Basic ideas and tools for projection-based model reduction of parametric partial differential equations - Gianluigi Rozza; Martin W. Hess; Giovanni Stabile; Marco Tezzele; F. Ballarin (Model Order Reduction, Volume 2 Snapshot-Based Methods and Algorithms, 2020)

Bayesian identification of a projection-based reduced order model for computational fluid dynamics - Giovanni Stabile; Bojana Rosic (Computers & Fluids, 2020)
In this paper we propose a Bayesian method as a numerical way to correct and stabilise projection-based reduced order models (ROM) in computational fluid dynamics problems. The approach is of hybrid type, and consists of the classical proper orthogonal decomposition driven Galerkin projection of the laminar part of the governing equations, and Bayesian identification of the correction term mimicking both the turbulence model and possible ROM-related instabilities given the full order data. In this manner the classical ROM approach is translated to the parameter identification problem on a set of nonlinear ordinary differential equations. Computationally the inverse problem is solved with the help of the Gauss-Markov-Kalman smoother in both ensemble and square-root polynomial chaos expansion forms. To reduce the dimension of the posterior space, a novel global variance based sensitivity analysis is proposed.

On the blow-up of GSBV functions under suitable geometric properties of the jump set - Emanuele Tasso (Advances in Calculus of Variations, 2020)

Certified Reduced Basis VMS-Smagorinsky model for natural convection flow in a cavity with variable height - F. Ballarin; Rebollo, T.C.; E.D. Ávila; Marmol, M.G.; Gianluigi Rozza (Computers and Mathematics with Applications, 2020)
In this work we present a Reduced Basis VMS-Smagorinsky Boussinesq model, applied to natural convection problems in a variable height cavity, in which the buoyancy forces are involved. We take into account in this problem both physical and geometrical parametrizations, considering the Rayleigh number as a parameter, so as the height of the cavity. We perform an Empirical Interpolation Method to approximate the sub-grid eddy viscosity term that lets us obtain an affine decomposition with respect to the parameters. We construct an a posteriori error estimator, based upon the Brezzi–Rappaz–Raviart theory, used in the greedy algorithm for the selection of the basis functions. Finally we present several numerical tests for different parameter configuration.

On coherent Hopf 2-algebras - Xiao Han (2020)
We construct a coherent Hopf 2-algebra as quantization of a coherent 2-group, which consists of two Hopf coquasigroups and a coassociator. For this constructive method, if we replace Hopf coquasigroups by Hopf algebras, we can construct a strict Hoft 2-algebra, which is a quantisation of 2-group. We also study the crossed comodule of Hopf algebras, which is shown to be a strict Hopf 2-algebra under some conditions. As an example, a quasi coassociative Hopf coquasigroup is employed to build a special coherent Hopf 2-algebra with nontrivial coassociator. Following this we study functions on Cayley algebra basis.

On the continuity of the trace operator in GSBV (Ω) and GSBD (Ω) - Emanuele Tasso (ESAIM: COCV, 2020)

Convergence of an adaptive discontinuous Galerkin method for elliptic interface problems - Andrea Cangiani; E.H. Georgoulis; Sabawi, Younis A. (J. Comput. Appl. Math., 2020)

Data-driven POD-Galerkin reduced order model for turbulent flows - Saddam Hijazi; Giovanni Stabile; Andrea Mola; Gianluigi Rozza (Journal of Computational Physics, 2020)
In this work we present a Reduced Order Model which is specifically designed to deal with turbulent flows in a finite volume setting. The method used to build the reduced order model is based on the idea of merging/combining projection-based techniques with data-driven reduction strategies. In particular, the work presents a mixed strategy that exploits a data-driven reduction method to approximate the eddy viscosity solution manifold and a classical POD-Galerkin projection approach for the velocity and the pressure fields, respectively. The newly proposed reduced order model has been validated on benchmark test cases in both steady and unsteady settings with Reynolds up to $Re=O(10^5)$.

The deal.II finite element library: Design, features, and insights - Daniel Arndt; Wolfgang Bangerth; Denis Davydov; Timo Heister; Luca Heltai; Martin Kronbichler; Matthias Maier; Jean-Paul Pelteret; Bruno Turcksin; David Wells (Computers and Mathematics with Applications, 2020)

The deal.II library, Version 9.2 - Daniel Arndt; Wolfgang Bangerth; Bruno Blais; Thomas C. Clevenger; Marc Fehling; Alexander V. Grayver; Timo Heister; Luca Heltai; Martin Kronbichler; Matthias Maier; Peter Munch; Jean-Paul Pelteret; et al. (Journal of Numerical Mathematics, 2020)

A dynamic model for viscoelastic materials with prescribed growing cracks - Maicol Caponi; Francesco Sapio (Annali di Matematica Pura ed Applicata (1923 -), 2020)
In this paper, we prove the existence of solutions for a class of viscoelastic dynamic systems on time-dependent cracked domains, with possibly degenerate viscosity coefficients. Under stronger regularity assumptions, we also show a uniqueness result. Finally, we exhibit an example where the energy-dissipation balance is not satisfied, showing there is an additional dissipation due to the crack growth.

