Rigidity and trace properties of divergence-measure vector fields - Leonardi, G. P.; Saracco, G. (Adv. Calc. Var.)

Breaking the left-right symmetry in fluttering artificial cilia that perform nonreciprocal oscillations - Boiardi, Ariel Surya; Marchello, Roberto (2024)
Recent investigations on active materials have introduced a new paradigm for soft robotics by showing that a complex response can be obtained from simple stimuli by harnessing dynamic instabilities. In particular, polyelectrolyte hydrogel filaments actuated by a constant electric field have been shown to exhibit self-sustained oscillations as a consequence of flutter instability. Owing to the nonreciprocal nature of the emerging oscillations, these artificial cilia are able to generate flows along the stimulus. Building upon these findings, in this paper we propose a design strategy to break the left-right symmetry in the generated flows, by endowing the filament with a natural curvature at the fabrication stage. We develop a mathematical model based on morphoelastic rod theory to characterize the stability of the equilibrium configurations of the filament, proving the persistence of flutter instability. We show that the emerging oscillations are nonreciprocal and generate thrust at an angle with the stimulus. The results we find at the level of the single cilium open new perspectives on the possible applications of artificial ciliary arrays in soft robotics and microfluidics.

A non local model for cell migration in response to mechanical stimuli - Marchello, Roberto; Colombi, Annachiara; Preziosi, Luigi; Giverso, Chiara (Mathematical Biosciences, 2024)

Optimisation–Based Coupling of Finite Element Model and Reduced Order Model for Computational Fluid Dynamics - Ivan Prusak; Davide Torlo; Monica Nonino; Gianluigi Rozza (2024)

Applicable Methodologies for the Mass Transfer Phenomenon in Tumble Dryers: A Review - Sajad Salavatidezfouli; Sajad Hajisharifi; Michele Girfoglio; Giovanni Stabile; Gianluigi Rozza (2023)
Tumble dryers offer a fast and convenient way of drying textiles independent of weather conditions and therefore are frequently used in ordinary households. However, artificial drying of textiles consumes considerable amounts of energy, approximately 8.2 percent of the residential electricity consumption is for drying of textiles in northern European countries (Cranston et al., 2019). Several authors have investigated the aspects of the clothes drying cycle with experimental and numerical methods to understand and improve the process. The first turning point study on understanding the physics of evaporation for tumble dryers was presented by Lambert et al. (1991) in the early 90s. With the aid of Chilton_Colburn analogy, they introduced the concept of area-mass transfer coefficient to address evaporation rate. Afterwards, several experimental or numerical studies were published based on this concept, and furthermore, the model was then developed into 0-dimensional (Deans, 2001) and 1-dimensional (Wei et al., 2017) to gain more accuracy. The evaporation rate is considered to be the main system parameter for dryers with which other performance parameters including drying time, effectiveness, moisture content and efficiency can be estimated. More recent literature focused on utilizing dimensional analysis or image processing techniques to correlate drying indices with system parameters. However, the validity of these regressed models is machine-specific, and hence, cannot be generalized yet. All the previous models for estimating the evaporation rate in tumble dryers are discussed. The review of the related literature showed that all of the previous models for the prediction of the evaporation rate in the clothes dryers have some limitations in terms of accuracy and applicability.

Benjamin-Feir Instability of Stokes Waves in Finite Depth - Berti, Massimiliano; Maspero, Alberto; Ventura, Paolo (ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS, 2023)
Whitham and Benjamin predicted in 1967 that small-amplitude periodic traveling Stokes waves of the 2d-gravity water waves equations are linearly unstable with respect to long-wave perturbations, if the depth h is larger than a critical threshold h(WB) approximate to 1.363. In this paper, we completely describe, for any finite value of h > 0, the four eigenvalues close to zero of the linearized equations at the Stokes wave, as the Floquet exponent mu is turned on. We prove, in particular, the existence of a unique depth h(WB), which coincides with the one predicted by Whitham and Benjamin, such that, for any 0 < h < h(WB), the eigenvalues close to zero are purely imaginary and, for any h > h(WB), a pair of non-purely imaginary eigenvalues depicts a closed figure ``8'', parameterized by the Floquet exponent. As h -> h(WB)(+) the ``8'' collapses to the origin of the complex plane. The complete bifurcation diagram of the spectrum is not deduced as in deep water, since the limits h -> +infinity (deep water) and mu -> 0 (long waves) do not commute. In finite depth, the four eigenvalues have all the same size O(mu), unlike in deep water, and the analysis of their splitting is much more delicate, requiring, as a new ingredient, a non-perturbative step of block-diagonalization. Along the whole proof, the explicit dependence of the matrix entries with respect to the depth h is carefully tracked.

On the distribution of the van der Corput sequences - Beretti, Thomas (Archiv der Mathematik, 2023)
For an integer $p\ge 2$, let $\{x_n\}_{n\in {\mathbb {N}}}\subset {\mathbb {T}}$ be the p-adic van der Corput sequence. For intervals $[0,\alpha )\subset {\mathbb {T}}$ and for positive integers N, consider the geometrically-shifted discrepancy function $D_{p,N,\alpha }(t)=\sum _{n=0}^{N-1}\mathcal {X}_{[0,\alpha )}(x_n+t)-N\alpha$. In this paper, we give a characterization of the asymptotic behavior of $\Vert D_{p,N,\alpha }(\cdot )\Vert _{L^2({\mathbb {T}})}$ for $N\rightarrow \infty$that depends on the p-adic expansion of $\alpha$.
Keywords: Diaphony; Discrepancy; Uniform distribution; Van der Corput sequence

Flutter instability in solids and structures, with a view on biomechanics and metamaterials - Davide Bigoni; Francesco Dal Corso; Oleg N. Kirillov; Diego Misseroni; Giovanni Noselli; Andrea Piccolroaz (Proceedings of the Royal Society A, 2023)
The phenomenon of oscillatory instability called ‘flutter’ was observed in aeroelasticity and rotor dynamics about a century ago. Driven by a series of applications involving non-conservative elasticity theory at different physical scales, ranging from nanomechanics to the mechanics of large space structures and including biomechanical problems of motility and growth, research on flutter is experiencing a new renaissance. A review is presented of the most notable applications and recent advances in fundamentals, both theoretical and experimental aspects, of flutter instability and Hopf bifurcation. Open problems, research gaps and new perspectives for investigations are indicated.

A general splitting principle on RCD spaces and applications to spaces with positive spectrum - Nicola Gigli; Fabio Marconi (2023)
In this paper we develop a general `analytic' splitting principle for RCD spaces: we show that if there is a function with suitable Laplacian and Hessian, then the space is (isomorphic to) a warped product. Our result covers most of the splitting-like results currently available in the literature about RCD spaces. We then apply it to extend to the non-smooth category some structural property of Riemannian manifolds obtained by Li and Wang.

On the Minimal Number of Solutions of the Equation φ(n+k)=Mφ(n), M=1,2 - Matteo Ferrari; Lorenzo Sillari (Journal of Integer Sequences, 2023)
We fix a positive integer $k$ and look for solutions $n \in \mathbb{N}$ of the equations $φ(n + k) = φ(n)$ and $φ(n + k) = 2 φ(n)$. For $k \le 12 · 10^{100}$, we prove that Fermat primes can be used to build five solutions for the first equation when $k$ is even, and five for the second one when $k$ is odd. Furthermore, for $k \le 4 · 10^{58}$, we show that for the second equation there are at least three solutions when $k$ is even. Our work increases the previously known minimal number of solutions for both equations.
Keywords: Euler’s phi function

Nonreciprocal oscillations of polyelectrolyte gel filaments subject to a steady and uniform electric field - Giancarlo Cicconofri; Valentina Damioli; Giovanni Noselli (Journal of the Mechanics and Physics of Solids, 2023)
Soft actuators typically require time-varying or spatially modulated control to be operationally effective. The scope of the present paper is to show, theoretically and experimentally, that a natural way to overcome this limitation is to exploit mechanical instabilities. We report experiments on active filaments of polyelectrolyte (PE) gels subject to a steady and uniform electric field. A large enough intensity of the field initiates the motion of the active filaments, leading to periodic oscillations. We develop a mathematical model based on morphoelasticity theory for PE gel filaments beating in a viscous fluid, and carry out the stability analysis of the governing equations to show the emergence of flutter and divergence instabilities for suitable values of the system’s parameters. We confirm the results of the stability analysis with numerical simulations for the nonlinear equations of motion to show that such instabilities may lead to periodic self-sustained oscillations, in agreement with experiments. The key mechanism that underlies such behaviour is the capability of the filament to undergo active shape changes depending on its local orientation relative to the external electric field, in striking similarity with gravitropism, the mechanism that drives shape changes in plants via differential growth induced by gravity. Interestingly, the resulting oscillations are nonreciprocal in nature, and hence able to generate thrust and directed flow at low Reynolds number. The exploitation of mechanical instabilities in soft actuators represents a new avenue for the advancement in engineering design in fields such as micro-robotics and micro-fluidics.

An optimisation-based domain-decomposition reduced order model for parameter-dependent non-stationary fluid dynamics problems - Ivan Prusak; Davide Torlo; Monica Nonino; Gianluigi Rozza (2023)

An optimisation–based domain–decomposition reduced order model for the incompressible Navier-Stokes equations - Ivan Prusak; Monica Nonino; Davide Torlo; Francesco Ballarin; Gianluigi Rozza (2023)
The aim of this work is to present a model reduction technique in the framework of optimal control problems for partial differential equations. We combine two approaches used for reducing the computational cost of the mathematical numerical models: domain–decomposition (DD) methods and reduced–order modelling (ROM). In particular, we consider an optimisation–based domain–decomposition algorithm for the parameter–dependent stationary incompressible Navier–Stokes equations. Firstly, the problem is described on the subdomains coupled at the interface and solved through an optimal control problem, which leads to the complete separation of the subdomain problems in the DD method. On top of that, a reduced model for the obtained optimal–control problem is built; the procedure is based on the Proper Orthogonal Decomposition technique and a further Galerkin projection. The presented methodology is tested on two fluid dynamics benchmarks: the stationary backward–facing step and lid-driven cavity flow. The numerical tests show a significant reduction of the computational costs in terms of both the problem dimensions and the number of optimisation iterations in the domain–decomposition algorithm.
Keywords: Computational fluid dynamics; Domain decomposition; Optimal control; Proper orthogonal decomposition; Reduced order modelling

Properties of Mixing BV Vector Fields - Stefano Bianchini; Martina Zizza (Communications in Mathematical Physics, 2023)

Relaxed area of $0$-homogeneous maps in the strict $BV$-convergence - Carano, Simone (2023)

Relaxed area of graphs of piecewise Lipschitz maps in the strict $BV$-convergence - Bellettini, Giovanni; Carano, Simone; Scala, Riccardo (2023)

Stokes waves at the critical depth are modulational unstable - Berti, Massimiliano; Maspero, Alberto; Ventura, Paolo (2023)

On the analyticity of the Dirichlet-Neumann operator and Stokes waves - Berti, Massimiliano; Maspero, Alberto; Ventura, Paolo (RENDICONTI LINCEI-MATEMATICA E APPLICAZIONI, 2022)
We prove an analyticity result for the Dirichlet-Neumann operator under space periodic boundary conditions in any dimension in an unbounded domain with infinite depth. We derive an analytic bifurcation result of analytic Stokes waves-i.e., space periodic traveling solutions-of the water waves equations in deep water.
Keywords: Bifurcation theory; Dirichlet-Neumann operator; traveling waves; water waves

A behavioral change model to assess vaccination-induced relaxation of social distancing during an epidemic - Buonomo, Bruno; Della Marca, Rossella; Sintayehu Sharbayta, Sileshi (Journal of Biological Systems, 2022)
The success of mass vaccination campaigns may be jeopardized by human risky behaviors. For example, high level of vaccination coverage may induce early relaxation of social distancing. In this paper, we focus on the mutual influence between the decline in prevalence, due to the rise in the overall immunization coverage, and the consequent decrease in the compliance to social distancing measures. We consider an epidemic model where both the vaccination rate and the disease transmission rate are influenced by human behavior, which in turn depends on the current and past information about the spread of the disease. We highlight the impact of the information-related parameters on the transient and asymptotic behavior of the system that is on the early stage of the epidemic and its final outcome. Among the main results, we evidence that sustained oscillations may be triggered by the behavioral memory in the prevalence-dependent vaccination rate. However, the relaxation of social distancing may induce a switch from a cyclic regime to damped oscillations.

