Dipartimento di Matematica

Petros Koumoutsakos is Herbert S. Winokur, Jr., Professor of Engineering and Applied Sciences and Area Chair for Applied Mathematics at Harvard’s John A. Paulson School of Engineering and Applied Sciences (SEAS). He studied Naval Architecture (Diploma-NTU of Athens, M.Eng.-U. of Michigan), Aeronautics, and Applied Mathematics (PhD-Caltech) and has served as the Chair of Computational Science at ETH Zurich (1997-2020). Petros is elected Fellow of the American Society of Mechanical Engineers (ASME), the American Physical Society (APS), and the Society of Industrial and Applied Mathematics (SIAM). He is the recipient of the Advanced Investigator Award by the European Research Council and the ACM Gordon Bell prize in Supercomputing. He is elected International Member of the US National Academy of Engineering (NAE). His research interests are on the fundamentals and applications of computing and artificial intelligence to understand, predict and optimize fluid flows in engineering, nanotechnology, and medicine.

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Over the last thirty years, we have experienced more than a billion-fold increase in hardware capabilities and a dizzying pace of acquiring and transmitting massive amounts of data. Artificial Intelligence (AI) has been the beneficiaries of these advances, and today it is increasingly embedded in technologies that touch every aspect of humanity. However, along with the abundance of promise, there is an ever increasing amount of hype, in particular regarding the capabilities of learning algorithms to model, predict and control complex fluid mechanics problems.
In this talk, I will offer a perspective on forming alloys of AI and simulations for the prediction and control of complex flow systems. I will present novel algorithms for learning the Effective Dynamics (LED) of complex flows and a fusion of multi-agent reinforcement learning and scientific computing (SciMARL) for modeling and control of complex flow-structure interactions. I will juxtapose successes and failures and argue that the proper fusion of fluid mechanics knowledge and AI expertise are essential to advance scientific frontiers.

Contatto: alfio.quarteroni@polimi.it

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The Casimir effect is a manifestation of van der Waals polarization forces acting between atoms and molecules, in a regime where relativistic retardation effects cannot be neglected. An effective description can be given in terms of quantum fields influenced by classical objects,
such as boundaries or external potentials. This leads to the emergence of ultraviolet divergences which can be treated by means of the so-called zeta-function regularization. In this seminar I first present a rigorous implementation of the zeta-function technique, using the language of canonical quantization for a scalar field. To this purpose, complex powers of the elliptic operator describing the dynamics are used to define smeared versions of the Wightman field and renormalization of expectation values is ultimately attained by analytic continuation. Next, I address the issue of boundary anomalies, discussing some explicitly solvable models with zero-range potentials modeling semi-transparent walls.
Based on joint works with Livio Pizzocchero (University of Milano).

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Environmental, social, and governance (ESG) factors have acquired great relevance for investors in recent years. This paper examines the intersection between ESG and crowdfunding, one of the most popular types of fintech. Using data on the population of 508 platforms established in the 37 OECD countries between 2007 and 2020, we document that platforms with higher levels of ESG criteria are more likely to survive over time. Results are robust to different measures of ESG and endogeneity concerns. The importance of ESG criteria depends on cultural differences, as it is more pronounced for those platforms operating in countries where the level of power distance is lower.

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How to identify the geometry of a nanostructure given the desired effective thermal conductivity tensor? While several topology optimization methods have been developed for macroscale scenarios, they can’t be readily applied to nanoscale devices. In fact, when the feature size of the material becomes comparable with the particles’ mean-free path, flux becomes nondiffusive, and a momentum-resolved model is needed. On the other side, standard material interpolation methods, such as SIMP, are mostly applied to averaged quantities, such as the local thermal conductivity. In this talk, I will discuss our recent attempt at solving this issue [1], based on a novel method, termed the “Transmission Interpolation Method” (TIM). Instead of working with locally resolved variables, TIM acts on the interfacial phonon flux and links it to the material distribution. The talk will show the application of this approach to two examples: Tuning the effective thermal conductivity tensor of a nanostructure and maximizing phonon size effects. The talk will conclude with a demo and remarks for future developments.
[1] Giuseppe Romano and Steven G Johnson, Structural and Multidisciplinary Optimization 65, 297 (2022)