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

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

The Effort of Increasing Reynolds Number in Projection-Based Reduced Order Methods: from Laminar to Turbulent Flows - Saddam Hijazi; Shafqat Ali; Giovanni Stabile; F. Ballarin; Gianluigi Rozza (Lecture Notes in Computational Science and Engineering, 2020)
We present in this double contribution two different reduced order strategies for incompressible parameterized Navier-Stokes equations characterized by varying Reynolds numbers. The first strategy deals with low Reynolds number (laminar flow) and is based on a stabilized finite element method during the offline stage followed by a Galerkin projection on reduced basis spaces generated by a greedy algorithm. The second methodology is based on a full order finite volume discretization. The latter methodology will be used for flows with moderate to high Reynolds number characterized by turbulent patterns. For the treatment of the mentioned turbulent flows at the reduced order level, a new POD-Galerkin approach is proposed. The new approach takes into consideration the contribution of the eddy viscosity also during the online stage and is based on the use of interpolation. The two methodologies are tested on classic benchmark test cases.

Energy-dissipation balance of a smooth moving crack - Maicol Caponi; Ilaria Lucardesi; Emanuele Tasso (Journal of Mathematical Analysis and Applications, 2020)
In this paper we provide necessary and sufficient conditions in order to guarantee the energy-dissipation balance of a Mode III crack, growing on a prescribed smooth path. Moreover, we characterize the singularity of the displacement near the crack tip, generalizing the result in [10] valid for straight fractures.
Keywords: Energy-dissipation balance; Fracture dynamics; Wave equation in time-dependent domains

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

Existence of Riemannian metrics with positive biorthogonal curvature on simply connected 5-manifolds - Boris Stupovski; Rafael Torres (Archiv der Mathematik, 2020)
Using the recent work of Bettiol, we show that a first-order conformal deformation of Wilking’s metric of almost-positive sectional curvature on $S2× S3$ yields a family of metrics with strictly positive average of sectional curvatures of any pair of 2-planes that are separated by a minimal distance in the 2-Grassmanian. A result of Smale allows us to conclude that every closed simply connected 5-manifold with torsion-free homology and trivial second Stiefel–Whitney class admits a Riemannian metric with a strictly positive average of sectional curvatures of any pair of orthogonal 2-planes.

Existence of solutions to a phase–field model of dynamic fracture with a crack–dependent dissipation - Maicol Caponi (Nonlinear Differential Equations and Applications NoDEA, 2020)
We propose a phase–field model of dynamic fracture based on the Ambrosio–Tortorelli’s approximation, which takes into account dissipative effects due to the speed of the crack tips. By adapting the time discretization scheme contained in Larsen et al. (Math Models Methods Appl Sci 20:1021–1048, 2010), we show the existence of a dynamic crack evolution satisfying an energy–dissipation balance, according to Griffith’s criterion. Finally, we analyze the dynamic phase–field model of Bourdin et al. (Int J Fract 168:133–143, 2011) and Larsen (in: Hackl (ed) IUTAM symposium on variational concepts with applications to the mechanics of materials, IUTAM Bookseries, vol 21. Springer, Dordrecht, 2010, pp 131–140) with no dissipative terms.

Finite element approximation of an obstacle problem for a class of integro–differential operators - Bonito, Andrea; Wenyu Lei; Salgado, Abner J (ESAIM: Mathematical Modelling and Numerical Analysis, 2020)

On functions having coincident p-norms - Giuliano Klun (Annali di Matematica Pura ed Applicata (1923 -), 2020)
In a measure space $(X,{\mathcal {A}},\mu )$, we consider two measurable functions $f,g:E\rightarrow {\mathbb {R}}$, for some $E\in {\mathcal {A}}$. We prove that the property of having equal p-norms when p varies in some infinite set $P\subseteq [1,+\infty )$ is equivalent to the following condition: $\begin{aligned} \mu (\{x\in E:|f(x)|>\alpha \})=\mu (\{x\in E:|g(x)|>\alpha \})\quad \text { for all } \alpha \ge 0. \end{aligned}$

On the gauge group of Galois objects - Xiao Han; Giovanni Landi (2020)
We study the Ehresmann--Schauenburg bialgebroid of a noncommutative principal bundle as a quantization of the classical gauge groupoid of a principal bundle. When the base algebra is in the centre of the total space algebra, the gauge group of the noncommutative principal bundle is isomorphic to the group of bisections of the bialgebroid. In particular we consider Galois objects (non-trivial noncommutative bundles over a point in a sense) for which the bialgebroid is a Hopf algebra. For these we give a crossed module structure for the bisections and the automorphisms of the bialgebroid. Examples include Galois objects of group Hopf algebras and of Taft algebras.