Benjamin-Feir instability of Stokes waves - Berti, Massimiliano; Maspero, Alberto; Ventura, Paolo (RENDICONTI LINCEI-MATEMATICA E APPLICAZIONI, 2022)
We present the recent results in Berti et al. [Invent. Math. (2022), to appear] regarding the Benjamin-Feir instability of small amplitude Stokes waves in deep water. We completely describe the behavior of the four eigenvalues close to zero of the linearized water waves equations at the Stokes solution, as the Floquet exponent is turned on, proving the conjecture that a pair of non-purely imaginary eigenvalues depicts a closed figure ``8'', in full agreement with numerical simulations.
Keywords: Kato perturbation theory; modulational instability; traveling waves; water waves

A comparison of reduced-order modeling approaches using artificial neural networks for PDEs with bifurcating solutions - Martin W. Hess; Annalisa Quaini; Gianluigi Rozza (ETNA - Electronic Transactions on Numerical Analysis, 2022)

Data-Driven Enhanced Model Reduction for Bifurcating Models in Computational Fluid Dynamics - Martin W. Hess; Annalisa Quaini; Gianluigi Rozza (2022)

A Data-Driven Surrogate Modeling Approach for Time-Dependent Incompressible Navier-Stokes Equations with Dynamic Mode Decomposition and Manifold Interpolation - Martin W. Hess; Annalisa Quaini; Gianluigi Rozza (2022)

Doubly Intermittent Full Branch Maps with Critical Points and Singularities - Douglas Coates; Stefano Luzzatto; Muhammad Mubarak (2022)
Keywords: 37E05; Dynamical Systems (math.DS); FOS: Mathematics

Doubly Intermittent Maps with Critical Points, Unbounded Derivatives and Regularly Varying Tail - Muhammad Mubarak; Tanja I. Schindler (2022)
Keywords: 37A05; 37A25; 37A50; Dynamical Systems (math.DS); FOS: Mathematics

Driving bifurcating parametrized nonlinear PDEs by optimal control strategies: application to Navier–Stokes equations with model order reduction - Federico Pichi; Maria Strazzullo; F. Ballarin; Gianluigi Rozza (ESAIM: M2AN, 2022)

An example of a weakly mixing BV vector field which is not strongly mixing - Martina Zizza (2022)

Full description of Benjamin-Feir instability of stokes waves in deep water - Berti, Massimiliano; Maspero, Alberto; Ventura, Paolo (2022)
Small-amplitude, traveling, space periodic solutions –called Stokes waves– of the 2 dimensional gravity water waves equations in deep water are linearly unstable with respect to long-wave perturbations, as predicted by Benjamin and Feir in 1967. We completely describe the behavior of the four eigenvalues close to zero of the linearized equations at the Stokes wave, as the Floquet exponent is turned on. We prove in particular the conjecture that a pair of non-purely imaginary eigenvalues depicts a closed figure “8”, parameterized by the Floquet exponent, in full agreement with numerical simulations. Our new spectral approach to the Benjamin-Feir instability phenomenon uses a symplectic version of Kato’s theory of similarity transformation to reduce the problem to determine the eigenvalues of a $ 4 × 4 $ complex Hamiltonian and reversible matrix. Applying a procedure inspired by KAM theory, we block-diagonalize such matrix into a pair of $2 × 2 $ Hamiltonian and reversible matrices, thus obtaining the full description of its eigenvalues.

A Gradient Flow Equation for Optimal Control Problems With End-point Cost - Alessandro Scagliotti (2022)
In this paper, we consider a control system of the form $\dot x = F(x)u$, linear in the control variable u. Given a fixed starting point, we study a finite-horizon optimal control problem, where we want to minimize a weighted sum of an end-point cost and the squared 2-norm of the control. This functional induces a gradient flow on the Hilbert space of admissible controls, and we prove a convergence result by means of the Lojasiewicz-Simon inequality. Finally, we show that, if we let the weight of the end-point cost tend to infinity, the resulting family of functionals is Γ-convergent, and it turns out that the limiting problem consists in joining the starting point and a minimizer of the end-point cost with a horizontal length-minimizer path.

Indeterminacy estimates, eigenfunctions and lower bounds on Wasserstein distances - Nicolò De Ponti; Sara Farinelli (2022)
In the paper we prove two inequalities in the setting of $$\mathsf {RCD}(K,∞ )$$spaces using similar techniques. The first one is an indeterminacy estimate involving the p-Wasserstein distance between the positive part and the negative part of an $$L^{∞ }$$function and the measure of the interface between the positive part and the negative part. The second one is a conjectured lower bound on the p-Wasserstein distance between the positive and negative parts of a Laplace eigenfunction.

Isoperimetric inequality for Finsler manifolds with non-negative Ricci curvature - Manini, Davide (2022)
We prove a sharp isoperimetric inequality for Finsler manifolds having non-negative Ricci curvature and Euclidean volume growth. We also prove a rigidity result for this inequality, under the additional hypotheses of boundedness of the isoperimetric set and the finite reversibility of the space. An application to the weighed anisotropic isoperimetric problem in Euclidean cones is presented.

Isoperimetric inequality in noncompact MCP spaces - Cavalletti, Fabio; Manini, Davide (Proc. Am. Math. Soc., 2022)
We prove a sharp isoperimetric inequality for the class of metric measure spaces verifying the synthetic Ricci curvature lower bounds Measure Contraction property (MCP(0, N)) and having Euclidean volume growth at infinity. We avoid the classical use of the Brunn-Minkowski inequality, not available for MCP(0, N), and of the PDE approach, not available in the singular setting. Our approach will be carried over by using a scaling limit of localization.

Kernel-based active subspaces with application to computational fluid dynamics parametric problems using discontinuous Galerkin method - Francesco Romor; Marco Tezzele; Andrea Lario; Gianluigi Rozza (International Journal for Numerical Methods in Engineering, 2022)

Long-time stability of the quantum hydrodynamic system on irrational tori - Roberto Feola; Felice Iandoli; Federico Murgante (Mathematics in Engineering, 2022)
Keywords: Euler-Korteweg; irrational tori; long time stability; QHD system; Small divisors

Mathematical modelling of oscillating patterns for chronic autoimmune diseases - Rossella Della Marca; Maria da Piedade Machado Ramos; Carolina Ribeiro; Ana Jacinta Soares (Mathematical Methods in the Applied SciencesMathematical Methods in the Applied SciencesMath Meth Appl Sci, 2022)
Many autoimmune diseases are chronic in nature, so that in general, patients experience periods of recurrence and remission of the symptoms characterizing their specific autoimmune ailment. In order to describe this very important feature of autoimmunity, we construct a mathematical model of kinetic type describing the immune system cellular interactions in the context of autoimmunity exhibiting recurrent dynamics. The model equations constitute a nonlinear system of integro-differential equations with quadratic terms that describe the interactions between self-antigen presenting cells, self-reactive T cells, and immunosuppressive cells. We consider a constant input of self-antigen presenting cells, due to external environmental factors that are believed to trigger autoimmunity in people with predisposition for this condition. We also consider the natural death of all cell populations involved in our model, caused by their interaction with cells of the host environment. We derive the macroscopic analogue and show positivity and well-posedness of the solution and then we study the equilibria of the corresponding dynamical system and their stability properties. By applying dynamical system theory, we prove that steady oscillations may arise due to the occurrence of a Hopf bifurcation. We perform some numerical simulations for our model, and we observe a recurrent pattern in the solutions of both the kinetic description and its macroscopic analogue, which leads us to conclude that this model is able to capture the chronic behaviour of many autoimmune diseases.
Keywords: autoimmune diseases; cellular interactions; Dynamical systems; Hopf bifurcation; kinetic theory; mathematical biology

Model hierarchies and higher-order discretisation of time-dependent thin-film free boundary problems with dynamic contact angle - Dirk Peschka; Luca Heltai (Journal of Computational Physics, 2022)

Model order reduction for bifurcating phenomena in fluid-structure interaction problems - Moaad Khamlich; Federico Pichi; Gianluigi Rozza (International Journal for Numerical Methods in FluidsInternational Journal for Numerical Methods in FluidsInt J Numer Meth Fluids, 2022)
Abstract This work explores the development and the analysis of an efficient reduced order model for the study of a bifurcating phenomenon, known as the Coand? effect, in a multi-physics setting involving fluid and solid media. Taking into consideration a fluid-structure interaction problem, we aim at generalizing previous works towards a more reliable description of the physics involved. In particular, we provide several insights on how the introduction of an elastic structure influences the bifurcating behavior. We have addressed the computational burden by developing a reduced order branch-wise algorithm based on a monolithic proper orthogonal decomposition. We compared different constitutive relations for the solid, and we observed that a nonlinear hyper-elastic law delays the bifurcation w.r.t. the standard model, while the same effect is even magnified when considering linear elastic solid.
Keywords: Bifurcation theory; Coandă effect; continuum mechanics; fluid dynamics; monolithic method; parametrized fluid-structure interaction problem; Proper orthogonal decomposition; reduced order modeling

Model Reduction Using Sparse Polynomial Interpolation for the Incompressible Navier-Stokes Equations - Martin W. Hess; Gianluigi Rozza (2022)

The Neural Network shifted-proper orthogonal decomposition: A machine learning approach for non-linear reduction of hyperbolic equations - Davide Papapicco; Nicola Demo; Michele Girfoglio; Giovanni Stabile; Gianluigi Rozza (Computer Methods in Applied Mechanics and Engineering, 2022)
Models with dominant advection always posed a difficult challenge for projection-based reduced order modelling. Many methodologies that have recently been proposed are based on the pre-processing of the full-order solutions to accelerate the Kolmogorov N−width decay thereby obtaining smaller linear subspaces with improved accuracy. These methods however must rely on the knowledge of the characteristic speeds in phase space of the solution, limiting their range of applicability to problems with explicit functional form for the advection field. In this work we approach the problem of automatically detecting the correct pre-processing transformation in a statistical learning framework by implementing a deep-learning architecture. The purely data-driven method allowed us to generalise the existing approaches of linear subspace manipulation to non-linear hyperbolic problems with unknown advection fields. The proposed algorithm has been validated against simple test cases to benchmark its performances and later successfully applied to a multiphase simulation. © 2022 Elsevier B.V.
Keywords: Advection; Computational complexity; Deep neural network; Deep neural networks; Linear subspace; Multiphase simulations; Non linear; Nonlinear hyperbolic equation; Partial differential equations; Phase space methods; Pre-processing; Principal component analysis; reduced order modeling; Reduced order modelling; Reduced-order model; Shifted-POD

The $N$-Link Swimmer in Three Dimensions: Controllability and Optimality Results - Marchello, Roberto; Morandotti, Marco; Shum, Henry; Zoppello, Marta (2022)
The controllability of a fully three-dimensional $N$-link swimmer is studied. After deriving the equations of motion in a low Reynolds number fluid by means of Resistive Force Theory, the controllability of the minimal 2-link swimmer is tackled using techniques from Geometric Control Theory. The shape of the 2-link swimmer is described by two angle parameters. It is shown that the associated vector fields that govern the dynamics generate, via taking their Lie brackets, all eight linearly independent directions in the combined configuration and shape space, leading to controllability; the swimmer can move from any starting configuration and shape to any target configuration and shape by operating on the two shape variables. The result is subsequently extended to the $N$-link swimmer. Finally, the minimal time optimal control problem and the minimization of the power expended are addressed and a qualitative description of the optimal strategies is provided.