Contatto: giulia.pozzi@polimi.it

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Il seminario si propone di analizzare rappresentazioni esterne spontanee o indotte messe in atto da studenti durante la risoluzione di problemi. Le rappresentazioni esterne possono fare da ponte per l’elaborazione di una strategia risolutiva. A volte si trovano schemi personali che riflettono il processo di pensiero utilizzato. Spesso esse sostituiscono anche un’esplicitazione verbale del processo risolutivo, o comunque coesistono con questa in un intreccio tra registro verbale e iconico, o grafico-simbolico. Durante il seminario saranno analizzate le risposte degli studenti con lo scopo di fornire indicazioni didattiche durante attività di problem-solving.

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Il seminario prende in considerazione il ragionamento diagrammatico in geometria elementare e le sue origini nella pratica matematica greca. Si discutono alcuni limiti ed alcune potenzialità di questo approccio ala geometria. Si mostra che il ruolo delle inferenze diagrammatiche diminuì nel corso della storia della geometria moderna anche prima di arrivare alla matematica astratta del diciannovesimo secolo. Le ragioni dell'abbandono delle pratiche diagrammatiche sembra legato alla trasformazione della geometria da una scienza delle figure in una scienza dello spazio. Si mostra allora che tale trasformazione ha preceduto e reso possibile la nascita della matematica astratta e di un ideale dimostrativo non-diagrammatico.

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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 [1], 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.
References
[1] 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).
[2] 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).
[3] 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).
Contatto: anna.scotti@polimi.it

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This talk presents a framework for forward uncertainty quantification problems in spatially-extended neurobiological networks. We will consider networks in which the cortex is represented as a continuum domain, and local neuronal activity evolves according to an integro-differential equation, collecting inputs nonlocally, from the whole cortex. These models are sometimes referred to as neural field equations.
Large-scale brain simulations of such models are currently performed heuristically, and the numerical analysis of these problems is largely unexplored. In the first part of the talk I will summarise recent developments for the rigorous numerical analysis of projection schemes [1] for deterministic neural fields, which sets the foundation for developing Finite-Element and Spectral schemes for large-scale problems.
The second part of the talk will discuss the case of systems in the presence of uncertainties modelled with random data, in particular: random synaptic connections, external stimuli, neuronal firing rates, and initial conditions (and any combination thereof). Such problems give rise to random solutions, whose mean, variance, or other quantities of interest have to be estimated using numerical simulations. This so-called forward uncertainty quantification problem is challenging because it couples spatially nonlocal, nonlinear problems to large-dimensional random data.
I will present a family of schemes that couple a spatial projector for the spatial discretisation, to stochastic collocation for the random data. We will analyse the time-dependent problem with random data and the schemes from a functional analytic viewpoint, and show that the proposed methods can achieve spectral accuracy, provided the random data is sufficiently regular.
Acknowledgements. This talk presents joint work with Francesca Cavallini (VU Amsterdam), Svetlana Dubinkina (VU Amsterdam), and Gabriel Lord (Radboud University).
References
[1] Avitabile D, Projection methods for Neural Field Equations arXiv e-prints,
arXiv:2112.03244 doi.org/10.48550/arXiv.2112.03244

Contatto: simona.perotto@polimi.it

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L’idea di “democrazia” che è ormai ben radicata dentro ciascuno di noi, spesso ci porta a considerare la questione dei sistemi elettorali (ovvero metodi per ottenere, a partire dalle preferenze degli individui di una certa popolazione, una lista di preferenze unica) con una certa leggerezza. Invece, appena si approfondisce la teoria dei sistemi di voto, ci si imbatte immediatamente in situazioni paradossali e problemi imprevisti, quasi sempre di difficile soluzione. Inizieremo questa conferenza discutendo alcuni dei principali problemi e paradossi che si presentano nella progettazione di un sistema elettorale. Nella parte finale ci occuperemo di un problema di diversa natura, ma di notevole importanza, ovvero quello della definizione “imparziale” dei collegi elettorali.

