Dipartimento di Matematica

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We consider a controlled reaction-diffusion equation, modeling the spreading of an invasive population. A simpler model is then derived, describing the controlled evolution of a contaminated set.
The first part of the talk will focus on the optimal control of 1-dimensional traveling wave profiles.
Namely, we seek a control with minimum norm which produces a traveling profile with a given speed.
In turn, this leads to a family of optimization problems for a moving set, related to the original reaction-diffusion equation via a sharp interface limit. In connection with moving sets, the second part of the talk will present some results on controllability, existence of optimal strategies, and optimality conditions. Some open questions will be discussed.

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The interaction between learning theory and harmonic analysis was emphasized by mathematics of quantum computing. One of the outstanding open problems in this area concerns the Bohnenblust--Hille inequality
that generalizes a celebrated Littlewood’s 4/3 lemma. How to learn (approximately and with large probability) a very large matrix in a relatively small number of random quantum quarries? Motivated by this question, a non-commutative counterpart of Bohnenblust--Hille inequality for Boolean cubes was recently conjectured in Cambyse Rouz, Melchior Wirth, and Haonan Zhang. By waving the hands I will explain the proof of non-commutative Bohnenblust--Hille inequalities with constants that are dimension-free. As applications, we study learning problems of quantum observables. Using Heisenberg—Weyl basis one
reduces the quantum problem to commutative problem: the Bohnenblust—Hille inequality for cyclic groups (joint with Haonan Zhang, Joe Slote). To prove the Bohnenblust—Hille inequality for cyclic groups turned out to be a challenging problem. I will explain the progress in this area.

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For problems that allow statistical interpretation, such as parameter estimation or information transmission, whenever different strategies are possible, it is natural to compare them on the basis of their statistical performance with respect to the problem at hand. The theory of statistical comparison provides a rigorous formalization of this scenario. In this colloquium, I will introduce the basic ideas of the classical theory, including the theory of majorization and the Blackwell-Shermal-Stein theorem, present their extensions to the quantum setting, and discuss applications in quantum measurement theory and quantum statistical mechanics.

Giuseppe Savaré started his scientific career as a researcher at the CNR Institute for Numerical Analysis in Pavia and has been Professor of Mathematical Analysis since 2000, first at the University of Pavia and then, since 2020, at Bocconi University in Milan.
During 2019-2023 he was Hans-Fischer Senior Fellow at the Institute for Advanced Study of the Technical University of Munich. In 2011 he received the Ennio De Giorgi prize awarded by the Italian Mathematical Union.
His research activity concerns variational methods for evolution problems, in particular gradient flows and rate-independent models, and their interplay with optimal transport, measure theoretic tools and analysis in metric spaces.
In collaboration with Luigi Ambrosio and Nicola Gigli, he has written a monograph on Gradient Flows in Metric Spaces and in the Space of Probability Measures.

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Optimal transport theory studies the optimal strategies for transferring a given initial distribution of resources (mathematically represented by a measure) to a final configuration in a way that minimizes the cost of transport.
First addressed mathematically by G. Monge in 1781, its first modern formulation (1942) earned L.V. Kantorovich the Nobel Prize in Economics in 1975.
Later, thanks to many outstanding theoretical, applied and numerical contributions, the theory has been developed in many directions including, in addition to economics, linear and stochastic programming, statistics and probability, mean field models, functional analysis, PDEs, Riemannian geometry, machine learning.
The seminar aims to give a quick overview of some of the classical results and more recent contributions, with applications to evolution problems.

Contatto: marco.verani@polimi.it

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In this talk, we will review some recent results on the modelling, analysis and numerical analysis of Koiter’s model for linearly elastic shells in the case where the shell under consideration is constrained to remain confined in a prescribed half-space.

To begin with, we will see how to recover the constrained Koiter’s model starting from the classical three-dimensional equations of linearized elasticity in the case where the shell under consideration is an elliptic membrane or a flexural shell.

Contatti: paola.antonietti@polimi.it e vittorino.pata@polimi.it

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We provide an overview and propose elementary proofs of energy conservation for weak solutions to the Euler and Navier-Stokes
equations in the class of Holder continuous functions. Our focus is on exploring the interplay between space and time regularity.

Additionally, we delve into the potential extension of these results to the Navier-Stokes equations in the presence of a solid boundary. Specifically, we consider the case of Dirichlet boundary conditions and our approach avoids any additional assumptions on the kinematic pressure.

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In this talk we present a novel family of high order accurate numerical schemes for the solution of hyperbolic partial differential equations (PDEs) which combines several geometrical and physical structure preserving properties.