Optimal design of planar shapes with active materials - Dario Andrini; Giovanni Noselli; Alessandro Lucantonio (Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2022)
Active materials have emerged as valuable candidates for shape morphing applications, where a body reconfiguration is achieved upon triggering its active response. Given a desired shape change, a natural question is to compare different morphing mechanisms to select the most effective one with respect to an optimality criterion. We introduce an optimal control problem to determine the active strains suitable to attain a target equilibrium shape while minimizing the complexity of the activation. Specifically, we discuss the planar morphing of active, hyperelastic bodies in the absence of external forces and exploit the notion of target metric to encompass a broad set of active materials in a unifying approach. For the case of affine shape changes, we derive explicit conditions on the body reference configuration for the optimality of homogeneous target metrics. More complex shape changes are analysed via numerical simulations to explore the impact on optimal solutions of different objective functionals inspired by features of existing materials. We show how stresses arising from incompatibilities contribute to reduce the complexity of the controls. We believe that our approach may be exploited for the optimal design of active systems and may contribute to gather insight into the morphing strategies of biological systems.

A piecewise conservative method for unconstrained convex optimization - Alessandro Scagliotti; Colli Franzone, P. (2022)
We consider a continuous-time optimization method based on a dynamical system, where a massive particle starting at rest moves in the conservative force field generated by the objective function, without any kind of friction. We formulate a restart criterion based on the mean dissipation of the kinetic energy, and we prove a global convergence result for strongly-convex functions. Using the Symplectic Euler discretization scheme, we obtain an iterative optimization algorithm. We have considered a discrete mean dissipation restart scheme, but we have also introduced a new restart procedure based on ensuring at each iteration a decrease of the objective function greater than the one achieved by a step of the classical gradient method. For the discrete conservative algorithm, this last restart criterion is capable of guaranteeing a qualitative convergence result. We apply the same restart scheme to the Nesterov Accelerated Gradient (NAG-C), and we use this restarted NAG-C as benchmark in the numerical experiments. In the smooth convex problems considered, our method shows a faster convergence rate than the restarted NAG-C. We propose an extension of our discrete conservative algorithm to composite optimization: in the numerical tests involving non-strongly convex functions with $$\ell ^1$$-regularization, it has better performances than the well known efficient Fast Iterative Shrinkage-Thresholding Algorithm, accelerated with an adaptive restart scheme.

A POD-Galerkin reduced order model for the Navier–Stokes equations in stream function-vorticity formulation - Michele Girfoglio; Annalisa Quaini; Gianluigi Rozza (2022)
We develop a Proper Orthogonal Decomposition (POD)-Galerkin based Reduced Order Model (ROM) for the efficient numerical simulation of the parametric Navier–Stokes equations in the stream function-vorticity formulation. Unlike previous works, we choose different reduced coefficients for the vorticity and stream function fields. In addition, for parametric studies we use a global POD basis space obtained from a database of time dependent full order snapshots related to sample points in the parameter space. We test the performance of our ROM strategy with the well-known vortex merger benchmark and a more complex case study featuring the geometry of the North Atlantic Ocean. Accuracy and efficiency are assessed for both time reconstruction and physical parametrization.
Keywords: Galerkin projection; Navier–Stokes equations; Proper orthogonal decomposition; Reduced order model; Stream function-vorticity formulation

Projection based semi–implicit partitioned Reduced Basis Method for non parametrized and parametrized Fluid–Structure Interaction problems - Monica Nonino; Francesco Ballarin; Gianluigi Rozza; Yvon Maday (2022)
The goal of this manuscript is to present a partitioned Model Order Reduction method that is based on a semi-implicit projection scheme to solve multiphysics problems. We implement a Reduced Order Method based on a Proper Orthogonal Decomposition, with the aim of addressing both time-dependent and time-dependent, parametrized Fluid-Structure Interaction problems, where the fluid is incompressible and the structure is thick and two dimensional.

A Proper Orthogonal Decomposition Approach for Parameters Reduction of Single Shot Detector Networks - Laura Meneghetti; Nicola Demo; Gianluigi Rozza (2022 IEEE International Conference on Image Processing (ICIP), 2022)

The relaxed area of $S^1$-valued singular maps in the strict $BV$-convergence - Bellettini, Giovanni; Carano, Simone; Scala, Riccardo (ESAIM: Control, Optimization and Calculus of Variations, 2022)

Rigidities of Isoperimetric inequality under nonnegative Ricci curvature - Cavalletti, Fabio; Manini, Davide (2022)
The sharp isoperimetric inequality for non-compact Riemannian manifolds with non-negative Ricci curvature and Euclidean volume growth has been obtained in increasing generality with different approaches in a number of contributions [arXiv:1812.05022, arXiv:2012.09490, arXiv:2009.13717, arXiv:2103.08496] culminated by Balogh and Kristaly [arXiv:2012.11862] covering also m.m.s.'s verifying the non-negative Ricci curvature condition in the synthetic sense of Lott, Sturm and Villani. In sharp contrast with the compact case of positive Ricci curvature, for a large class of spaces including weighted Riemannian manifolds, no complete characterisation of the equality cases is present in the literature. The scope of this note is to settle this problem by proving, in the same generality of [arXiv:2012.11862], that the equality in the isoperimetric inequality can be attained only by metric balls. Whenever this happens the space is forced, in a measure theoretic sense, to be a cone. Our result applies to different frameworks yielding as corollaries new rigidity results: it extend to weighted Riemannian manifold the rigidity results of [arXiv:2009.13717], it extend to general RCD spaces the rigidity results of [arXiv:2201.04916] and finally applies also to the Euclidean setting by proving that that optimisers in the anisotropic and weighted isoperimetric inequality for Euclidean cones are necessarily the Wulff shapes.

An SIR–like kinetic model tracking individuals' viral load - Rossella Della Marca; Nadia Loy; Andrea Tosin (Networks and Heterogeneous Media, 2022)

The \textttdeal.II Library, Version 9.4 - Daniel Arndt; Wolfgang Bangerth Marco Feder; Marc Fehling; Rene Gassmöller; Timo Heister; Luca Heltai; Martin Kronbichler; Matthias Maier; Peter Munch; Jean-Paul Pelteret; Simon Sticko; Bruno Turcksin; et al. (Journal of Numerical Mathematics, 2022)

Variational Approach to Fluid–Structure Interaction via GENERIC - Dirk Peschka; Andrea Zafferi; Luca Heltai; Marita Thomas (Journal of Non-Equilibrium Thermodynamics, 2022)

Vibration Analysis of Piezoelectric Kirchhoff-Love shells based on Catmull-Clark Subdivision Surfaces - Zhaowei Liu; Andrew McBride; Prashant Saxena; Luca Heltai; Yilin Qu; Paul Steinmann (International Journal for Numerical Methods in Engineering, 2022)

Adaptive non-hierarchical Galerkin methods for parabolic problems with application to moving mesh and virtual element methods - Andrea Cangiani; E.H. Georgoulis; Sutton, Oliver J. (Mathematical Models and Methods in Applied Sciences, 2021)
We present a posteriori error estimates for inconsistent and non-hierarchical Galerkin methods for linear parabolic problems, allowing them to be used in conjunction with very general mesh modification for the first time. We treat schemes which are non-hierarchical in the sense that the spatial Galerkin spaces between time-steps may be completely unrelated from one another. The practical interest of this setting is demonstrated by applying our results to finite element methods on moving meshes and using the estimators to drive an adaptive algorithm based on a virtual element method on a mesh of arbitrary polygons. The a posteriori error estimates, for the error measured in the L2(H1) and L∞(L2) norms, are derived using the elliptic reconstruction technique in an abstract framework designed to precisely encapsulate our notion of inconsistency and non-hierarchicality and requiring no particular compatibility between the computational meshes used on consecutive time-steps, thereby significantly relaxing this basic assumption underlying previous estimates.

Approximation of the spectral fractional powers of the Laplace-Beltrami Operator - Bonito, Andrea; Wenyu Lei (arXiv preprint arXiv:2101.05141, 2021)

An artificial neural network approach to bifurcating phenomena in computational fluid dynamics - Federico Pichi; Francesco Ballarin; Gianluigi Rozza; Jan S Hesthaven (2021)

Asymptotic approach to a rotational Taylor swimming sheet - Giovanni Corsi (Comptes Rendus. Mécanique, 2021)

ATHENA: Advanced Techniques for High dimensional parameter spaces to Enhance Numerical Analysis - Francesco Romor; Marco Tezzele; Gianluigi Rozza (Software Impacts, 2021)

BisPy: Bisimulation in Python - Francesco Andreuzzi (Journal of Open Source Software, 2021)

A CERTIFIED REDUCED BASIS Method FOR LINEAR PARAMETRIZED PARABOLIC OPTIMAL CONTROL PROBLEMS IN SPACE-TIME FORMULATION - Maria Strazzullo; Francesco Ballarin; Gianluigi Rozza (2021)

On the comparison of LES data-driven reduced order approaches for hydroacoustic analysis - Mahmoud Gadalla; Marta Cianferra; Marco Tezzele; Giovanni Stabile; Andrea Mola; Gianluigi Rozza (Computers & Fluids, 2021)
In this work, Dynamic Mode Decomposition (DMD) and Proper Orthogonal Decomposition (POD) methodologies are applied to hydroacoustic dataset computed using Large Eddy Simulation (LES) coupled with Ffowcs Williams and Hawkings (FWH) analogy. First, a low-dimensional description of the flow fields is presented with modal decomposition analysis. Sensitivity towards the DMD and POD bases truncation rank is discussed, and extensive dataset is provided to demonstrate the ability of both algorithms to reconstruct the flow fields with all the spatial and temporal frequencies necessary to support accurate noise evaluation. Results show that while DMD is capable to capture finer coherent structures in the wake region for the same amount of employed modes, reconstructed flow fields using POD exhibit smaller magnitudes of global spatiotemporal errors compared with DMD counterparts. Second, a separate set of DMD and POD modes generated using half the snapshots is employed into two data-driven reduced models respectively, based on DMD mid cast and POD with Interpolation (PODI). In that regard, results confirm that the predictive character of both reduced approaches on the flow fields is sufficiently accurate, with a relative superiority of PODI results over DMD ones. This infers that, discrepancies induced due to interpolation errors in PODI is relatively low compared with errors induced by integration and linear regression operations in DMD, for the present setup. Finally, a post processing analysis on the evaluation of FWH acoustic signals utilizing reduced fluid dynamic fields as input demonstrates that both DMD and PODI data-driven reduced models are efficient and sufficiently accurate in predicting acoustic noises.
Keywords: Dynamic mode decomposition; Ffowcs Williams and Hawkings; Hydroacoustics; Large eddy simulation; Model reduction; Proper orthogonal decomposition

Consistency of the full and reduced order models for Evolve-Filter-Relax Regularization of Convection-Dominated, Marginally-Resolved Flows - Maria Strazzullo; Michele Girfoglio; Francesco Ballarin; T. Iliescu; Gianluigi Rozza (2021)

Data-driven parameter and model order reduction for industrial optimisation problems with applications in naval engineering - Marco Tezzele (SISSA - International School for Advanced Studies, 2021)

A data-driven partitioned approach for the resolution of time-dependent optimal control problems with dynamic mode decomposition - Eleonora Donadini; Maria Strazzullo; Marco Tezzele; Gianluigi Rozza (2021)

The deal.II Library, Version 9.3 - Daniel Arndt; Wolfgang Bangerth; Bruno Blais; Marc Fehling; Rene Gassmöller; Timo Heister; Luca Heltai; Uwe Köcher; Martin Kronbichler; Matthias Maier; Peter Munch; Jean-Paul Pelteret; et al. (Journal of Numerical Mathematics, 2021)

Deep Learning Approximation of Diffeomorphisms via Linear-Control Systems - Alessandro Scagliotti (2021)
Keywords: FOS: Computer and information sciences; FOS: Mathematics; Machine Learning (cs.LG); Optimization and Control (math.OC)

A Differential Perspective on Gradient Flows on CAT(K)-Spaces and Applications - Nicola Gigli; Francesco Nobili (2021)
We review the theory of Gradient Flows in the framework of convex and lower semicontinuous functionals on $$CAT (κ )$$-spaces and prove that they can be characterized by the same differential inclusion $$y_t'\in -\partial ^-E (y_t)$$one uses in the smooth setting and more precisely that $$y_t'$$selects the element of minimal norm in $$-\partial ^-E (y_t)$$. This generalizes previous results in this direction where the energy was also assumed to be Lipschitz. We then apply such result to the Korevaar–Schoen energy functional on the space of $$L^2$$and CAT(0) valued maps: we define the Laplacian of such $$L^2$$map as the element of minimal norm in $$-\partial ^-E (u)$$, provided it is not empty. The theory of gradient flows ensures that the set of maps admitting a Laplacian is $$L^2$$-dense. Basic properties of this Laplacian are then studied.