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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.

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“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.

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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

(joint work with Stefano Costa and others)

Contatto: paola.antonietti@polimi.it

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Budding is the main process by which a cell externalizes material and also the main mechanism of virus reproduction [1]. It requires the cell membrane to wrap around a cargo to conform to its surface. The cargo surface (CS), e.g. the nucleocapsid of a virus, defines the shape and size of the bud, thereby providing an important geometrical constraint. The cargo can be described with rigid sphere, with its radius defining the energetic barrier crated by cell membrane bending and stretching. The energetic driving force is then provided by the affinity of the membrane with the CS [2-3]. The tradeoff between bending energy and surface energy provides the minimum radius of the bud that can be successfully completed, hence the minimum size of the virus progeny.


References:

[1] N.J. Dimmock et al. (2007) "Introduction to Modern Virology, 6th edition." Blackwell Publishing.

[2] H. Gao, W. Shi, L.B. Freund (2005), “Mechanics of receptor-mediated endocytosis”, PNAS 102 (27), 9469–9474.

[3] D. Agostinelli, G.J. Elfring, M. Bacca (2021), “Morphological control of receptor-mediated endocytosis”, bioRxiv 2021.10.01.462837.

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The class of holomorphic Hardy Sobolev spaces in the unit ball of $\mathbb C^n$ is a family of spaces including the Hardy, Drury Arveson, Bergman and Dirichlet space. In this talk we will focus on questions regarding exceptional sets both from function theoretic and a functional analytic perspective. These two approaches have led to the past in two different notions of exceptional sets. From the point of view of function theory, exceptional sets are sets where a function in the corresponding Hardy Sobolev space fails to have admissible limits and can be characterized as null sets for some appropriately defined capacity. While from the functional analysis perspective null sets, called totally null sets, play the role of Lebesgue measure zero sets in the Sz.-Nagy-Foias $H^\infty(D)$ functional calculus. The problem of bridging these two approaches will be the main topic of the talk. The talk is based on joint work with Michael Hartz.

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The talk will contain some informal reflections on the roles manifolds have played in statistics historically, Including what I like to call the “Evil Manifold”, the “Casino Manifold” and the "Surprisal Manifold.” I will also alert us to some exciting developments in recent years on optimization over manifolds, including over linear finite element surfaces. It will be argued that we have hardly begun to escape “Decartes/Hilbert Box”, and that exciting times are just around the corner.

Gianluigi Rozza is Full Professor in Numerical Analysis and Scientific Computing at SISSA MathLab – International School for Advanced Studies, Trieste, Italy. Master of Science in Aerospace Engineering (2002) at Politecnico di Milano, PhD in Applied Mathematics (2005) at Ecole Polytechnique Fédérale de Lausanne, Switzerland. Research Assistant (2002-06), Researcher and Lecturer (2008-12) at École Polytechnique Fédérale de Lausanne; Post Doctoral Associate Researcher (2006- 08) at Massachusetts Institute of Technology, Boston MA, USA; Researcher (2012- 14) and Associate Tenured Professor (2014-17) at International School for Advanced Studies, Trieste, Italy.
His research interests include: Numerical Analysis, Numerical Simulation, Scientific Computing; Reduced Order Modelling and Methods with special focus on viscous flows and complex geometrical parametrizations.
Recipient of Bill Morton CFD Prize (2004) by Institute of Computational Fluid Dynamics, University of Oxford (UK), ECCOMAS Ph.D Award (2005) by European Community on Computational Methods in Applied Sciences, Springer Computational Science and Engineering Prize (2009), ECCOMAS Jacques Louis Lions Award in Computational Mathematics (2014), ERC consolidator grant ‘Advanced Reduced Order Methods with Applications in Computational Fluid Dynamics’ (AROMA-CFD, 2016-2021), ERC-Proof of Concept Grant ‘Advanced Reduced Groupware Online Simulation’(ARGOS, 2022). In recent years he has led several computational sciences industrial projects with partners like Fincantieri,Danieli, Electrolux, Wartsila, Cetena, MonteCarloYachts.
He is SISSA director’s delegate for innovation, valorisation and technology transfer since 2016, responsible of SISSA mathLab group since 2018, coordinator of Mathematics area of SISSA since 2020 and recently he has been appointed as president of the advisory board of SMACT, industry 4.0 competence center of North-East regions, grouping 9 universities and other 40 companies.