Indeed, first, we settle in the Lagrangian framework, where each element of the mesh evolves following as close as possible the local fluid flow, so to reduce the numerical dissipation at contact waves and moving interfaces and to respect the Galilean and rotational invariance of the studied PDEs system. In particular, we choose the direct Arbitrary-Lagrangian-Eulerian setting which, in order to always guarantee the high quality of the moving mesh, allows to combine the Lagrangian motion with mesh optimization techniques.
The employed Voronoi tessellation is thus regenerated at each time step, the previous one is connected with the new one by space-time control volumes, including hole-like sliver elements in correspondence of topology changes, over which we integrate a space-time divergence form of the original PDEs through a high order accurate ADER discontinuous Galerkin (DG) scheme. Mass conservation and the respect of the GCL condition are guaranteed by construction thanks to the integration over closed control volumes, and robustness over shock discontinuities is ensured by the use of an a posteriori sub-cell finite volume (FV) limiter. On the top of this effective moving mesh framework, we have also made, for the first time in literature, the full ALE ADER DG scheme with a posteriori sub-cell FV limiter well balanced in order to increase our resolution on small perturbations of equilibrium solutions.

The presentation is closed by a wide set of numerical results, including simulations of Keplerian disks, which demonstrate all the claimed properties and the increased accuracy and robustness of our novel family of schemes. We will also give an outlook on future works on the Einstein field equations, based on our recent results obtained with Cartesian methods.

Acknowledgement:

E. Gaburro gratefully acknowledges the support received from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement SuPerMan (No. 101025563).


Contatti: luca.formaggia@polimi.it

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Questo seminario prende spunto dai risultati ottenuti in due lavori, il primo del 2014 in collaborazione con Elvise Berchio e Gabriele Grillo ed il secondo del 2023 in collaborazione ancora con Elvise Berchio e con Debdip Ganguly e Prasun Roychowdhury. Essi trattano le proprietà di stabilità per le soluzioni di equazioni ellittiche con non linearità di tipo potenza o di tipo esponenziale su varietà Riemanniane con un polo di simmetria.
Prima di entrare nello specifico dei problemi trattati nei due lavori sopra citati, si procederà con un'introduzione sul significato in ambito astrofisico di tali equazioni nello spazio euclideo tridimensionale. Si passerà poi alla presentazione di alcuni modelli nell'ambito della fluidodinamica che si riconducono allo studio di varianti delle suddette equazioni su sfere dello spazio euclideo.
Infine si passerà a trattare le equazioni su varietà Riemanniane essenzialmente con curvatura definitivamente negativa per grandi distanze dal polo di simmetria, tra le quali rientra come caso particolare lo spazio iperbolico; seguirà poi la trattazione dei principali risultati presenti sull'argomento in letteratura ed in particolare nei due articoli del sottoscritto.

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The well-posedness of relevant physical models expressed in terms of partial differential equations (PDE) hinges on subtle analytical, homological, and algebraic properties underlying Hilbert complexes [1]. The best-known example is the de Rham complex which, in practically relevant situations, can be expressed through vector proxies as the sequence of Hilbert spaces H1, H(curl), H(div), and L2 connected by the vector calculus operators grad, curl, div. In the first part of this presentation, we illustrate the role of the de Rham complex in the well-posedness of several PDE problems arising in the modelling of nuclear facilities.

The design of efficient numerical methods for such problems is challenging for several reasons: on one hand, stability requires to mimic, at the discrete level, the homological and analytical properties of the de Rham complex (leading to the notion of “compatible method”); on the other hand, the complicated geometrical features of the domain and behaviours of the solution require a great flexibility in terms of supported meshes and approximation orders. In the second part of this presentation we provide an introduction to the recently introduced Discrete de Rham (DDR) paradigm [3,4] for the design and analysis of compatible discretization methods supporting general polyhedral meshes and arbitrary orders.

The general principle of DDR methods is to replace both spaces and operators by discrete counterparts designed so as to be compatible with the cohomology properties of the continuous complex. Specifically:

- The discrete spaces are spanned by vectors of polynomials with components attached to mesh entities in order to mimic, through their single-valuedness, global (complete or partial) continuity properties of the continuous spaces. The local polynomial spaces can be either full or incomplete.

- The discrete operators are obtained in two steps: first, reconstructions in full polynomial spaces are built mimicking an approximate version of the Stokes formula; second, whenever needed, the L2-orthogonal projection on the appropriate incomplete polynomial space is taken.

- A full set of results for DDR methods have been recently proved [3], including cohomology-related properties, Poincaré inequalities, as well as primal and adjoint consistency of the discrete vector calculus operators. An overview of such results, along with examples of applications, is provided. In [2], the algebraic results have been generalized through the used of differential forms, leading to a Polytopal Exterior Calculus framework.

The material in this presentation is mostly taken from joint works with F. Bonaldi, J. Droniou, K. Hu, F. Rapetti.