A Dimensionality Reduction Approach for Convolutional Neural Networks - Laura Meneghetti; Nicola Demo; Gianluigi Rozza (2021)

On Dini derivatives of real functions - Giuliano Klun; Alessandro Fonda; Andrea Sfecci (2021)

Discontinuous Galerkin Model Order Reduction of Geometrically Parametrized Stokes Equation - Nirav Shah; Martin W. Hess; Gianluigi Rozza (Numerical Mathematics and Advanced Applications ENUMATH 2019, 2021)
The present work focuses on the geometric parametrization and the reduced order modeling of the Stokes equation. We discuss the concept of a parametrized geometry and its application within a reduced order modeling technique. The full order model is based on the discontinuous Galerkin method with an interior penalty formulation. We introduce the broken Sobolev spaces as well as the weak formulation required for an affine parameter dependency. The operators are transformed from a fixed domain to a parameter dependent domain using the affine parameter dependency. The proper orthogonal decomposition is used to obtain the basis of functions of the reduced order model. By using the Galerkin projection the linear system is projected onto the reduced space. During this process, the offline-online decomposition is used to separate parameter dependent operations from parameter independent operations. Finally this technique is applied to an obstacle test problem.The numerical outcomes presented include experimental error analysis, eigenvalue decay and measurement of online simulation time.

Displacement convexity of Entropy and the distance cost Optimal Transportation - Fabio Cavalletti; Nicola Gigli; Flavia Santarcangelo (Annales de la Faculté des sciences de Toulouse : Mathématiques, 2021)

Doulbeault and J-invariant Cohomologies on Almost Complex Manifolds - Lorenzo Sillari; Adriano Tomassini (Complex Analysis and Operator Theory, 2021)
In this paper we relate the cohomology of $J$-invariant forms to the Dolbeault cohomology of an almost complex manifold. We find necessary and sufficient condition for the inclusion of the former into the latter to be true up to isomorphism. We also extend some results obtained by J. Cirici and S. O. Wilson about the computation of the left-invariant cohomology of nilmanifolds to the setting of solvmanifolds. Several examples are given.
Keywords: Almost Complex Manifolds; Cohomology of Lie Algebra; Compact four-manifold; Dolbeault Cohomology; Frölicher Spectral Sequence; Solvmanifold

A dynamic mode decomposition extension for the forecasting of parametric dynamical systems - Francesco Andreuzzi; Nicola Demo; Gianluigi Rozza (arXiv preprint arXiv:2110.09155, 2021)

A dynamic model for viscoelasticity in domains with time-dependent cracks - Francesco Sapio (2021)
In this paper, we prove the existence of solutions for a class of viscoelastic dynamic systems on time-dependent cracking domains.

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, 2021)
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.

An efficient computational framework for naval shape design and optimization problems by means of data-driven reduced order modeling techniques - Nicola Demo; Giulio Ortali; Gianluca Gustin; Gianluigi Rozza; Gianpiero Lavini (Bolletino dell Unione Matematica Italiana, 2021)
This contribution describes the implementation of a data-driven shape optimization pipeline in a naval architecture application. We adopt reduced order models in order to improve the efficiency of the overall optimization, keeping a modular and equation-free nature to target the industrial demand. We applied the above mentioned pipeline to a realistic cruise ship in order to reduce the total drag. We begin by defining the design space, generated by deforming an initial shape in a parametric way using free form deformation. The evaluation of the performance of each new hull is determined by simulating the flux via finite volume discretization of a two-phase (water and air) fluid. Since the fluid dynamics model can result very expensive—especially dealing with complex industrial geometries—we propose also a dynamic mode decomposition enhancement to reduce the computational cost of a single numerical simulation. The real-time computation is finally achieved by means of proper orthogonal decomposition with Gaussian process regression technique. Thanks to the quick approximation, a genetic optimization algorithm becomes feasible to converge towards the optimal shape.

Equilibrium measure for a nonlocal dislocation energy with physical confinement - Mora, Maria Giovanna; Alessandro Scagliotti (2021)

Exactness of Linear Response in the Quantum Hall Effect - Sven Bachmann; De Roeck, Wojciech; Fraas, Martin; Markus Lange (Annales Henri Poincaré, 2021)

An existence result for the fractional Kelvin–Voigt’s model on time-dependent cracked domains - Maicol Caponi; Francesco Sapio (2021)
We prove an existence result for the fractional Kelvin–Voigt’s model involving Caputo’s derivative on time-dependent cracked domains. We first show the existence of a solution to a regularized version of this problem. Then, we use a compactness argument to derive that the fractional Kelvin–Voigt’s model admits a solution which satisfies an energy-dissipation inequality. Finally, we prove that when the crack is not moving, the solution is unique.

AN EXTENDED PHYSICS INFORMED NEURAL NETWORK FOR PRELIMINARY ANALYSIS OF PARAMETRIC OPTIMAL CONTROL PROBLEMS - Nicola Demo; Maria Strazzullo; Gianluigi Rozza (2021)

A first-order condition for the independence on p of weak gradients - Nicola Gigli; Francesco Nobili (2021)

Hierarchical model reduction techniques for flow modeling in a parametrized setting - Matteo Zancanaro; F. Ballarin; Simona Perotto; Gianluigi Rozza (Multiscale Modeling and Simulation, 2021)
In this work we focus on two different methods to deal with parametrized partial differential equations in an efficient and accurate way. Starting from high fidelity approximations built via the hierarchical model reduction discretization, we consider two approaches, both based on a projection model reduction technique. The two methods differ for the algorithm employed during the construction of the reduced basis. In particular, the former employs the proper orthogonal decomposition, while the latter relies on a greedy algorithm according to the certified reduced basis technique. The two approaches are preliminarily compared on two-dimensional scalar and vector test cases.

Hull Shape Design Optimization with Parameter Space and Model Reductions, and Self-Learning Mesh Morphing - Nicola Demo; Marco Tezzele; Andrea Mola; Gianluigi Rozza (Journal of Marine Science and Engineering, 2021)
In the field of parametric partial differential equations, shape optimization represents a challenging problem due to the required computational resources. In this contribution, a data-driven framework involving multiple reduction techniques is proposed to reduce such computational burden. Proper orthogonal decomposition (POD) and active subspace genetic algorithm (ASGA) are applied for a dimensional reduction of the original (high fidelity) model and for an efficient genetic optimization based on active subspace property. The parameterization of the shape is applied directly to the computational mesh, propagating the generic deformation map applied to the surface (of the object to optimize) to the mesh nodes using a radial basis function (RBF) interpolation. Thus, topology and quality of the original mesh are preserved, enabling application of POD-based reduced order modeling techniques, and avoiding the necessity of additional meshing steps. Model order reduction is performed coupling POD and Gaussian process regression (GPR) in a data-driven fashion. The framework is validated on a benchmark ship.

Hybrid Neural Network Reduced Order Modelling for Turbulent Flows with Geometric Parameters - Matteo Zancanaro; Markus Mrosek; Giovanni Stabile; Carsten Othmer; Gianluigi Rozza (Fluids, 2021)

Independence of synthetic curvature dimension conditions on transport distance exponent - Afiny Akdemir; Andrew Colinet; Robert McCann; Fabio Cavalletti; Flavia Santarcangelo (Trans. Amer. Math. Soc., 2021)

A local approach to parameter space reduction for regression and classification tasks - Francesco Romor; Marco Tezzele; Gianluigi Rozza (arXiv preprint arXiv:2107.10867, 2021)

Local Well Posedness of the Euler–Korteweg Equations on $$\mathbb T}^d}$$ - Massimiliano Berti; Alberto Maspero; Federico Murgante (2021)
We consider the Euler–Korteweg system with space periodic boundary conditions $$ x \in {\mathbb {T}}^d$$. We prove a local in time existence result of classical solutions for irrotational velocity fields requiring natural minimal regularity assumptions on the initial data.

On master test plans for the space of BV functions - Francesco Nobili; Enrico Pasqualetto; Timo Schultz (2021)

Model Order Reduction for Nonlinear and Time-Dependent Parametric Optimal Flow Control Problems - Maria Strazzullo (SISSA, 2021)

A Monolithic and a Partitioned, Reduced Basis Method for Fluid–Structure Interaction Problems - Monica Nonino; F. Ballarin; Gianluigi Rozza (Fluids, 2021)
The aim of this work is to present an overview about the combination of the Reduced Basis Method (RBM) with two different approaches for Fluid–Structure Interaction (FSI) problems, namely a monolithic and a partitioned approach. We provide the details of implementation of two reduction procedures, and we then apply them to the same test case of interest. We first implement a reduction technique that is based on a monolithic procedure where we solve the fluid and the solid problems all at once. We then present another reduction technique that is based on a partitioned (or segregated) procedure: the fluid and the solid problems are solved separately and then coupled using a fixed point strategy. The toy problem that we consider is based on the Turek–Hron benchmark test case, with a fluid Reynolds number Re=100.

Monotonicity formulas for harmonic functions in RCD(0,N) spaces - Nicola Gigli; Ivan Yuri Violo (2021)
We generalize to the <span id="MathJax-Element-2-Frame" class="MathJax" style="display: inline; line-height: normal; font-size: 13.608px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; font-family: 'Lucida Grande', Helvetica, Arial, sans-serif;"><span id="MathJax-Span-13" class="math" style="transition: none 0s ease 0s; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 5.909em;"><span style="transition: none 0s ease 0s; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 4.825em; height: 0px; font-size: 16.6018px;"><span style="transition: none 0s ease 0s; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(1.329em, 1004.7em, 2.655em, -999.997em); top: -2.227em; left: 0em;"><span id="MathJax-Span-17" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Main;">R<span id="MathJax-Span-18" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Main;">C<span id="MathJax-Span-19" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Main;">D<span id="MathJax-Span-20" class="mo" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Main;">(<span id="MathJax-Span-21" class="mn" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Main;">0<span id="MathJax-Span-22" class="mo" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Main;">,<span id="MathJax-Span-23" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.184em; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Math-italic;">N<span id="MathJax-Span-24" class="mo" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Main;">) setting a family of monotonicity formulas by Colding and Minicozzi for positive harmonic functions in Riemannian manifolds with non-negative Ricci curvature. Rigidity and almost rigidity statements are also proven, the second appearing to be new even in the smooth setting. Motivated by the recent work in [AFM] we also introduce the notion of electrostatic potential in <span id="MathJax-Element-3-Frame" class="MathJax" style="display: inline; line-height: normal; font-size: 13.608px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; font-family: 'Lucida Grande', Helvetica, Arial, sans-serif;"><span id="MathJax-Span-25" class="math" style="transition: none 0s ease 0s; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 2.715em;"><span style="transition: none 0s ease 0s; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 2.233em; height: 0px; font-size: 16.6018px;"><span style="transition: none 0s ease 0s; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(1.329em, 1002.17em, 2.414em, -999.997em); top: -2.227em; left: 0em;"><span id="MathJax-Span-29" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Main;">R<span id="MathJax-Span-30" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Main;">C<span id="MathJax-Span-31" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Main;">D spaces, which also satisfies our monotonicity formulas. 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Multi-fidelity data fusion for the approximation of scalar functions with low intrinsic dimensionality through active subspaces - Francesco Romor; Marco Tezzele; Gianluigi Rozza (Proceedings in Applied Mathematics & Mechanics, 2021)