Gauge theories on compact toric manifolds - Giulio Bonelli; Francesco Fucito; Morales, Jose Francisco; Massimiliano Ronzani; Sysoeva, Ekaterina; Alessandro Tanzini (2020)

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

Indeterminacy estimates and the size of nodal sets in singular spaces - Fabio Cavalletti; Sara Farinelli (2020)
Keywords: Differential Geometry (math.DG); FOS: Mathematics; Metric Geometry (math.MG)

\it A posteriori error analysis for implicit-explicit $hp$-discontinuous Galerkin timestepping methods for semilinear parabolic problems - Andrea Cangiani; E.H. Georgoulis; Sabawi, Mohammad (J. Sci. Comput., 2020)

Kernel-based Active Subspaces with application to CFD parametric problems using Discontinuous Galerkin method - Francesco Romor; Marco Tezzele; Lario, Andrea; Gianluigi Rozza (2020)

Matematica ed elezioni, paradossi e problemi elettorali - Saracco, A.; Saracco, G. (Mat. Cult. Soc. Riv. Unione Mat. Ital. (I), 2020)

MicroMotility: State of the art, recent accomplishments and perspectives on the mathematical modeling of bio-motility at microscopic scales - Daniele Agostinelli; Roberto Cerbino; Del Alamo, Juan C; Antonio DeSimone; Stephanie Höhn; Cristian Micheletti; Giovanni Noselli; Eran Sharon; Julia Yeomans (Mathematics in Engineering, 2020)
Mathematical modeling and quantitative study of biological motility (in particular, of motility at microscopic scales) is producing new biophysical insight and is offering opportunities for new discoveries at the level of both fundamental science and technology. These range from the explanation of how complex behavior at the level of a single organism emerges from body architecture, to the understanding of collective phenomena in groups of organisms and tissues, and of how these forms of swarm intelligence can be controlled and harnessed in engineering applications, to the elucidation of processes of fundamental biological relevance at the cellular and sub-cellular level. In this paper, some of the most exciting new developments in the fields of locomotion of unicellular organisms, of soft adhesive locomotion across scales, of the study of pore translocation properties of knotted DNA, of the development of synthetic active solid sheets, of the mechanics of the unjamming transition in dense cell collectives, of the mechanics of cell sheet folding in volvocalean algae, and of the self-propulsion of topological defects in active matter are discussed. For each of these topics, we provide a brief state of the art, an example of recent achievements, and some directions for future research.
Keywords: active matter; adhesive locomotion; cell motility; cell sheet folding; knotted DNA; topological defects; unicellular swimmers; unjamming transition

MicroROM: An Efficient and Accurate Reduced Order Method to Solve Many-Query Problems in Micro-Motility - Nicola Giuliani; Martin W. Hess; Antonio DeSimone; Gianluigi Rozza (2020)
Keywords: FOS: Mathematics; Numerical Analysis (math.NA)

Minimality of the ball for a model of charged liquid droplets - Ekaterina Mukoseeva; Giulia Vescovo (2020)

Minimizers of the prescribed mean curvature functional in a Jordan domain with no necks - Leonardi, G. P.; Saracco, G. (ESAIM Control Optim. Calc. Var., 2020)

Multiscale modeling of fiber reinforced materials via non-matching immersed methods - Giovanni Alzetta; Luca Heltai (Computers & Structures, 2020)
To appear

Non-intrusive Polynomial Chaos Method Applied to Full-Order and Reduced Problems in Computational Fluid Dynamics: A Comparison and Perspectives - Saddam Hijazi; Giovanni Stabile; Andrea Mola; Gianluigi Rozza (Quantification of Uncertainty: Improving Efficiency and Technology: QUIET selected contributions, 2020)
In this work, Uncertainty Quantification (UQ) based on non-intrusive Polynomial Chaos Expansion (PCE) is applied to the CFD problem of the flow past an airfoil with parameterized angle of attack and inflow velocity. To limit the computational cost associated with each of the simulations required by the non-intrusive UQ algorithm used, we resort to a Reduced Order Model (ROM) based on Proper Orthogonal Decomposition (POD)-Galerkin approach. A first set of results is presented to characterize the accuracy of the POD-Galerkin ROM developed approach with respect to the Full Order Model (FOM) solver (OpenFOAM). A further analysis is then presented to assess how the UQ results are affected by substituting the FOM predictions with the surrogate ROM ones.

A numerical study of the jerky crack growth in elastoplastic materials with localized plasticity - Gianni Dal Maso; Luca Heltai (Journal of Convex Analysis, 2020)
To appear

Periodic solutions of nearly integrable Hamiltonian systems bifurcating from infinite-dimensional tori - Alessandro Fonda; Giuliano Klun; Andrea Sfecci (NONLINEAR ANALYSIS, 2020)
<span style="color: #2e2e2e; font-family: NexusSerif, Georgia, 'Times New Roman', Times, STIXGeneral, 'Cambria Math', 'Lucida Sans Unicode', 'Microsoft Sans Serif', 'Segoe UI Symbol', 'Arial Unicode MS', serif; font-size: 18px;">We prove the existence of periodic solutions of some infinite-dimensional nearly integrable Hamiltonian systems, bifurcating from infinite-dimensional tori, by the use of a generalization of the Poincaré–Birkhoff Theorem.