[1] D. Arnold. Finite Element Exterior Calculus. SIAM, 2018.
[2] F. Bonaldi, D. A. Di Pietro, J. Droniou, and K. Hu. An exterior calculus framework for polytopal methods. Mar. 2023.
[3] D. A. Di Pietro and J. Droniou. “An arbitrary-order discrete de Rham complex on polyhedral meshes: Exactness, Poincaré inequalities, and consistency”. In: Found. Comput. Math. 23 (2023), pp. 85–164.
[4] D. A. Di Pietro, J. Droniou, and F. Rapetti. “Fully discrete polynomial de Rham sequences of arbitrary degree on polygons and polyhedra”. In: Math. Models Methods Appl. Sci. 30.9 (2020), pp. 1809–1855.


Contatto: paola.antonietti@polimi.it

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I this talk I will present some results on the resolvent expansions of magnetic Hamiltonians at the threshold of the essential spectrum. I will show, in particular, that the nature of the expansion of a two-dimensional magnetic Hamiltonian is completely determined by the flux of the associated magnetic field. Applications to time decay of the wave functions will be discussed as well.

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The growth of online technology companies heavily relies on their ability to innovate and improve their products.
To rigorously measure the positive impact of new interventions, such as newly introduced features, and enable effective data-driven innovation, practitioners routinely rely on A/B tests (or online randomized experiments). In this talk, we will discuss the main challenges and open research questions faced by teams running large-scale experiments in online A/B testing.
After outlining the classic model for online experiments, we will show how recent advances in the field have allowed to improve over this classical approach. In particular, we will discuss how modeling improvements (e.g., Bayesian inference and regression models leveraging covariate adjustments), as well as predictions methods aimed to determine the optimal duration of an experiment have impacted the theory and practice of online experimentation.
We will conclude our discussion with open research questions and challenges, with an emphasis of limitation of experimentation under interference, a common issue in online experimentation.

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The seminar will address general, sharp estimates of the singular values of quantum differentials of Connes spectral triples, on which Noncommutative Geometry is based. Special attention will concern the natural spectral triple of a Dirichlet form and those arising from negative definite functions on countable discrete groups.

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NOSAS is a family of adaptive Schwarz preconditioners designed originally for solving elliptic problems with highly heterogeneous coefficients. It has flavors from both overlapping and nonoverlapping methods. Different from the overlapping methods, like GDSW and GeoEo, NOSAS has no local problems on extended subdomains, and different from the nonoverlapping methods, like BDD, BDDC, FETI and FETI-DP, NOSAS has no weighting operator to average local solutions on the interface. In RASHO, all the local factorizations and eigensolvers are done inside the nonoverlapping subdomains, and like BDD the coarse degrees of freedom are related to the bad nodes inside each of these subdomains, however, with a better corresponding coarse stiffness in terms of sparsity. RASHO is very flexible, can be designed to have condition number from (1+log H/h)^2 to (1+H/h)^2 without solving any eigenvalue problem. Finally, RASHO is easy to implement. In this talk we introduce the predecessor of NOSAS, the Average Additive Schwarz Method introduced by Bjorstad and Dryja 99', and then move to the vanilla version of NOSAS and then to the adaptive versions, economical versions and three-level versions. We show numerical experiments and some comparisons with GeoEo and Adaptive BDDC.

Contatto: paola.antonietti@polimi.it

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The talk discusses the implications of the synchronization of interest rates and credit growth for central bank policy in the Euro Area. The bedrock that sustains the importance of such an issue is found in the growing evidence that monetary policy in the Euro Area has heterogeneous impacts across several dimensions, e.g. across countries, within countries, and across different bank market segments. Given policy asymmetries, it becomes essential for policy makers to quantify the extent of synchronization and divergence across interest rates and credit growth and, accordingly, tailor central bank policy to ensure its effectiveness. We investigate the synchronization of the Eurozone’s government bond yields at different maturities. For this purpose, we combine principal component analysis with random matrix theory. We describe the dynamic of credit granted across Italian provinces over the last two decades. We focus on the synchronization and divergence of the credit growth to identify for which provinces credit grows or decreases similarly and which provinces show a diverging behavior. Moreover, by decomposing credit growth into a common component and components capturing heterogeneity in supply and demand we introduce a framework to support policymaker’s choice between different types of macroprudential instruments.

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We will study the equilibrium configurations of a fluid-structure interaction problem where a body is immersed in a fluid confined in a bounded planar channel and governed by the stationary Navier-Stokes equations with laminar inflow and outflow. The body is subject to both the lift force from the fluid and to some external elastic force. Motivated by an application to suspension bridges, asymmetry is taken into account. This requires the introduction of suitable assumptions to prevent collisions of the body with the boundary. We will present an existence and uniqueness result for sufficiently small inflow/outflow. This talk is based on a recent joint work with F. Gazzola.