Multi-fidelity data fusion through parameter space reduction with applications to automotive engineering - Francesco Romor; Marco Tezzele; Markus Mrosek; Carsten Othmer; Gianluigi Rozza (arXiv preprint arXiv:2110.14396, 2021)

Multiscale coupling of one-dimensional vascular models and elastic tissues - Luca Heltai; Alfonso Caiazzo; Lucas Müeller (Annals of Biomedical Engineering, 2021)

The Neural Network shifted-Proper Orthogonal Decomposition: a Machine Learning Approach for Non-linear Reduction of Hyperbolic Equations - Davide Papapicco; Nicola Demo; Michele Girfoglio; Giovanni Stabile; Gianluigi Rozza (2021)

Non-intrusive data-driven ROM framework for hemodynamics problems - Michele Girfoglio; Leonardo Scandurra; Francesco Ballarin; Giuseppe Infantino; Francesca Nicolò; Andrea Montalto; Gianluigi Rozza; Roberto Scrofani; Marina Comisso; Francesco Musumeci (Acta Mechanica Sinica, 2021)

Non-well-ordered lower and upper solutions for semilinear systems of PDEs - Alessandro Fonda; Giuliano Klun; Andrea Sfecci (Communications in Contemporary MathematicsCommunications in Contemporary Mathematics, 2021)
We prove existence results for systems of boundary value problems involving elliptic second-order differential operators. The assumptions involve lower and upper solutions, which may be either well-ordered, or not at all. The results are stated in an abstract framework, and can be translated also for systems of parabolic type.We prove existence results for systems of boundary value problems involving elliptic second-order differential operators. The assumptions involve lower and upper solutions, which may be either well-ordered, or not at all. The results are stated in an abstract framework, and can be translated also for systems of parabolic type.

A novel iterative penalty method to enforce boundary conditions in Finite Volume POD-Galerkin reduced order models for fluid dynamics problems - Kelbij Star; Giovanni Stabile; Francesco Belloni; Gianluigi Rozza; Joris Degroote (Communications in Computational Physics, 2021)
A Finite-Volume based POD-Galerkin reduced order model is developed for fluid dynamic problems where the (time-dependent) boundary conditions are controlled using two different boundary control strategies: the control function method, whose aim is to obtain homogeneous basis functions for the reduced basis space and the penalty method where the boundary conditions are enforced in the reduced order model using a penalty factor. The penalty method is improved by using an iterative solver for the determination of the penalty factor rather than tuning the factor with a sensitivity analysis or numerical experimentation. The boundary control methods are compared and tested for two cases: the classical lid driven cavity benchmark problem and a Y-junction flow case with two inlet channels and one outlet channel. The results show that the boundaries of the reduced order model can be controlled with the boundary control methods and the same order of accuracy is achieved for the velocity and pressure fields. Finally, the speedup ratio between the reduced order models and the full order model is of the order 1000 for the lid driven cavity case and of the order 100 for the Y-junction test case.

A numerical approach for heat flux estimation in thin slabs continuous casting molds using data assimilation - Umberto Emil Morelli; Patricia Barral; Peregrina Quintela; Gianluigi Rozza; Giovanni Stabile (International Journal for Numerical Methods in Engineering, 2021)

Nutations in growing plant shoots as a morphoelastic flutter instability - Daniele Agostinelli; Giovanni Noselli; Antonio DeSimone (Phil. Trans. R. Soc. A, 2021)
Growing plant shoots exhibit spontaneous oscillations that Darwin observed, and termed "circumnutations". Recently, they have received renewed attention for the design and optimal actuation of bioinspired robotic devices. We discuss a possible interpretation of these spontaneous oscillations as a Hopf-type bifurcation in a growing morphoelastic rod. Using a three-dimensional model and numerical simulations, we analyse the salient features of this flutter-like phenomenon (e.g. the characteristic period of the oscillations) and their dependence on the model details (in particular, the impact of choosing different growth models) finding that, overall, these features are robust with respect to changes in the details of the growth model adopted.

Nutations in plant shoots: Endogenous and exogenous factors in the presence of mechanical deformations - Daniele Agostinelli; Antonio DeSimone; Giovanni Noselli (Frontiers in Plant Science, 2021)
We present a three-dimensional morphoelastic rod model capable to describe the morphogenesis of growing plant shoots driven by differential growth. We discuss the evolution laws for endogenous oscillators, straightening mechanisms, and reorientations to directional cues, such as gravitropic reactions governed by the avalanche dynamics of statoliths. We use this model to investigate the role of elastic deflections due to gravity loading in circumnutating plant shoots. We show that, in the absence of endogenous cues, pendular and circular oscillations arise as a critical length is attained, thus suggesting the occurrence of an instability triggered by exogenous factors. When also oscillations due to endogenous cues are present, their weight relative to those associated with the instability varies in time as the shoot length and other biomechanical properties change. Thanks to the simultaneous occurrence of these two oscillatory mechanisms, we are able to reproduce a variety of complex behaviors, including trochoid-like patterns, which evolve into circular orbits as the shoot length increases, and the amplitude of the exogenous oscillations becomes dominant.

Parallel transport on non-collapsed $\mathsfRCD(K,N)$ spaces - Emanuele Caputo; Nicola Gigli; Enrico Pasqualetto (2021)
We provide a general theory for parallel transport on non-collapsed <span id="MathJax-Element-2-Frame" class="MathJax" style="display: inline; line-height: normal; font-size: 13.608px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; font-family: 'Lucida Grande', Helvetica, Arial, sans-serif;"><span id="MathJax-Span-13" class="math" style="transition: none 0s ease 0s; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 2.474em;"><span style="transition: none 0s ease 0s; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 1.992em; height: 0px; font-size: 16.6018px;"><span style="transition: none 0s ease 0s; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(1.088em, 1001.93em, 2.173em, -999.997em); top: -1.986em; left: 0em;"><span id="MathJax-Span-17" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_SansSerif;">R<span id="MathJax-Span-18" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_SansSerif;">C<span id="MathJax-Span-19" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_SansSerif;">D spaces obtaining both existence and uniqueness results. Our theory covers the case of geodesics and, more generally, of curves obtained via the flow of sufficiently regular time dependent vector fields: the price that we pay for this generality is that we cannot study parallel transport along a single such curve, but only along almost all of these (in a sense related to the notions of Sobolev vector calculus and Regular Lagrangian Flow in the nonsmooth setting). The class of <span id="MathJax-Element-3-Frame" class="MathJax" style="display: inline; line-height: normal; font-size: 13.608px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; font-family: 'Lucida Grande', Helvetica, Arial, sans-serif;"><span id="MathJax-Span-20" class="math" style="transition: none 0s ease 0s; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 3.619em;"><span style="transition: none 0s ease 0s; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 2.956em; height: 0px; font-size: 16.6018px;"><span style="transition: none 0s ease 0s; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(1.088em, 1002.9em, 2.173em, -999.997em); top: -1.986em; left: 0em;"><span id="MathJax-Span-24" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_SansSerif;">n<span id="MathJax-Span-25" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_SansSerif;">c<span id="MathJax-Span-26" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_SansSerif;">R<span id="MathJax-Span-27" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_SansSerif;">C<span id="MathJax-Span-28" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_SansSerif;">D spaces contains finite dimensional Alexandrov spaces with curvature bounded from below, thus our construction provides a way of speaking about parallel transport in this latter setting alternative to the one proposed by Petrunin (1998). The precise relation between the two approaches is yet to be understood.

Periodic Solutions of Second-Order Differential Equations in Hilbert Spaces - Alessandro Fonda; Giuliano Klun; Andrea Sfecci (2021)
We prove the existence of periodic solutions of some infinite-dimensional systems by the use of the lower/upper solutions method. Both the well-ordered and non-well-ordered cases are treated, thus generalizing to systems some well-established results for scalar equations.

A POD-Galerkin reduced order model for a LES filtering approach - Michele Girfoglio; Annalisa Quaini; Gianluigi Rozza (Journal of Computational Physics, 2021)
We propose a Proper Orthogonal Decomposition (POD)-Galerkin based Reduced Order Model (ROM) for an implementation of the Leray model that combines a two-step algorithm called Evolve-Filter (EF) with a computationally efficient finite volume method. The main novelty of the proposed approach relies in applying spatial filtering both for the collection of the snapshots and in the reduced order model, as well as in considering the pressure field at reduced level. In both steps of the EF algorithm, velocity and pressure fields are approximated by using different POD basis and coefficients. For the reconstruction of the pressures fields, we use a pressure Poisson equation approach. We test our ROM on two benchmark problems: 2D and 3D unsteady flow past a cylinder at Reynolds number 0≤Re≤100. The accuracy of the reduced order model is assessed against results obtained with the full order model. For the 2D case, a parametric study with respect to the filtering radius is also presented.

A POD-Galerkin reduced order model of a turbulent convective buoyant flow of sodium over a backward-facing step - Kelbij Star; Giovanni Stabile; Gianluigi Rozza; Joris Degroote (Applied Mathematical Modelling, 2021)
A Finite-Volume based POD-Galerkin reduced order modeling strategy for steady-state Reynolds averaged Navier–Stokes (RANS) simulation is extended for low-Prandtl number flow. The reduced order model is based on a full order model for which the effects of buoyancy on the flow and heat transfer are characterized by varying the Richardson number. The Reynolds stresses are computed with a linear eddy viscosity model. A single gradient diffusion hypothesis, together with a local correlation for the evaluation of the turbulent Prandtl number, is used to model the turbulent heat fluxes. The contribution of the eddy viscosity and turbulent thermal diffusivity fields are considered in the reduced order model with an interpolation based data-driven method. The reduced order model is tested for buoyancy-aided turbulent liquid sodium flow over a vertical backward-facing step with a uniform heat flux applied on the wall downstream of the step. The wall heat flux is incorporated with a Neumann boundary condition in both the full order model and the reduced order model. The velocity and temperature profiles predicted with the reduced order model for the same and new Richardson numbers inside the range of parameter values are in good agreement with the RANS simulations. Also, the local Stanton number and skin friction distribution at the heated wall are qualitatively well captured. Finally, the reduced order simulations, performed on a single core, are about 105 times faster than the RANS simulations that are performed on eight cores.

Propagating geometry information to finite element computations - Luca Heltai; Wolfgang Bangerth; Martin Kronbichler; Andrea Mola (Transactions on Mathematical Software, 2021)

PyGeM: Python Geometrical Morphing - Marco Tezzele; Nicola Demo; Andrea Mola; Gianluigi Rozza (Software Impacts, 2021)
PyGeM is an open source Python package which allows to easily parametrize and deform 3D object described by CAD files or 3D meshes. It implements several morphing techniques such as free form deformation, radial basis function interpolation, and inverse distance weighting. Due to its versatility in dealing with different file formats it is particularly suited for researchers and practitioners both in academia and in industry interested in computational engineering simulations and optimization studies.
Keywords: Free form deformation; Geometrical morphing; Inverse distance weighting; Python; Radial basis functions interpolation

Quadratic Life Span of Periodic Gravity-capillary Water Waves - Massimiliano Berti; Roberto Feola; Luca Franzoi (2021)
We consider the gravity-capillary water waves equations for a bi-dimensional fluid with a periodic one-dimensional free surface. We prove a rigorous reduction of this system to Birkhoff normal form up to cubic degree. Due to the possible presence of three-wave resonances for general values of gravity, surface tension, and depth, such normal form may be not trivial and exhibit a chaotic dynamics (Wilton ripples). Nevertheless, we prove that for all the values of gravity, surface tension, and depth, initial data that are of size $$ ε $$in a sufficiently smooth Sobolev space leads to a solution that remains in an $$ ε $$-ball of the same Sobolev space up times of order $$ ε ^{-2}$$. We exploit that the three-wave resonances are finitely many, and the Hamiltonian nature of the Birkhoff normal form.