POD-Galerkin Model Order Reduction for Parametrized Nonlinear Time Dependent Optimal Flow Control: an Application to Shallow Water Equations - Maria Strazzullo; F. Ballarin; Gianluigi Rozza (2020)
In this work we propose reduced order methods as a reliable strategy to efficiently solve parametrized optimal control problems governed by shallow waters equations in a solution tracking setting. The physical parametrized model we deal with is nonlinear and time dependent: this leads to very time consuming simulations which can be unbearable e.g. in a marine environmental monitoring plan application. Our aim is to show how reduced order modelling could help in studying different configurations and phenomena in a fast way. After building the optimality system, we rely on a POD-Galerkin reduction in order to solve the optimal control problem in a low dimensional reduced space. The presented theoretical framework is actually suited to general nonlinear time dependent optimal control problems. The proposed methodology is finally tested with a numerical experiment: the reduced optimal control problem governed by shallow waters equations reproduces the desired velocity and height profiles faster than the standard model, still remaining accurate.

POD–Galerkin Model Order Reduction for Parametrized Time Dependent Linear Quadratic Optimal Control Problems in Saddle Point Formulation - Maria Strazzullo; F. Ballarin; Gianluigi Rozza (Journal of Scientific Computing, 2020)
In this work we deal with parametrized time dependent optimal control problems governed by partial differential equations. We aim at extending the standard saddle point framework of steady constraints to time dependent cases. We provide an analysis of the well-posedness of this formulation both for parametrized scalar parabolic constraint and Stokes governing equations and we propose reduced order methods as an effective strategy to solve them. Indeed, on one hand, parametrized time dependent optimal control is a very powerful mathematical model which is able to describe several physical phenomena, on the other, it requires a huge computational effort. Reduced order methods are a suitable approach to have rapid and accurate simulations. We rely on POD–Galerkin reduction over the physical and geometrical parameters of the optimality system in a space-time formulation. Our theoretical results and our methodology are tested on two examples: a boundary time dependent optimal control for a Graetz flow and a distributed optimal control governed by time dependent Stokes equations. With these two test cases the convenience of the reduced order modelling is further extended to the field of time dependent optimal control.

POD–Galerkin Model Order Reduction for Parametrized Time Dependent Linear Quadratic Optimal Control Problems in Saddle Point Formulation - Maria Strazzullo; Francesco Ballarin; Gianluigi Rozza (Journal of Scientific Computing, 2020)

POD–Galerkin reduced order methods for combined Navier–Stokes transport equations based on a hybrid FV-FE solver - S. Busto; Giovanni Stabile; Gianluigi Rozza; M.E. Vázquez-Cendón (Computers and Mathematics with Applications, 2020)
The purpose of this work is to introduce a novel POD–Galerkin strategy for the semi-implicit hybrid high order finite volume/finite element solver introduced in Bermúdez et al. (2014) and Busto et al. (2018). The interest is into the incompressible Navier–Stokes equations coupled with an additional transport equation. The full order model employed in this article makes use of staggered meshes. This feature will be conveyed to the reduced order model leading to the definition of reduced basis spaces in both meshes. The reduced order model presented herein accounts for velocity, pressure, and a transport-related variable. The pressure term at both the full order and the reduced order level is reconstructed making use of a projection method. More precisely, a Poisson equation for pressure is considered within the reduced order model. Results are verified against three-dimensional manufactured test cases. Moreover a modified version of the classical cavity test benchmark including the transport of a species is analysed.

A priori error estimates of regularized elliptic problems - Luca Heltai; Wenyu Lei (Numerische Mathematik, 2020)

A priori error estimates of regularized elliptic problems - Luca Heltai; Wenyu Lei (Numerische Mathematik, 2020)
Approximations of the Dirac delta distribution are commonly used to create sequences of smooth functions approximating nonsmooth (generalized) functions, via convolution. In this work we show a-priori rates of convergence of this approximation process in standard Sobolev norms, with minimal regularity assumptions on the approximation of the Dirac delta distribution. The application of these estimates to the numerical solution of elliptic problems with singularly supported forcing terms allows us to provide sharp \$\$H\^1\$\$and \$\$L\^2\$\$error estimates for the corresponding regularized problem. As an application, we show how finite element approximations of a regularized immersed interface method results in the same rates of convergence of its non-regularized counterpart, provided that the support of the Dirac delta approximation is set to a multiple of the mesh size, at a fraction of the implementation complexity. Numerical experiments are provided to support our theories.

Projection-based reduced order models for a cut finite element method in parametrized domains - Efthymios N Karatzas; F. Ballarin; Gianluigi Rozza (Computers and Mathematics with Applications, 2020)
This work presents a reduced order modeling technique built on a high fidelity embedded mesh finite element method. Such methods, and in particular the CutFEM method, are attractive in the generation of projection-based reduced order models thanks to their capabilities to seamlessly handle large deformations of parametrized domains and in general to handle topological changes. The combination of embedded methods and reduced order models allows us to obtain fast evaluation of parametrized problems, avoiding remeshing as well as the reference domain formulation, often used in the reduced order modeling for boundary fitted finite element formulations. The resulting novel methodology is presented on linear elliptic and Stokes problems, together with several test cases to assess its capability. The role of a proper extension and transport of embedded solutions to a common background is analyzed in detail.