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We provide a state of the art of Reduced Order Methods (ROM) for parametric Partial Differential Equations (PDEs), and we focus on some perspectives in their current trends and developments, with a special interest in parametric problems arising in offline-online Computational Fluid Dynamics (CFD) with application in industry, medicine and environment. Efficient parametrisations (random inputs, geometry, physics) are very important to be able to properly address an offline-online decoupling of the computational procedures and to allow competitive computational performances. Current ROM developments in CFD include: a better use of stable high fidelity methods, considering also spectral element method, discontinuous Galerkin and finite volume discretisations, to enhance the quality of the reduced model too, and allowing to incorporate turbulent patterns, as well as increasing the Reynolds number and considering compressibility effects; more efficient sampling techniques to reduce the number of the basis functions, retained as snapshots, as well as the dimension of online systems; potential enhancements provided by automatic learning especially in turbulence modelling and in dealing with fast transient dynamics, as well as the guarantee of the stability of the approximation with proper space enrichments or ad hoc segregated solvers.
Contact: paola.antonietti@polimi.it

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In the last two decades great progress has been made in the study of dispersive and wave equations. Over the years the toolbox used in order to attack highly nontrivial problems related to these equations has developed to include a collection of techniques: Fourier and harmonic analysis, analytic number theory, math physics, dynamical systems, probability and symplectic geometry. In this talk I will introduce a variety of results using as model problem mainly the periodic 2D cubic nonlinear Schrödinger equation. I will start by giving a physical derivation of the equation from a quantum many-particles system, I will introduce periodic Strichartz estimates along with some remarkable connections to analytic number theory, I will move on to the concept of energy transfer and its connection to dynamical systems, and I will end with some results on the derivation of a wave kinetic equation.

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In this talk, I will present several functional analysis tools that are instrumental to establish the stability and convergence analysis of nonconforming approximation methods on polyhedral meshes for a wide class of nonlinear problems. Starting from some preliminary results concerning broken Sobolev spaces, I will develop novel arguments to prove the broken version of Poincaré, Korn, trace inequalities and Sobolev embeddings. The main ingredient is a generalization of the local continuous trace inequality allowing to establish the results without restricting to piecewise polynomials or other discrete functional spaces and without requiring the definition of an interpolator mapping the discontinuous functions to continuous ones.

Contatto: paola.antonietti@polimi.it

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Natural muscles provide billions of work cycles involving tension development or movement and convert the chemical energy to mechanical energy with an exceptional high efficiency. The contractile material of the cardiomyocyte may be primarily affected by gene mutations, with detrimental mechanical and energetic consequences. Faster cross-bridge detachment and increased tension cost, for instance, are hallmarks of hypertrophic cardiomyopathy, identified in human myocardial samples and recapitulated by in vitro models. In selected contexts of HCM, cellular mechanisms leading to enhanced excitability may be as, or more, important than reentry macro-substrates in predicting clinical manifestations.

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We formulate the problem of approach to equilibrium in algebraic quantum statistical mechanics and study some of its structural aspects, focusing on the relation between the zeroth law of thermodynamics (approach to equilibrium) and the second law (increase of entropy). Our main result is that approach to equilibrium is necessarily accompanied by a strict increase of the specific (mean) entropy. In the course of our analysis, we introduce the concept of quantum weak Gibbs state which is of independent interest.

This is a joint work with C. Tauber and C.-A. Pillet