Quantitative lower bounds to the Euclidean and the Gaussian Cheeger constants - Julin, V.; Saracco, G. (Ann. Fenn. Math., 2021)

Quantum Systems at The Brink. Existence and Decay Rates of Bound States at Thresholds; Critical Potentials and dimensionality - Dirk Hundertmark; Michal Jex; Markus Lange (arXiv:2107.14128, 2021)

Quasi-optimal mesh sequence construction through Smoothed Adaptive Finite Element Methods - Ornela Mulita; Stefano Giani; Luca Heltai (SIAM Journal on Scientific Computing, 2021)

Quasistatic Limit of a Dynamic Viscoelastic Model with Memory - Gianni Dal Maso; Francesco Sapio (2021)
We study the behaviour of the solutions to a dynamic evolution problem for a viscoelastic model with long memory, when the rate of change of the data tends to zero. We prove that a suitably rescaled version of the solutions converges to the solution of the corresponding stationary problem.

Rectifiability of the free boundary for varifolds - Luigi De Masi (Indiana Univ. Math. J., 2021)

A Reduced Order Cut Finite Element method for geometrically parametrized steady and unsteady Navier–Stokes problems - Efthymios N Karatzas; Monica Nonino; F. Ballarin; Gianluigi Rozza (Computer & Mathematics With Applications, 2021)
We focus on steady and unsteady Navier–Stokes flow systems in a reduced-order modeling framework based on Proper Orthogonal Decomposition within a levelset geometry description and discretized by an unfitted mesh Finite Element Method. This work extends the approaches of [1], [2], [3] to nonlinear CutFEM discretization. We construct and investigate a unified and geometry independent reduced basis which overcomes many barriers and complications of the past, that may occur whenever geometrical morphings are taking place. By employing a geometry independent reduced basis, we are able to avoid remeshing and transformation to reference configurations, and we are able to handle complex geometries. This combination of a fixed background mesh in a fixed extended background geometry with reduced order techniques appears beneficial and advantageous in many industrial and engineering applications, which could not be resolved efficiently in the past.
Keywords: Cut Finite Element Method; Navier–Stokes equations; Parameter–dependent shape geometry; Reduced Order Models; Unfitted mesh

Reduced Order Methods for Parametrized Non-linear and Time Dependent Optimal Flow Control Problems, Towards Applications in Biomedical and Environmental Sciences - Maria Strazzullo; Zakia Zainib; F. Ballarin; Gianluigi Rozza (Numerical Mathematics and Advanced Applications ENUMATH 2019, 2021)
We introduce reduced order methods as an efficient strategy to solve parametrized non-linear and time dependent optimal flow control problems governed by partial differential equations. Indeed, the optimal control problems require a huge computational effort in order to be solved, most of all in physical and/or geometrical parametrized settings. Reduced order methods are a reliable and suitable approach, increasingly gaining popularity, to achieve rapid and accurate optimal solutions in several fields, such as in biomedical and environmental sciences. In this work, we employ a POD-Galerkin reduction approach over a parametrized optimality system, derived from the Karush-Kuhn-Tucker conditions. The methodology presented is tested on two boundary control problems, governed respectively by (1) time dependent Stokes equations and (2) steady non-linear Navier-Stokes equations.

Reduced Order Methods for Parametrized Non-linear and Time Dependent Optimal Flow Control Problems, Towards Applications in Biomedical and Environmental Sciences - Maria Strazzullo; Zakia Zainib; F. Ballarin; Gianluigi Rozza (Numerical Mathematics and Advanced Applications ENUMATH 2019, 2021)

Reduced order models for the incompressible Navier-Stokes equations on collocated grids using a `discretize-then-project' approach - Kelbij Star; Benjamin Sanderse; Giovanni Stabile; Gianluigi Rozza; Joris Degroote (International Journal for Numerical Methods in Fluids, 2021)

A remark on two notions of flatness for sets in the Euclidean space - Ivan Yuri Violo (2021)
In this note we compare two ways of measuring the <span id="MathJax-Element-1-Frame" class="MathJax" style="display: inline; line-height: normal; font-size: 13.608px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; font-family: 'Lucida Grande', Helvetica, Arial, sans-serif;"><span id="MathJax-Span-1" class="math" style="transition: none 0s ease 0s; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0.787em;"><span style="transition: none 0s ease 0s; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0.606em; height: 0px; font-size: 16.6018px;"><span style="transition: none 0s ease 0s; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(1.389em, 1000.61em, 2.173em, -999.997em); 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display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 2.534em; height: 0px; font-size: 16.6018px;"><span style="transition: none 0s ease 0s; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(1.329em, 1002.53em, 2.474em, -999.997em); top: -2.227em; left: 0em;"><span id="MathJax-Span-35" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Math-italic;">x<span id="MathJax-Span-36" class="mo" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.304em; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Main;">∈<span id="MathJax-Span-37" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.304em; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Math-italic;">S, <span id="MathJax-Element-7-Frame" class="MathJax" style="display: inline; line-height: normal; font-size: 13.608px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; font-family: 'Lucida Grande', Helvetica, Arial, sans-serif;"><span id="MathJax-Span-38" class="math" style="transition: none 0s ease 0s; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 2.896em;"><span style="transition: none 0s ease 0s; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 2.354em; height: 0px; font-size: 16.6018px;"><span style="transition: none 0s ease 0s; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(1.389em, 1002.29em, 2.474em, -999.997em); top: -2.227em; left: 0em;"><span id="MathJax-Span-40" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Math-italic;">r<span id="MathJax-Span-41" class="mo" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.304em; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Main;">><span id="MathJax-Span-42" class="mn" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.304em; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Main;">0), which measure the minimal scaling-invariant Hausdorff distances in <span id="MathJax-Element-8-Frame" class="MathJax" style="display: inline; line-height: normal; font-size: 13.608px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; font-family: 'Lucida Grande', Helvetica, Arial, sans-serif;"><span id="MathJax-Span-43" class="math" style="transition: none 0s ease 0s; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 3.017em;"><span style="transition: none 0s ease 0s; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 2.474em; height: 0px; font-size: 16.6018px;"><span style="transition: none 0s ease 0s; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(1.329em, 1002.35em, 2.655em, -999.997em); top: -2.227em; left: 0em;"><span style="transition: none 0s ease 0s; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(3.137em, 1000.79em, 4.162em, -999.997em); top: -3.975em; left: 0em;"><span id="MathJax-Span-46" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Math-italic;">B<span id="MathJax-Span-47" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-size: 11.7374px; font-family: MathJax_Math-italic;">r<span id="MathJax-Span-48" class="mo" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Main;">(<span id="MathJax-Span-49" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Math-italic;">x<span id="MathJax-Span-50" class="mo" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Main;">) between <span id="MathJax-Element-9-Frame" class="MathJax" style="display: inline; line-height: normal; font-size: 13.608px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; font-family: 'Lucida Grande', Helvetica, Arial, sans-serif;"><span id="MathJax-Span-51" class="math" style="transition: none 0s ease 0s; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0.847em;"><span style="transition: none 0s ease 0s; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0.666em; height: 0px; font-size: 16.6018px;"><span style="transition: none 0s ease 0s; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(1.088em, 1000.67em, 2.173em, -999.997em); top: -1.986em; left: 0em;"><span id="MathJax-Span-53" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Math-italic;">S and <span id="MathJax-Element-10-Frame" class="MathJax" style="display: inline; line-height: normal; font-size: 13.608px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; font-family: 'Lucida Grande', Helvetica, Arial, sans-serif;"><span id="MathJax-Span-54" class="math" style="transition: none 0s ease 0s; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0.787em;"><span style="transition: none 0s ease 0s; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0.606em; height: 0px; font-size: 16.6018px;"><span style="transition: none 0s ease 0s; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(1.389em, 1000.61em, 2.173em, -999.997em); top: -1.986em; left: 0em;"><span id="MathJax-Span-56" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Math-italic;">n-dimensional affine subspaces of <span id="MathJax-Element-11-Frame" class="MathJax" style="display: inline; line-height: normal; font-size: 13.608px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; font-family: 'Lucida Grande', Helvetica, Arial, sans-serif;"><span id="MathJax-Span-57" class="math" style="transition: none 0s ease 0s; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 1.389em;"><span style="transition: none 0s ease 0s; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 1.148em; height: 0px; font-size: 16.6018px;"><span style="transition: none 0s ease 0s; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(-0.057em, 1001.15em, 1.208em, -999.997em); top: -1.022em; left: 0em;"><span style="transition: none 0s ease 0s; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(3.137em, 1000.73em, 4.162em, -999.997em); top: -3.975em; left: 0em;"><span id="MathJax-Span-62" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_AMS;">R<span id="MathJax-Span-63" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-size: 11.7374px; font-family: MathJax_Math-italic;">d. The second is an `intrinsic' approach in which we view the same set <span id="MathJax-Element-12-Frame" class="MathJax" style="display: inline; line-height: normal; font-size: 13.608px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; font-family: 'Lucida Grande', Helvetica, Arial, sans-serif;"><span id="MathJax-Span-64" class="math" style="transition: none 0s ease 0s; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0.847em;"><span style="transition: none 0s ease 0s; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0.666em; height: 0px; font-size: 16.6018px;"><span style="transition: none 0s ease 0s; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(1.088em, 1000.67em, 2.173em, -999.997em); top: -1.986em; left: 0em;"><span id="MathJax-Span-66" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Math-italic;">S as a metric space (endowed with the induced Euclidean distance). Then we consider numbers <span id="MathJax-Element-13-Frame" class="MathJax" style="display: inline; line-height: normal; font-size: 13.608px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; font-family: 'Lucida Grande', Helvetica, Arial, sans-serif;"><span id="MathJax-Span-67" class="math" style="transition: none 0s ease 0s; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 3.378em;"><span style="transition: none 0s ease 0s; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 2.775em; height: 0px; font-size: 16.6018px;"><span style="transition: none 0s ease 0s; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(1.329em, 1002.65em, 2.655em, -999.997em); top: -2.227em; left: 0em;"><span id="MathJax-Span-71" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_SansSerif;">a<span id="MathJax-Span-72" class="mo" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Main;">(<span id="MathJax-Span-73" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Math-italic;">x<span id="MathJax-Span-74" class="mo" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Main;">,<span id="MathJax-Span-75" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.184em; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Math-italic;">r<span id="MathJax-Span-76" class="mo" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Main;">)'s, that are the scaling-invariant Gromov-Hausdorff distances between balls centered at <span id="MathJax-Element-14-Frame" class="MathJax" style="display: inline; line-height: normal; font-size: 13.608px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; font-family: 'Lucida Grande', Helvetica, Arial, sans-serif;"><span id="MathJax-Span-77" class="math" style="transition: none 0s ease 0s; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0.787em;"><span style="transition: none 0s ease 0s; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0.606em; height: 0px; font-size: 16.6018px;"><span style="transition: none 0s ease 0s; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(1.389em, 1000.54em, 2.173em, -999.997em); top: -1.986em; left: 0em;"><span id="MathJax-Span-79" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Math-italic;">x of radius <span id="MathJax-Element-15-Frame" class="MathJax" style="display: inline; line-height: normal; font-size: 13.608px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; font-family: 'Lucida Grande', Helvetica, Arial, sans-serif;"><span id="MathJax-Span-80" class="math" style="transition: none 0s ease 0s; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0.545em;"><span style="transition: none 0s ease 0s; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0.425em; height: 0px; font-size: 16.6018px;"><span style="transition: none 0s ease 0s; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(1.389em, 1000.42em, 2.173em, -999.997em); top: -1.986em; left: 0em;"><span id="MathJax-Span-82" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Math-italic;">r in <span id="MathJax-Element-16-Frame" class="MathJax" style="display: inline; line-height: normal; font-size: 13.608px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; font-family: 'Lucida Grande', Helvetica, Arial, sans-serif;"><span id="MathJax-Span-83" class="math" style="transition: none 0s ease 0s; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0.847em;"><span style="transition: none 0s ease 0s; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0.666em; height: 0px; font-size: 16.6018px;"><span style="transition: none 0s ease 0s; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(1.088em, 1000.67em, 2.173em, -999.997em); top: -1.986em; left: 0em;"><span id="MathJax-Span-85" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Math-italic;">S and the <span id="MathJax-Element-17-Frame" class="MathJax" style="display: inline; line-height: normal; font-size: 13.608px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; font-family: 'Lucida Grande', Helvetica, Arial, sans-serif;"><span id="MathJax-Span-86" class="math" style="transition: none 0s ease 0s; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0.787em;"><span style="transition: none 0s ease 0s; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0.606em; height: 0px; font-size: 16.6018px;"><span style="transition: none 0s ease 0s; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(1.389em, 1000.61em, 2.173em, -999.997em); top: -1.986em; left: 0em;"><span id="MathJax-Span-88" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Math-italic;">n-dimensional Euclidean ball of the same radius. As main result of our analysis we make rigorous a phenomenon, first noted by David and Toro, for which the numbers <span id="MathJax-Element-18-Frame" class="MathJax" style="display: inline; line-height: normal; font-size: 13.608px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; font-family: 'Lucida Grande', Helvetica, Arial, sans-serif;"><span id="MathJax-Span-89" class="math" style="transition: none 0s ease 0s; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 3.378em;"><span style="transition: none 0s ease 0s; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 2.775em; height: 0px; font-size: 16.6018px;"><span style="transition: none 0s ease 0s; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(1.329em, 1002.65em, 2.655em, -999.997em); top: -2.227em; left: 0em;"><span id="MathJax-Span-93" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_SansSerif;">a<span id="MathJax-Span-94" class="mo" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Main;">(<span id="MathJax-Span-95" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Math-italic;">x<span id="MathJax-Span-96" class="mo" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Main;">,<span id="MathJax-Span-97" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.184em; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Math-italic;">r<span id="MathJax-Span-98" class="mo" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Main;">)'s behaves as the square of the numbers <span id="MathJax-Element-19-Frame" class="MathJax" style="display: inline; line-height: normal; font-size: 13.608px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; font-family: 'Lucida Grande', Helvetica, Arial, sans-serif;"><span id="MathJax-Span-99" class="math" style="transition: none 0s ease 0s; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 3.559em;"><span style="transition: none 0s ease 0s; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 2.896em; height: 0px; font-size: 16.6018px;"><span style="transition: none 0s ease 0s; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(1.329em, 1002.77em, 2.655em, -999.997em); top: -2.227em; left: 0em;"><span id="MathJax-Span-101" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Math-italic;">α<span id="MathJax-Span-102" class="mo" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Main;">(<span id="MathJax-Span-103" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Math-italic;">x<span id="MathJax-Span-104" class="mo" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Main;">,<span id="MathJax-Span-105" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.184em; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Math-italic;">r<span id="MathJax-Span-106" class="mo" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Main;">)'s. Moreover we show how this result finds application in extending the Cheeger-Colding intrinsic-Reifenberg theorem to the biLipschitz case. As a by-product of our arguments, we deduce analogous results also for the Jones' numbers <span id="MathJax-Element-20-Frame" class="MathJax" style="display: inline; line-height: normal; font-size: 13.608px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; font-family: 'Lucida Grande', Helvetica, Arial, sans-serif;"><span id="MathJax-Span-107" class="math" style="transition: none 0s ease 0s; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0.666em;"><span style="transition: none 0s ease 0s; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0.545em; height: 0px; font-size: 16.6018px;"><span style="transition: none 0s ease 0s; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(1.088em, 1000.54em, 2.354em, -999.997em); top: -1.986em; left: 0em;"><span id="MathJax-Span-109" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Math-italic;">β's (i.e. the one-sided version of the numbers <span id="MathJax-Element-21-Frame" class="MathJax" style="display: inline; line-height: normal; font-size: 13.608px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; font-family: 'Lucida Grande', Helvetica, Arial, sans-serif;"><span id="MathJax-Span-110" class="math" style="transition: none 0s ease 0s; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0.847em;"><span style="transition: none 0s ease 0s; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0.666em; height: 0px; font-size: 16.6018px;"><span style="transition: none 0s ease 0s; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(1.389em, 1000.61em, 2.173em, -999.997em); top: -1.986em; left: 0em;"><span id="MathJax-Span-112" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Math-italic;">α's).