Quantum Systems at The Brink: Properties of Atomic Bound States at The Ionization Threshold - Dirk Hundertmark; Michal Jex; Markus Lange (2020)

Quasi-periodic solutions of nonlinear wave equations on the $d$-dimensional torus - Massimiliano Berti; Bolle, Philippe (EMS Monographs in Mathematics, 2020)

Quasi-periodic standing wave solutions of gravity-capillary water waves - Massimiliano Berti; Montalto, Riccardo (Mem. Amer. Math. Soc., 2020)

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

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

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

Reduced order isogeometric analysis approach for pdes in parametrized domains - Fabrizio Garotta; Nicola Demo; Marco Tezzele; Massimo Carraturo; Alessandro Reali; Gianluigi Rozza (Lecture Notes in Computational Science and Engineering, 2020)
In this contribution, we coupled the isogeometric analysis to a reduced order modelling technique in order to provide a computationally efficient solution in parametric domains. In details, we adopt the free-form deformation method to obtain the parametric formulation of the domain and proper orthogonal decomposition with interpolation for the computational reduction of the model. This technique provides a real-time solution for any parameter by combining several solutions, in this case computed using isogeometric analysis on different geometrical configurations of the domain, properly mapped into a reference configuration. We underline that this reduced order model requires only the full-order solutions, making this approach non-intrusive. We present in this work the results of the application of this methodology to a heat conduction problem inside a deformable collector pipe.

Reduced order methods for parametric optimal flow control in coronary bypass grafts, toward patient-specific data assimilation - Zakia Zainib; F. Ballarin; Stephen E. Fremes; Piero Triverio; Laura Jiménez-Juan; Gianluigi Rozza (International Journal for Numerical Methods in Biomedical EngineeringInternational Journal for Numerical Methods in Biomedical EngineeringInt J Numer Meth Biomed Engng, 2020)
Abstract Coronary artery bypass grafts (CABG) surgery is an invasive procedure performed to circumvent partial or complete blood flow blockage in coronary artery disease. In this work, we apply a numerical optimal flow control model to patient-specific geometries of CABG, reconstructed from clinical images of real-life surgical cases, in parameterized settings. The aim of these applications is to match known physiological data with numerical hemodynamics corresponding to different scenarios, arisen by tuning some parameters. Such applications are an initial step toward matching patient-specific physiological data in patient-specific vascular geometries as best as possible. Two critical challenges that reportedly arise in such problems are: (a) lack of robust quantification of meaningful boundary conditions required to match known data as best as possible and (b) high computational cost. In this work, we utilize unknown control variables in the optimal flow control problems to take care of the first challenge. Moreover, to address the second challenge, we propose a time-efficient and reliable computational environment for such parameterized problems by projecting them onto a low-dimensional solution manifold through proper orthogonal decomposition-Galerkin.
Keywords: coronary artery bypass grafts; data assimilation; flow control; Galerkin methods; hemodynamics modeling; Optimization; patient-specific simulations; Proper orthogonal decomposition; reduced order methods

A reduced order modeling technique to study bifurcating phenomena: Application to the gross-pitaevskii equation - Federico Pichi; Annalisa Quaini; Gianluigi Rozza (SIAM Journal on Scientific Computing, 2020)
We propose a computationally efficient framework to treat nonlinear partial differential equations having bifurcating solutions as one or more physical control parameters are varied. Our focus is on steady bifurcations. Plotting a bifurcation diagram entails computing multiple solutions of a parametrized, nonlinear problem, which can be extremely expensive in terms of computational time. In order to reduce these demanding computational costs, our approach combines a continuation technique and Newton's method with a reduced order modeling (ROM) technique, suitably supplemented with a hyperreduction method. To demonstrate the effectiveness of our ROM approach, we trace the steady solution branches of a nonlinear Schrödinger equation, called the Gross{Pitaevskii equation, as one or two physical parameters are varied. In the two-parameter study, we show that our approach is 60 times faster in constructing a bifurcation diagram than a standard full order method.

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

A reduced-order shifted boundary method for parametrized incompressible Navier–Stokes equations - Efthymios N Karatzas; Giovanni Stabile; Leo Nouveau; Guglielmo Scovazzi; Gianluigi Rozza (Computer Methods in Applied Mechanics and Engineering, 2020)
We investigate a projection-based reduced order model of the steady incompressible Navier–Stokes equations for moderate Reynolds numbers. In particular, we construct an “embedded” reduced basis space, by applying proper orthogonal decomposition to the Shifted Boundary Method, a high-fidelity embedded method recently developed. We focus on the geometrical parametrization through level-set geometries, using a fixed Cartesian background geometry and the associated mesh. This approach avoids both remeshing and the development of a reference domain formulation, as typically done in fitted mesh finite element formulations. Two-dimensional computational examples for one and three parameter dimensions are presented to validate the convergence and the efficacy of the proposed approach.

