Tra il XVI e il XVIII secolo è avvenuta, in Europa, una rivoluzione cosmologica. Si tratta di molto più di un semplice cambiamento “geometrico” (il Sole al centro invece della Terra al centro); si tratta della sostituzione di tutto un insieme di conoscenze e di concezioni. Le nuove concezioni sono state sviluppate in modo non sincrono, talvolta secondo le osservazioni empiriche ma talvolta anche contro le osservazioni empiriche, e hanno preso il sopravvento quando sono arrivate a costituire un insieme sufficientemente completo e coerente.
Vengono percorsi i passaggi principali, attraverso i protagonisti chiave (Copernico, Tycho Brahe, Keplero, Galileo, Newton), sia astronomi osservatori sia puramente speculativi, le loro scoperte e i loro tentativi di interpretazione, cercando di mettere in luce alcuni snodi che sono particolarmente interessanti dal punto di vista epistemologico.
The Material Point Method for the simulation of water-related hazards and their interaction with critical structures
Antonia Larese, Universita degli studi di Padova & Technical University of Munich
In recent years, natural hazards involving large mass movements such as landslides, debris flows, and mud flows have been increasing their frequency and intensity as a consequence of climate change and other related factors. These phenomena often carry huge rocks and heavy materials that may, directly or indirectly, cause damage to our structures resulting in a relevant socio-economic impact.
The numerical simulation of the above events still represents a big challenge mainly for two reasons: the need to deal with large strain regimes and the intrinsic multiphysics nature of such events.
While the Finite Element Method (FEM) represents a recognized, well established and widely used technique in many engineering fields, unfortunately it shows some limitation when dealing with problems where large deformation occurs. In the last decades many possible alternatives have been proposed and developed to overcome this drawback, such as the use of the so called particle-based methods. Among these, the Material Point Method (MPM) blends the advantages of both mesh-based and mesh-less methods. MPM avoids the problems of mesh tangling while preserving the accuracy of Lagrangian FEM and it is especially suited for non linear problems in solid mechanics and fluid dynamics.
The talk will show some recent advances in MPM formulations , presenting both an irreducible and mixed formulation stabilized using variational multiscale techniques, as well as the partitioned strategies to couple MPM with other techniques such as FEM or DEM [2, 3]. All algorithms are implemented within the Kratos-Multiphysics open-source framework and available under the BSD license.
 Iaconeta, I., Larese, A., Rossi, R. and Guo, Z., Comparison of a Material Point Method and
a Meshfree Galerkin Method for the simulation of cohesive-frictional materials, Materials, 10 , 1150, (2017).
 Chandra, B., Singer, V., Teschemacher, T., Wuechner, R. and Larese, A., Nonconforming
Dirichlet boundary conditions in Implicit Material Point Method by means of penalty augmentation,
Acta Geotechnica, 16(8), 2315-2335 (2021).
 Singer, V., Sautter, K.B., Larese, A., Wuchner, R. and Bletzinger, K.U.,, A Partitioned
Material Point Method and Discrete Element Method Coupling Scheme , Under revision in Advanced Modeling and Simulation in Engineering Sciences (2022).
Long-time behavior for local and nonlocal porous medium equations with small initial energy
Bruno Volzone, Università degli Studi di Napoli 'Parthenope'
In the first part of the talk, we will describe some aspects of a study developed in a joint paper with L. Brasco concerning the long-time behavior for the solution of the Porous Medium Equation in an open bounded connected set, with smooth boundary and sign-changing initial datum. Homogeneous Dirichlet boundary conditions are considered. We prove that if the initial datum has sufficiently small energy, then the solution converges to a nontrivial constant-sign solution of a sublinear Lane-Emden equation, once suitably rescaled.
We also give a sufficient energetic criterion on the initial datum, which permits to decide whether convergence takes place towards the positive solution or to the negative one. The second part of the talk will be devoted to some new advances obtained in collaboration with G. Franzina, in the spirit of the ones explained above, for the study of the asymptotics of signed solutions for the Fractional Porous Medium Equation.
Optimal design of planar shapes with active materials
Active materials (e.g., polymer gels, liquid crystal elastomers) have emerged as suitable candidates for shape morphing applications, where the configuration of a body is varied in a controlled fashion upon triggering the active response. Given the large variety of these materials, a natural question is to compare different morphing mechanisms for a desired functional shape change and select the most effective one with respect to a certain optimality criterion. To address such a question, we set an optimal control problem that allows to determine the active strains suitable to attain a desired equilibrium transformation, while minimizing the complexity of the activation. Specifically, we discuss the planar morphing of active, hyperelastic bodies in the plain-strain regime, in the absence of external forces.
