

Linearisation of multiwell energies - Roberto Alicandro; Gianni Dal Maso; Giuliano Lazzaroni; Mariapia Palombaro (2017)
Linear elasticity can be rigorously derived from finite elasticity under the assumption of small loadings in terms of Gamma-convergence. This was first done in the case of one-well energies with super-quadratic growth and later generalised to different settings, in particular to the case of multi-well energies where the distance between the wells is very small (comparable to the size of the load). In this paper we study the case when the distance between the wells is independent of the size of the load. In this context linear elasticity can be derived by adding to the multi-well energy a singular higher order term which penalises jumps from one well to another. The size of the singular term has to satisfy certain scaling assumptions whose optimality is shown in most of the cases. Finally, the derivation of linear elasticty from a two-well discrete model is provided, showing that the role of the singular perturbation term is played in this setting by interactions beyond nearest neighbours.

Lower semicontinuity of a class of integral functionals on the space of functions of bounded deformation - Gianni Dal Maso; Gianluca Orlando; Rodica Toader (Advances in Calculus of Variations, 2017)
We study the lower semicontinuity of some free discontinuity functionals with linear growth defined on the space of functions with bounded deformation. The volume term is convex and depends only on the Euclidean norm of the symmetrized gradient. We introduce a suitable class of surface terms, which make the functional lower semicontinuous with respect to $L^1$ convergence.

A lower semicontinuity result for a free discontinuity functional with a boundary term - Stefano Almi; Gianni Dal Maso; Rodica Toader (Journal de Mathématiques Pures et Appliquées, 2017)
We study the lower semicontinuity in $GSBV^{p}(Ω;\mathbb{R}^{m})$ of a free discontinuity functional $\mathcal{F}(u)$ that can be written as the sum of a crack term, depending only on the jump set $S_{u}$, and of a boundary term, depending on the trace of $u$ on $\partialΩ$. We give sufficient conditions on the integrands for the lower semicontinuity of $\mathcal{F}$. Moreover, we prove a relaxation result, which shows that, if these conditions are not satisfied, the lower semicontinuous envelope of $\mathcal{F}$ can be represented by the sum of two integrals on $S_{u}$ and $\partialΩ$, respectively.

The Malgrange form and Fredholm determinants - Marco Bertola (SIGMA Symmetry Integrability Geom. Methods Appl., 2017)