Rigidity and almost rigidity of Sobolev inequalities on compact spaces with lower Ricci curvature bounds - Francesco Nobili; Ivan Yuri Violo (2021)
<blockquote class="abstract mathjax" style="line-height: 1.55; font-size: 1.05em; margin-bottom: 21.6px; margin-block: 14.4px 21.6px; margin-inline: 40px; background-color: white; border-left: 0px; padding: 0px; color: #000000; font-family: 'Lucida Grande', Helvetica, Arial, sans-serif; font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration-thickness: initial; text-decoration-style: initial; text-decoration-color: initial;">We prove that if <span id="MathJax-Element-1-Frame" class="MathJax" style="display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 13.608px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px;"><span id="MathJax-Span-1" class="math" style="transition: none 0s ease 0s; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; width: 1.269em;"><span style="transition: none 0s ease 0s; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; width: 1.028em; height: 0px; font-size: 16.6018px;"><span style="transition: none 0s ease 0s; display: inline; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; clip: rect(1.148em, 1001.03em, 2.173em, -999.997em); top: -1.986em; left: 0em;"><span id="MathJax-Span-2" class="mrow" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none;"><span id="MathJax-Span-3" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; font-family: MathJax_Math-italic;">M is a closed <span id="MathJax-Element-2-Frame" class="MathJax" style="display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 13.608px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px;"><span id="MathJax-Span-4" class="math" style="transition: none 0s ease 0s; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; width: 0.787em;"><span style="transition: none 0s ease 0s; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; width: 0.606em; height: 0px; font-size: 16.6018px;"><span style="transition: none 0s ease 0s; display: inline; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; clip: rect(1.389em, 1000.61em, 2.173em, -999.997em); top: -1.986em; left: 0em;"><span id="MathJax-Span-5" class="mrow" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none;"><span id="MathJax-Span-6" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; font-family: MathJax_Math-italic;">n-dimensional Riemannian manifold, <span id="MathJax-Element-3-Frame" class="MathJax" style="display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 13.608px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px;"><span id="MathJax-Span-7" class="math" style="transition: none 0s ease 0s; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; width: 3.017em;"><span style="transition: none 0s ease 0s; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; width: 2.474em; height: 0px; font-size: 16.6018px;"><span style="transition: none 0s ease 0s; display: inline; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; clip: rect(1.389em, 1002.41em, 2.534em, -999.997em); top: -2.227em; left: 0em;"><span id="MathJax-Span-8" class="mrow" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none;"><span id="MathJax-Span-9" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; font-family: MathJax_Math-italic;">n<span id="MathJax-Span-10" class="mo" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.304em; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; font-family: MathJax_Main;">≥<span id="MathJax-Span-11" class="mn" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.304em; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; font-family: MathJax_Main;">3, with <span id="MathJax-Element-4-Frame" class="MathJax" style="display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 13.608px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px;"><span id="MathJax-Span-12" class="math" style="transition: none 0s ease 0s; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; width: 6.392em;"><span style="transition: none 0s ease 0s; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; width: 5.247em; height: 0px; font-size: 16.6018px;"><span style="transition: none 0s ease 0s; display: inline; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; clip: rect(1.389em, 1005.19em, 2.534em, -999.997em); top: -2.227em; left: 0em;"><span id="MathJax-Span-13" class="mrow" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none;"><span id="MathJax-Span-14" class="texatom" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none;"><span id="MathJax-Span-15" class="mrow" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none;"><span id="MathJax-Span-16" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; font-family: MathJax_Main;">R<span id="MathJax-Span-17" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; font-family: MathJax_Main;">i<span id="MathJax-Span-18" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; font-family: MathJax_Main;">c<span id="MathJax-Span-19" class="mo" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.304em; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; font-family: MathJax_Main;">≥<span id="MathJax-Span-20" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.304em; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; font-family: MathJax_Math-italic;">n<span id="MathJax-Span-21" class="mo" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.244em; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; font-family: MathJax_Main;">−<span id="MathJax-Span-22" class="mn" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.244em; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; font-family: MathJax_Main;">1 and for which the optimal constant in the critical Sobolev inequality equals the one of the <span id="MathJax-Element-5-Frame" class="MathJax" style="display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 13.608px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px;"><span id="MathJax-Span-23" class="math" style="transition: none 0s ease 0s; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; width: 0.787em;"><span style="transition: none 0s ease 0s; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; width: 0.606em; height: 0px; font-size: 16.6018px;"><span style="transition: none 0s ease 0s; display: inline; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; clip: rect(1.389em, 1000.61em, 2.173em, -999.997em); top: -1.986em; left: 0em;"><span id="MathJax-Span-24" class="mrow" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none;"><span id="MathJax-Span-25" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; font-family: MathJax_Math-italic;">n-dimensional sphere <span id="MathJax-Element-6-Frame" class="MathJax" style="display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 13.608px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px;"><span id="MathJax-Span-26" class="math" style="transition: none 0s ease 0s; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; width: 1.329em;"><span style="transition: none 0s ease 0s; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; width: 1.088em; height: 0px; font-size: 16.6018px;"><span style="transition: none 0s ease 0s; display: inline; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; clip: rect(0.124em, 1001.09em, 1.208em, -999.997em); top: -1.022em; left: 0em;"><span id="MathJax-Span-27" class="mrow" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none;"><span id="MathJax-Span-28" class="msubsup" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none;"><span style="transition: none 0s ease 0s; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; width: 1.088em; height: 0px;"><span style="transition: none 0s ease 0s; display: inline; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; clip: rect(3.077em, 1000.54em, 4.162em, -999.997em); top: -3.975em; left: 0em;"><span id="MathJax-Span-29" class="texatom" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none;"><span id="MathJax-Span-30" class="mrow" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none;"><span id="MathJax-Span-31" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; font-family: MathJax_AMS;">S<span style="transition: none 0s ease 0s; display: inline; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; top: -4.397em; left: 0.545em;"><span id="MathJax-Span-32" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; font-size: 11.7374px; font-family: MathJax_Math-italic;">n, then <span id="MathJax-Element-7-Frame" class="MathJax" style="display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 13.608px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px;"><span id="MathJax-Span-33" class="math" style="transition: none 0s ease 0s; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; width: 1.269em;"><span style="transition: none 0s ease 0s; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; width: 1.028em; height: 0px; font-size: 16.6018px;"><span style="transition: none 0s ease 0s; display: inline; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; clip: rect(1.148em, 1001.03em, 2.173em, -999.997em); top: -1.986em; left: 0em;"><span id="MathJax-Span-34" class="mrow" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none;"><span id="MathJax-Span-35" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; font-family: MathJax_Math-italic;">M is isometric to <span id="MathJax-Element-8-Frame" class="MathJax" style="display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 13.608px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px;"><span id="MathJax-Span-36" class="math" style="transition: none 0s ease 0s; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; width: 1.329em;"><span style="transition: none 0s ease 0s; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; width: 1.088em; height: 0px; font-size: 16.6018px;"><span style="transition: none 0s ease 0s; display: inline; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; clip: rect(0.124em, 1001.09em, 1.208em, -999.997em); top: -1.022em; left: 0em;"><span id="MathJax-Span-37" class="mrow" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none;"><span id="MathJax-Span-38" class="msubsup" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none;"><span style="transition: none 0s ease 0s; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; width: 1.088em; height: 0px;"><span style="transition: none 0s ease 0s; display: inline; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; clip: rect(3.077em, 1000.54em, 4.162em, -999.997em); top: -3.975em; left: 0em;"><span id="MathJax-Span-39" class="texatom" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none;"><span id="MathJax-Span-40" class="mrow" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none;"><span id="MathJax-Span-41" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; font-family: MathJax_AMS;">S<span style="transition: none 0s ease 0s; display: inline; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; top: -4.397em; left: 0.545em;"><span id="MathJax-Span-42" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; font-size: 11.7374px; font-family: MathJax_Math-italic;">n. An almost-rigidity result is also established, saying that if equality is almost achieved, then <span id="MathJax-Element-9-Frame" class="MathJax" style="display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 13.608px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px;"><span id="MathJax-Span-43" class="math" style="transition: none 0s ease 0s; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; width: 1.269em;"><span style="transition: none 0s ease 0s; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; width: 1.028em; height: 0px; font-size: 16.6018px;"><span style="transition: none 0s ease 0s; display: inline; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; clip: rect(1.148em, 1001.03em, 2.173em, -999.997em); top: -1.986em; left: 0em;"><span id="MathJax-Span-44" class="mrow" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none;"><span id="MathJax-Span-45" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; font-family: MathJax_Math-italic;">M is close in the measure Gromov-Hausdorff sense to a spherical suspension. These statements are obtained in the <span id="MathJax-Element-10-Frame" class="MathJax" style="display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 13.608px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px;"><span id="MathJax-Span-46" class="math" style="transition: none 0s ease 0s; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; width: 2.715em;"><span style="transition: none 0s ease 0s; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; width: 2.233em; height: 0px; font-size: 16.6018px;"><span style="transition: none 0s ease 0s; display: inline; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; clip: rect(1.329em, 1002.17em, 2.414em, -999.997em); top: -2.227em; left: 0em;"><span id="MathJax-Span-47" class="mrow" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none;"><span id="MathJax-Span-48" class="texatom" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none;"><span id="MathJax-Span-49" class="mrow" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none;"><span id="MathJax-Span-50" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; font-family: MathJax_Main;">R<span id="MathJax-Span-51" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; font-family: MathJax_Main;">C<span id="MathJax-Span-52" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; font-family: MathJax_Main;">D-setting of (possibly non-smooth) metric measure spaces satisfying synthetic lower Ricci curvature bounds. An independent result of our analysis is the characterization of the best constant in the Sobolev inequality on any compact <span id="MathJax-Element-11-Frame" class="MathJax" style="display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 13.608px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px;"><span id="MathJax-Span-53" class="math" style="transition: none 0s ease 0s; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; width: 1.871em;"><span style="transition: none 0s ease 0s; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; width: 1.51em; height: 0px; font-size: 16.6018px;"><span style="transition: none 0s ease 0s; display: inline; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; clip: rect(1.329em, 1001.45em, 2.414em, -999.997em); top: -2.227em; left: 0em;"><span id="MathJax-Span-54" class="mrow" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none;"><span id="MathJax-Span-55" class="texatom" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none;"><span id="MathJax-Span-56" class="mrow" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none;"><span id="MathJax-Span-57" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; font-family: MathJax_Main;">C<span id="MathJax-Span-58" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; font-family: MathJax_Main;">D space, extending to the non-smooth setting a classical result by Aubin. Our arguments are based on a new concentration compactness result for mGH-converging sequences of <span id="MathJax-Element-12-Frame" class="MathJax" style="display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 13.608px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px;"><span id="MathJax-Span-59" class="math" style="transition: none 0s ease 0s; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; width: 2.715em;"><span style="transition: none 0s ease 0s; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; width: 2.233em; height: 0px; font-size: 16.6018px;"><span style="transition: none 0s ease 0s; display: inline; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; clip: rect(1.329em, 1002.17em, 2.414em, -999.997em); top: -2.227em; left: 0em;"><span id="MathJax-Span-60" class="mrow" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none;"><span id="MathJax-Span-61" class="texatom" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none;"><span id="MathJax-Span-62" class="mrow" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none;"><span id="MathJax-Span-63" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; font-family: MathJax_Main;">R<span id="MathJax-Span-64" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; font-family: MathJax_Main;">C<span id="MathJax-Span-65" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; font-family: MathJax_Main;">D spaces and on a Polya-Szego inequality of Euclidean-type in <span id="MathJax-Element-13-Frame" class="MathJax" style="display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 13.608px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px;"><span id="MathJax-Span-66" class="math" style="transition: none 0s ease 0s; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; width: 1.871em;"><span style="transition: none 0s ease 0s; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; width: 1.51em; height: 0px; font-size: 16.6018px;"><span style="transition: none 0s ease 0s; display: inline; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; clip: rect(1.329em, 1001.45em, 2.414em, -999.997em); top: -2.227em; left: 0em;"><span id="MathJax-Span-67" class="mrow" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none;"><span id="MathJax-Span-68" class="texatom" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none;"><span id="MathJax-Span-69" class="mrow" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none;"><span id="MathJax-Span-70" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; font-family: MathJax_Main;">C<span id="MathJax-Span-71" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; font-family: MathJax_Main;">D spaces. As an application of the technical tools developed we prove both an existence result for the Yamabe equation and the continuity of the generalized Yamabe constant under measure Gromov-Hausdorff convergence, in the <span id="MathJax-Element-14-Frame" class="MathJax" style="display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 13.608px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px;"><span id="MathJax-Span-72" class="math" style="transition: none 0s ease 0s; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; width: 2.715em;"><span style="transition: none 0s ease 0s; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; width: 2.233em; height: 0px; font-size: 16.6018px;"><span style="transition: none 0s ease 0s; display: inline; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; clip: rect(1.329em, 1002.17em, 2.414em, -999.997em); top: -2.227em; left: 0em;"><span id="MathJax-Span-73" class="mrow" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none;"><span id="MathJax-Span-74" class="texatom" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none;"><span id="MathJax-Span-75" class="mrow" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none;"><span id="MathJax-Span-76" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; font-family: MathJax_Main;">R<span id="MathJax-Span-77" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; font-family: MathJax_Main;">C<span id="MathJax-Span-78" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; font-family: MathJax_Main;">D-setting.