A remark on vanishing geodesic distances in infinite dimensions - Magnani, Valentino; Tiberio, Daniele (Proceedings of the American Mathematical Society, 2020)

Special Issue on Reduced Order Models in CFD - Simona Perotto; Gianluigi Rozza (International Journal of Computational Fluid Dynamics, 2020)

A spectral element reduced basis method for navier–stokes equations with geometric variations - Martin W. Hess; Annalisa Quaini; Gianluigi Rozza (Lecture Notes in Computational Science and Engineering, 2020)
We consider the Navier-Stokes equations in a channel with a narrowing of varying height. The model is discretized with high-order spectral element ansatz functions, resulting in 6372 degrees of freedom. The steady-state snapshot solutions define a reduced order space through a standard POD procedure. The reduced order space allows to accurately and efficiently evaluate the steady-state solutions for different geometries. In particular, we detail different aspects of implementing the reduced order model in combination with a spectral element discretization. It is shown that an expansion in element-wise local degrees of freedom can be combined with a reduced order modelling approach to enhance computational times in parametric many-query scenarios.

Stabilized reduced basis methods for parametrized steady Stokes and Navier–Stokes equations - Shafqat Ali; F. Ballarin; Gianluigi Rozza (Computers and Mathematics with Applications, 2020)
It is well known in the Reduced Basis approximation of saddle point problems that the Galerkin projection on the reduced space does not guarantee the inf–sup approximation stability even if a stable high fidelity method was used to generate snapshots. For problems in computational fluid dynamics, the lack of inf–sup stability is reflected by the inability to accurately approximate the pressure field. In this context, inf–sup stability is usually recovered through the enrichment of the velocity space with suitable supremizer functions. The main goal of this work is to propose an alternative approach, which relies on the residual based stabilization techniques customarily employed in the Finite Element literature, such as Brezzi–Pitkaranta, Franca–Hughes, streamline upwind Petrov–Galerkin, Galerkin Least Square. In the spirit of offline–online reduced basis computational splitting, two such options are proposed, namely offline-only stabilization and offline–online stabilization. These approaches are then compared to (and combined with) the state of the art supremizer enrichment approach. Numerical results are discussed, highlighting that the proposed methodology allows to obtain smaller reduced basis spaces (i.e., neglecting supremizer enrichment) for which a modified inf–sup stability is still preserved at the reduced order level.

Stabilized reduced basis methods for parametrized steady Stokes and Navier–Stokes equations - Shafqat Ali; Francesco Ballarin; Gianluigi Rozza (Computers & Mathematics with Applications, 2020)
It is well known in the Reduced Basis approximation of saddle point problems that the Galerkin projection on the reduced space does not guarantee the inf–sup approximation stability even if a stable high fidelity method was used to generate snapshots. For problems in <a class="topic-link" title="Learn more about computational fluid dynamics from ScienceDirect's AI-generated Topic Pages" href="https://www.sciencedirect.com/topics/materials-science/computational-fluid-dynamics">computational fluid dynamics, the lack of inf–sup stability is reflected by the inability to accurately approximate the pressure field. In this context, inf–sup stability is usually recovered through the enrichment of the velocity space with suitable supremizer functions. The main goal of this work is to propose an alternative approach, which relies on the residual based stabilization techniques customarily employed in the Finite Element literature, such as Brezzi–Pitkaranta, Franca–Hughes, streamline upwind Petrov–Galerkin, Galerkin Least Square. In the spirit of offline–online reduced basis computational splitting, two such options are proposed, namely offline-only stabilizationand offline–online stabilization. These approaches are then compared to (and combined with) the state of the art supremizer enrichment approach. Numerical results are discussed, highlighting that the proposed methodology allows to obtain smaller reduced basis spaces (i.e., neglecting supremizer enrichment) for which a modified inf–sup stability is still preserved at the reduced order level.

Stable vector bundles on the families of curves - Fedor Bogomolov; Elena Lukzen (2020)
We offer a new approach to proving the Chen-Donaldson-Sun theorem which we demonstrate with a series of examples. We discuss the existence of a construction of a special metric on stable vector bundles over the surfaces formed by a families of curves and its relation to the one-dimensional cycles in the moduli space of stable bundles on curves.

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

A Theoretical Study on the Transient Morphing of Linear Poroelastic Plates - Dario Andrini; Alessandro Lucantonio; Giovanni Noselli (Journal of Applied Mechanics, 2020)
Based on their shape-shifting capabilities, soft active materials have enabled new possibilities for the engineering of sensing and actuation devices. While the relation between active strains and emergent equilibrium shapes has been fully characterized, the transient morphing of thin structures is a rather unexplored topic. Here, we focus on polymer gel plates and derive a reduced linear model to study their time-dependent response to changes in the fluid environment. We show that independent control of stretching and bending deformations in stress-free conditions allows to realize spherical shapes with prescribed geometry of the mid-plane. Furthermore, we demonstrate that tensile (compressive) membrane stresses delay (accelerate) swelling-induced shape transitions compared to the stress-free evolution. We believe that these effects should be considered for the accurate design of smart systems and may contribute to explain the complexity of natural shapes.