Our approach aims to be general enough to account for a broad set of active materials through the notion of target metric. For the case of affine shape changes, we derive explicit conditions on the geometry of the reference configuration for the optimality of homogeneous target metrics.
More complex shape changes are then analyzed via numerical simulations. We explore the impact on optimal solutions of different objective functionals, some of them inspired by features of existing active materials. Further, 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 accurate design of active systems and may also contribute to gather insight into the morphing strategies adopted by biological systems, as a result of natural selection.
“A cosa serve la matematica?” Qual è quel docente che non si è mai sentito rivolgere questa domanda? Spesso è una domanda provocatoria, posta dallo studente più "simpatico" della classe. Ma a volte è una domanda sincera, posta da uno sconosciuto in treno. Solitamente siamo troppo stanchi o demoralizzati per rispondere a questa domanda. Ovviamente la matematica è utilissima, nel mondo di oggi, pieno di dati e in cui ogni aspetto della vita è intrecciato ad applicazioni, banali o profonde della matematica. Forse però vale la pena di soffermarsi un attimo in più su tale domanda e cercare di sviscerarla meglio...
Cosa vuol dire "a cosa serve la matematica?"?
La matematica pura è utile?
Perché i politici devono sapere la matematica?
A cosa serve a me la matematica?
L'utilità della matematica è interessante per chi la studia?
Nel seminario più che fornire le mie personali risposte proverò a dare un po' di materiale per pensarci su. Spero possa essere utile per avere qualche idea in più la prossima volta che sentirete queste o altre domande...
Nick Trefethen is Professor of Numerical Analysis and head of the Numerical Analysis Group at Oxford University. He was educated at Harvard and Stanford and held positions at NYU, MIT, and Cornell before moving to Oxford in 1997. He is a Fellow of the Royal Society and a member of the US National Academy of Engineering, and served during 2011-2012 as President of SIAM. He has won many prizes including the Gold Medal of the Institute for Mathematics and its Applications, the Naylor Prize of the London Mathematical Society, and the Polya and von Neumann Prizes from SIAM. He holds honorary doctorates from the University of Fribourg and Stellenbosch University.
As an author Trefethen is known for his books including Numerical Linear Algebra (1997), Spectral Methods in MATLAB (2000), Spectra and Pseudospectra (2005), Approximation Theory and Approximation Practice (2013/2019), Exploring ODEs (2018), and An Applied Mathematician's Apology (2022). He organized the SIAM 100-Dollar, 100-Digit Challenge in 2002 and is the inventor of Chebfun.
Applications of AAA rational approximation
Nick Trefethen, University of Oxford
giovedì 23 febbraio 2023 alle ore 14:00
Aula Consiglio VII piano - Dipartimento di Matematica
For the first time, a method has recently become available for fast computation of near-best rational approximations on arbitrary sets in the real line or complex plane: the AAA algorithm (Nakatsukasa-Sete-T. 2018). We will present the algorithm and then demonstrate a number of applications, including
* detection of singularities
* model order reduction
* analytic continuation
* functions of matrices
* nonlinear eigenvalue problems
* interpolation of equispaced data
* smooth extension of multivariate real functions
* extrapolation of ODE and PDE solutions into the complex plane
* solution of Laplace problems
* conformal mapping
* Wiener-Hopf factorization
Riflettere sugli obiettivi del percorso formativo è cruciale per sviluppare singole attività coerenti e funzionali. Durante il seminario rifletteremo e discuteremo su quali competenze l'educazione matematica dovrebbe promuovere, quali le difficoltà a raggiungere gli obiettivi formativi che ci poniamo con gli studenti rispetto a tali competenze, quali le possibili strategie didattiche.
Filtration of generators and an inverse Fekete--Szego problem
In this talk, based on joint works, we present some results connecting dynamic system with geometric function theory.
In the first part of the talk, we study the problem of characterizing membership of normalized holomorphic functions of the disk to the class of infinitesimal generators and some its subclasses as well as dynamical properties of generated semigroups. Presenting results include analytic extension in the semigroup parameter and the uniform convergence. Our approach is based on so-called `filtrations' of the class of infinitesimal generators.
In the second part we introduce and study a question that can be interpreted as `an inverse Fekete-Szego problem'. This problem links to the first part of the talk. We introduce new filtration classes using a suitable non-linear differential operator
and establish certain properties of these classes. Sharp upper bounds of the absolute value of the Fekete--Szego functional over some filtration classes are found. We also present open problems for further study.