The sharp quantitative isocapacitary inequality - Guido De Philippis; Michele Marini; Ekaterina Mukoseeva (Revista Matematica Iberoamericana, 2021)
We prove a sharp quantitative form of the classical isocapacitary inequality. Namely, we show that the difference between the capacity of a set and that of a ball with the same volume bounds the square of the Fraenkel asymmetry of the set. This provides a positive answer to a conjecture of Hall, Hayman, and Weitsman (J. Analyse Math.'91). © 2021 Real Sociedad Matemática Española
Keywords: Fraenkel asymmetry; isocapacitary inequality; Stability estimates

The sharp quantitative isocapacitary inequality (the case of p-capacity) - Ekaterina Mukoseeva (Advances in Calculus of Variations, 2021)
We prove a sharp quantitative form of isocapacitary inequality in the case of a general p. This work is a generalization of the author's paper with Guido De Philippis and Michele Marini, where we treated the case of 2-capacity. © 2021 Walter de Gruyter GmbH, Berlin/Boston 2021.
Keywords: isocapacitary inequality; Stability estimates

Smoothed-adaptive perturbed inverse iteration for elliptic eigenvalue problems - Stefano Giani; Luka Grubisic; Luca Heltai; Ornela Mulita (Computational Methods in Applied Mathematics, 2021)

A sufficient criterion to determine planar self-Cheeger sets - Saracco, G. (J. Convex Anal., 2021)

A supervised learning approach involving active subspaces for an efficient genetic algorithm in high-dimensional optimization problems - Nicola Demo; Marco Tezzele; Gianluigi Rozza (SIAM Journal on Scientific Computing, 2021)
In this work, we present an extension of the genetic algorithm (GA) which exploits the active subspaces (AS) property to evolve the individuals on a lower dimensional space. In many cases, GA requires in fact more function evaluations than others optimization method to converge to the optimum. Thus, complex and high-dimensional functions may result intractable with the standard algorithm. To address this issue, we propose to linearly map the input parameter space of the original function onto its AS before the evolution, performing the mutation and mate processes in a lower dimensional space. In this contribution, we describe the novel method called ASGA, presenting differences and similarities with the standard GA method. We test the proposed method over n-dimensional benchmark functions – Rosenbrock, Ackley, Bohachevsky, Rastrigin, Schaffer N. 7, and Zakharov – and finally we apply it to an aeronautical shape optimization problem.

Traveling Quasi-periodic Water Waves with Constant Vorticity - Massimiliano Berti; Luca Franzoi; Alberto Maspero (2021)
We prove the first bifurcation result of time quasi-periodic traveling wave solutions for space periodic water waves with vorticity. In particular, we prove the existence of small amplitude time quasi-periodic solutions of the gravity-capillary water waves equations with constant vorticity, for a bidimensional fluid over a flat bottom delimited by a space-periodic free interface. These quasi-periodic solutions exist for all the values of depth, gravity and vorticity, and restrict the surface tension to a Borel set of asymptotically full Lebesgue measure.

Trotter product formulae for $*$-automorphisms of quantum lattice systems - Sven Bachmann; Markus Lange (2021)

Uniqueness and continuous dependence for a viscoelastic problem with memory in domains with time dependent cracks - Federico Cianci; Gianni Dal Maso (2021)

A vanishing-inertia analysis for finite-dimensional rate-independent systems with nonautonomous dissipation and an application to soft crawlers - Paolo Gidoni; Filippo Riva (2021)
We study the approximation of finite-dimensional rate-independent quasistatic systems, via a vanishing-inertia asymptotic analysis of dynamic evolutions. We prove the uniform convergence of dynamic solutions to a rate-independent one, employing the variational concept of energetic solution. Motivated by applications in soft locomotion, we allow time-dependence of the dissipation potential, and translation invariance of the potential energy.

A weighted POD-reduction approach for parametrized PDE-constrained optimal control problems with random inputs and applications to environmental sciences - G. Carere; Maria Strazzullo; Francesco Ballarin; Gianluigi Rozza; R. Stevenson (Computers and Mathematics with Applications, 2021)

Well-Ordered and Non-Well-Ordered Lower and Upper Solutions for Periodic Planar Systems - Alessandro Fonda; Giuliano Klun; Andrea Sfecci (Advanced Nonlinear Studies, 2021)

Advances in reduced order methods for parametric industrial problems in computational fluid dynamics - Gianluigi Rozza; M.H. Malik; Nicola Demo; Marco Tezzele; Michele Girfoglio; Giovanni Stabile; Andrea Mola (Proceedings of the 6th European Conference on Computational Mechanics: Solids, Structures and Coupled Problems, ECCM 2018 and 7th European Conference on Computational Fluid Dynamics, ECFD 2018, 2020)
Reduced order modeling has gained considerable attention in recent decades owing to the advantages offered in reduced computational times and multiple solutions for parametric problems. The focus of this manuscript is the application of model order reduction techniques in various engineering and scientific applications including but not limited to mechanical, naval and aeronautical engineering. The focus here is kept limited to computational fluid mechanics and related applications. The advances in the reduced order modeling with proper orthogonal decomposition and reduced basis method are presented as well as a brief discussion of dynamic mode decomposition and also some present advances in the parameter space reduction. Here, an overview of the challenges faced and possible solutions are presented with examples from various problems.

On the Approximation of Quasistatic Evolutions for the Debonding of a Thin Film via Vanishing Inertia and Viscosity - Filippo Riva (2020)
In this paper, we contribute to studying the issue of quasistatic limit in the context of Griffith’s theory by investigating a one-dimensional debonding model. It describes the evolution of a thin film partially glued to a rigid substrate and subjected to a vertical loading. Taking viscosity into account and under suitable assumptions on the toughness of the glue, we prove that, in contrast to what happens in the undamped case, dynamic solutions converge to the quasistatic one when inertia and viscosity go to zero, except for a possible discontinuity at the initial time. We then characterise the size of the jump by means of an asymptotic analysis of the debonding front.

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 (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 (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 (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)

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)

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 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)

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.

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.

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 (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.