Topology change and selection rules for high-dimensional spin(1,n)0-Lorentzian cobordisms - Gleb Smirnov; Rafael Torres (Transactions of the american mathematical society, 2020)

Twisted Ehresmann Schauenburg bialgebroids - Xiao Han (2020)
We construct an invertible normalised 2 cocycle on the Ehresmann Schauenburg bialgebroid of a cleft Hopf Galois extension under the condition that the corresponding Hopf algebra is cocommutative and the image of the unital cocycle corresponding to this cleft Hopf Galois extension belongs to the centre of the coinvariant subalgebra. Moreover, we show that any Ehresmann Schauenburg bialgebroid of this kind is isomorphic to a 2-cocycle twist of the Ehresmann Schauenburg bialgebroid corresponding to a Hopf Galois extension without cocycle, where comodule algebra is an ordinary smash product of the coinvariant subalgebra and the Hopf algebra (i.e. $\C(B/#_{σ}H, H)\simeq \C(B#H, H)^{\tilde{σ}}$). We also study the theory in the case of a Galois object where the base is trivial but without requiring the Hopf algebra to be cocommutative.

The $ε-ε^β$ property in the isoperimetric problem with double density, and the regularity of isoperimetric sets - Pratelli, A.; Saracco, G. (Adv. Nonlinear Stud., 2020)

Weak formulation of elastodynamics in domains with growing cracks - Emanuele Tasso (Annali di Matematica Pura ed Applicata (1923 -), 2020)
In this paper, we formulate and study the system of elastodynamics on domains with arbitrary growing cracks. This includes homogeneous Neumann conditions on the crack sets and mixed general Dirichlet–Neumann conditions on the boundary. The only assumptions on the crack sets are to be $$(n-1)$$-rectifiable with finite surface measure, and increasing in the sense of set inclusions. In particular, they might be dense; hence, the weak formulation must fall outside the usual context of Sobolev spaces and Korn’s inequality. We prove existence of a solution for both the damped and undamped systems, while in the damped case we are also able to prove uniqueness and an energy balance.

Activation of a muscle as a mapping of stress–strain curves - Davide Riccobelli; D. Ambrosi (Extreme Mech. Lett., 2019)

Benamou–Brenier and duality formulas for the entropic cost on RCD*(K,N) spaces - Nicola Gigli; Luca Tamanini (Probability Theory and Related Fields, 2019)
In this paper we prove that, within the framework of $RCD^\star(K,N)$ spaces with $N<∞$, the entropic cost (i.e. the minimal value of the Schrödinger problem) admits:A threefold dynamical variational representation, in the spirit of the Benamou–Brenier formula for the Wasserstein distance; A Hamilton–Jacobi–Bellman dual representation, in line with Bobkov–Gentil–Ledoux and Otto–Villani results on the duality between Hamilton–Jacobi and continuity equation for optimal transport;A Kantorovich-type duality formula, where the Hopf–Lax semigroup is replaced by a suitable `entropic' counterpart.We thus provide a complete and unifying picture of the equivalent variational representations of the Schrödinger problem as well as a perfect parallelism with the analogous formulas for the Wasserstein distance. Riemannian manifolds with Ricci curvature bounded from below are a relevant class of $RCD^*(K,N)$ spaces and our results are new even in this setting.

BlackNUFFT: Modular customizable black box hybrid parallelization of type 3 NUFFT in 3D - Nicola Giuliani (Computer Physics Communications, 2019)
Many applications benefit from an efficient Discrete Fourier Transform (DFT) between arbitrarily spaced points. The Non Uniform Fast Fourier Transform reduces the computational cost of such operation from O(N2) to O(NlogN) exploiting gridding algorithms and a standard Fast Fourier Transform on an equi-spaced grid. The parallelization of the NUFFT of type 3 (between arbitrary points in space and frequency) still poses some challenges: we present a novel and flexible hybrid parallelization in a MPI-multithreaded environment exploiting existing HPC libraries on modern architectures. To ensure the reliability of the developed library, we exploit continuous integration strategies using Travis CI. We present performance analyses to prove the effectiveness of our implementation, possible extensions to the existing library, and an application of NUFFT type 3 to MRI image processing. Program summary Program Title: BlackNUFFT Program Files doi: http://dx.doi.org/10.17632/vxfj6x2p8x.1 Licensing provisions: LGPL Programming language: C++ External routines/libraries: deal.II , FFTW, PFFT Nature of problem: Provide a modular and extensible implementation of a parallel Non Uniform Fast Fourier Transform of type 3. Solution method: Use of hybrid shared distributed memory paradigm to achieve high level of efficiency. We exploit existing HPC library following best practices in scientific computing (as continuous integration via TravisCI) to reach higher complexities and guarantee the accuracy of the solution proposed.
Keywords: C++; Extensibility; FFT; Modularity; MPI; MRI image processing; NUFFT type 3; TBB