We construct a comprehensive dataset on a near universe of non-fungible token (NFT) transactions, create indices for the NFT market and its components, and analyze their properties. The NFT market return is significantly exposed to the cryptocurrency market return, but the majority of the NFT market variations remain unexplained. NFT market returns have low exposures to other cryptocurrency factors and factors from traditional asset markets. In the time-series, volatility and the NFT valuation ratio significantly predict NFT market returns. In the cross-section, NFT returns exhibit size and return reversal effects.
A free discontinuity approach to optimal profiles in Stokes flows
In the talk I will consider the problem of finding the optimal shape of an obstacle which minimizes the drag force in an incompressible Stokes flow under Navier conditions at the boundary. I will propose a relaxation of the problem within the framework of free discontinuity problems, modeling the obstacle as a set of finite perimeter and the velocity field as a special function of bounded deformation (SBD): within this approach, the optimal obstacle may develop naturally geometric features of co-dimension 1.
In this talk I will discuss recent results about the time evolution of the Fröhlich Hamiltonian in a mean-field limit in which many particles weakly couple to the quantized phonon field. For large particle number and initial data in which the particles are in a Bose-Einstein condensate and the excitations of the phonon field are in a coherent state I will show that the time evolved many-body state can be approximated in norm by an effective dynamics. The approximation is given by a product state which evolves according to the Landau–Pekar equations and which is corrected by a Bogoliubov dynamics.
If time permits I will, in addition, present a joint work with D. Mitrouskas, S. Rademacher, B. Schlein and R. Seiringer about the Fröhlich model in the strong coupling limit and compare the Bogoliubov dynamics in the strong coupling and mean-field regime.
Claudio Canuto was researcher at the CNR Istitute of Numerical Analysis in Pavia until 1986, when he became full professor of Numerical Analysis; he is at Politecnico di Torino since 1989. His scientific work is addressed to the methodological study of discretization methods for partial differential problems, with a particular emphasis on high-order methods, multi-level methods, adaptive methods, and applications to Fluid Dynamics.
The early part of his scientific production concerns the study of spectral methods, for which he co-authored three books and was co-founder with A. Quarteroni of the ICOSAHOM Conference, now at its 14th edition. Subsequently, his research interests have focused on the following topics: wavelet methods, coupling of models, uncertainty quantification, Eulerian models of dynamics on networks, numerical simulation in industry and society, virtual element methods, and physics informed neural networks.
He served for several terms as associate editor of SINUM, J. Scientific Computing, M3AS, CMAME, M2AN. He was the Head of the Department of Mathematics at Politecnico di Torino from 2003 to 2011; currently he is the President of the Scientific Board of the Italian National Institute for Advanced Mathematics (INdAM).
Variational Physics-Informed Neural Networks (VPINN) are an instance of application of deep learning to the solution of boundary-value problems. In VPINN, the network is trained by minimizing a functional of the weak residual of the problem. We look at VPINN from a Petrov-Galerkin perspective, where the spaces of test functions are built by finite elements.
We discuss several ways to enforce essential boundary conditions, and we introduce a variant of VPINN, named IVPINN, which incorporates a piecewise polynomial interpolation of the neural network. This guarantees the numerical stability of the resulting discretization, provided an inf-sup condition between polynomial spaces is fulfilled. The effect of the choice of numerical quadratures is also taken into account. As a consequence, we derive a priori and a posteriori error estimates in the energy norm. Numerical experiments highlight the performances of the method, which depend on the choice of the trial/test function spaces and the architecture of the network.
The presentation is based on joint papers with Stefano Berrone, Moreno Pintore, and N. Sukumar.
In che modo devo disporre una corda per abbracciare la più ampia area possibile? L'antico dilemma di Didone (Eneide Libro I, vv. 365-368) costituisce uno dei più antichi problemi del calcolo delle variazioni. Questo problema si nasconde sotto mentite spoglie in moltissimi contesti fisici, il più sorprendente di tutti: la geometria delle bolle di sapone. Nonostante l'intuito suggerisca che la soluzione è il cerchio, una dimostrazione sufficientemente generale è sfuggita per molti anni e solo in epoca recente si è potuti pervenire ad una prova completa di questo fatto. Il seminario si propone di illustrare in modo semplice il problema e le maggiori difficoltà che si incontrano nel cercare di provare questo risultato. In questo senso il problema costituisce un eccellente esempio per illustrare il concetto di "dimostrazione matematica" e la sua importanza come strumento di comprensione del mondo.