BladeX: Python Blade Morphing - Mahmoud Gadalla; Marco Tezzele; Andrea Mola; Gianluigi Rozza (The Journal of Open Source Software, 2019)

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

A complete data-driven framework for the efficient solution of parametric shape design and optimisation in naval engineering problems - Nicola Demo; Marco Tezzele; Andrea Mola; Gianluigi Rozza (8th International Conference on Computational Methods in Marine Engineering, MARINE 2019, 2019)
In the reduced order modeling (ROM) framework, the solution of a parametric partial differential equation is approximated by combining the high-fidelity solutions of the problem at hand for several properly chosen configurations. Examples of the ROM application, in the naval field, can be found in [31, 24]. Mandatory ingredient for the ROM methods is the relation between the high-fidelity solutions and the parameters. Dealing with geometrical parameters, especially in the industrial context, this relation may be unknown and not trivial (simulations over hand morphed geometries) or very complex (high number of parameters or many nested morphing techniques). To overcome these scenarios, we propose in this contribution an efficient and complete data-driven framework involving ROM techniques for shape design and optimization, extending the pipeline presented in [7]. By applying the singular value decomposition (SVD) to the points coordinates defining the hull geometry — assuming the topology is inaltered by the deformation —, we are able to compute the optimal space which the deformed geometries belong to, hence using the modal coefficients as the new parameters we can reconstruct the parametric formulation of the domain. Finally the output of interest is approximated using the proper orthogonal decomposition with interpolation technique. To conclude, we apply this framework to a naval shape design problem where the bulbous bow is morphed to reduce the total resistance of the ship advancing in calm water.

A continuous dependence result for a dynamic debonding model in dimension one - Filippo Riva (SISSA, 2019)
In this paper we address the problem of continuous dependence on initial and boundary data for a one-dimensional debonding model describing a thin ﬁlm peeled away from a substrate. The system underlying the process couples the weakly damped wave equation with a Griﬃth’s criterion which rules the evolution of the debonded region. We show that under general convergence assumptions on the data the corresponding solutions converge to the limit one with respect to diﬀerent natural topologies.

Convergence analysis of LSQR for compact operator equations - Noe Caruso; Paolo Novati (Linear Algebra and its Applications, 2019)
In this paper we analyze the behavior of the LSQR algorithm for the solution of compact operator equations in Hilbert spaces. We present results concerning existence of Krylov solutions and the rate of convergence in terms of an ℓp sequence where p depends on the summability of the singular values of the operator. Under stronger regularity requirements we also consider the decay of the error. Finally we study the approximation of the dominant singular values of the operator attainable with the bidiagonal matrices generated by the Lanczos bidiagonalization and the arising low rank approximations. Some numerical experiments on classical test problems are presented.
Keywords: Compact operator; Lanczos bidiagonalization; Linear ill-posed problem; LSQR

The deal.II Library, Version 9.1 - Daniel Arndt; Wolfgang Bangerth; Thomas C. Clevenger; Denis Davydov; Marc Fehling; Garcia-Sanchez, Daniel; Harper, Graham; Timo Heister; Luca Heltai; Martin Kronbichler; Maguire Kynch, Ross; Matthias Maier; et al. (Journal of Numerical Mathematics, 2019)

The deal.II Library, Version 9.1 - Daniel Arndt; Wolfgang Bangerth; Thomas C. Clevenger; Denis Davydov; Marc Fehling; Garcia-Sanchez, Daniel; Harper, Graham; Timo Heister; Luca Heltai; Martin Kronbichler; Maguire Kynch, Ross; Matthias Maier; et al. (Journal of Numerical Mathematics, 2019)
This paper provides an overview of the new features of the finite element library deal.II, version 9.1.

Differential structure associated to axiomatic Sobolev spaces - Nicola Gigli; Enrico Pasqualetto (Expositiones Mathematicae, 2019)
The aim of this note is to explain in which sense an axiomatic Sobolev space over a general metric measure space (à la Gol’dshtein–Troyanov) induces – under suitable locality assumptions – a first-order differential structure.
Keywords: Axiomatic Sobolev space; Cotangent module; Locality of differentials

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 & 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.

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.

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

KAM theory for partial differential equations - Massimiliano Berti (Anal. Theory Appl., 2019)

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.

Long time dynamics of Schrödinger and wave equations on flat tori - Berti, M.; Alberto Maspero (J. Differential Equations, 2019)

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; Atoms - Dirk Hundertmark; Michal Jex; Markus Lange (arXiv:1908.05016, 2019)

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)

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 (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

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.

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 (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.

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.

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)

Almost global solutions of capillary-gravity water waves equations on the circle - Massimiliano Berti; Delort, Jean-Marc (Lecture Notes of the Unione Matematica Italiana, 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)

KAM for gravity water waves in finite depth - Baldi, Pietro; Massimiliano Berti; Haus, Emanuele; Montalto, Riccardo (Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 2018)

Large KAM tori for perturbations of the defocusing NLS equation - Massimiliano Berti; Kappeler, Thomas; Montalto, Riccardo (Astérisque, 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)