Dipartimento di Matematica

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We reflect on the MSCA-IF project MiDiROM, presenting its successes and challenges over the past two years, and discussing the remaining open questions. The project's main objective was to develop efficient computational methods for modeling processes in fractured porous media, by combining mixed-dimensional modeling with deep learning approaches.

This talk focuses on two prominent research themes within MiDiROM. Firstly, we developed multipoint mixed finite element methods for rotation-based formulations of Stokes flow and Biot poroelasticity. By using exact discrete complexes and a carefully selected quadrature rule, we obtained efficient and stable discretization methods using the lowest-order Raviart-Thomas finite element pair.
Secondly, we introduced reduced order modeling techniques for Darcy flow systems capable of generating locally mass conservative flux fields. These methods combine concepts from reduced basis methods, exact discrete complexes, and trained neural networks. Mass conservation is then guaranteed through a "spanning tree solve" and appropriate projection operators.

To conduct numerical simulations, we simultaneously developed the Python package PyGeoN, in which discrete complexes are explicitly available. The conclusion of the talk will emphasize various capabilities of this open-source software, such as mixed-dimensional differential operators and a polygonal mesh generator using aperiodic monotiles.

Contatto:
alessio.fumagalli@polimi.it

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One of the deepest results in hypercomplex analysis is the Fueter extension theorem. It gives a two steps procedure to extend holomorphic functions to the hyperholomorphic setting. The first step gives the class of slice hyperholomorphic functions; their Cauchy formula allows to define the so-called S-functional calculus for noncommuting operators based on the S-spectrum. In the second step, this extension procedure generates monogenic functions; the related monogenic functional calculus, based on the monogenic spectrum, was widely studied by McIntosh and collaborators.
In this talk, I will discuss the main notions of the S-spectrum and some concepts of the monogenic functional calculus. Moreover, I will also give some ideas on the new research direction of the fine structures.

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La ricerca operativa (RO) è una branca della matematica applicata solitamente insegnata a livello universitario. Tuttavia, negli ultimi anni sono state sviluppate diverse iniziative per introdurre la RO agli studenti delle scuole secondarie. A queste iniziative si aggiunge ROAR (Ricerca Operativa Applicazioni Reali), un corso triennale composto da tre unità didattiche, pensato per gli studenti del triennio delle scuole secondarie. In questo seminario ci si concentrerà sulla prima unità didattica, mettendo in luce sia gli aspetti contenutistici sia le metodologie utilizzate per la sua implementazione con gli studenti. In particolare, si delineeranno gli aspetti teorici che hanno guidato la progettazione e poi si mostreranno esempi di problemi di ottimizzazione proposti agli studenti, che mirano a sviluppare le loro capacità di modellizzazione e problem-solving.

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The classification of minimal surfaces of general type is a classical and long-standing research topic. In this context fixing invariants turns out to be fundamental.
In this talk we will review some recent results on the case pg=q=2, which is still widely open in spite of several contributions by many authors over the last two decades. More specifically, given a minimal surface S of general type with pg=q=2, it turns out that the self-intersection K^2 of the canonical divisor K is 4

Ramon Codina is Professor of Continuum Mechanics and Structural Analysis at the Technical University of Catalonia (UPC), where he began his academic career in 1990. He teaches courses on continuum mechanics and mechanics of structures, as well as courses on computational mechanics and functional analysis in mechanics.

His research is concerned with the development, implementation and numerical analysis of finite element methods applied to physical problems in engineering, mainly in fluid mechanics, but also in solids and electromagnetism. He is responsible for the research group Numerical Analysis and Computation, identified as a group of excellence by the Generalitat de Catalunya, and leader of the Fluid Mechanics Group at the International Centre for Numerical Methods in Engineering (CIMNE).

In recognition to his research, he has received several awards, among which the Prandtl Medal from Eccomas (2018), the ICREA Academia Award (2011, 2016 and 2021) from the Catalan Government, the IACM Young Investigator Award (2004), the JL Lions Award to Young Scientists in Computational Mathematics from Eccomas (2000) and the Distinció de la Generalitat de Catalunya per a la Recerca Universitària (2000).

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In this talk we consider the finite element approximation of some mixed formulations of linear elasticity, namely, the displacement-pressure, the displacement-stress and the displacement-pressure-stress approaches. As for any mixed formulation, involving unknowns belonging to different functional spaces, the global inf-sup stability that ensures well posedness of the problem is a consequence of “little” inf-sup conditions that need to be satisfied between the interpolating spaces of the different unknowns. Approximations satisfying these conditions are sometimes difficult to implement, and even very rare, as in the case of the displacement-pressure-stress approach. The alternative is to modify the discrete problem by adding stabilisation terms to the Galerkin equations that yield stable approximations for any choice of the interpolating spaces. We discuss this possibility and show that one may prove stability and convergence in the same norms as for the continuous problem.

Contatto: simona.perotto@polimi.it

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Verranno osservati e discussi alcuni fenomeni di interazione fluido-struttura della vita quotidiana. I comportamenti sia dei fluidi che delle strutture sembrano mettere in discussione i principi elementari del calcolo vettoriale.

David Keyes directs the Extreme Computing Research Center at the King Abdullah University of Science and Technology (KAUST), where he was a founding Dean in 2009 and currently serves in the Office of the President as Senior Associate. He is a professor in the programs of Applied Mathematics, Computer Science, and Mechanical Engineering. He is also an Adjunct Professor of Applied Mathematics and Applied Physics at Columbia University, where he formerly held the Fu Foundation Chair. He works at the interface between parallel computing and PDEs and statistics, with a focus on scalable algorithms that exploit data sparsity. Before joining KAUST, Keyes led multi-institutional scalable solver software projects in the SciDAC and ASCI programs of the US Department of Energy (DoE), ran university collaboration programs at US DoE and NASA institutes, and taught at Columbia, Old Dominion, and Yale Universities. He is a Fellow of SIAM, the AMS, and the AAAS. He has been awarded the Gordon Bell Prize from the ACM, the Sidney Fernbach Award from the IEEE Computer Society, and the SIAM Prize for Distinguished Service to the Profession. He earned a B.S.E. in Aerospace and Mechanical Sciences from Princeton in 1978 and a Ph.D. in Applied Mathematics from Harvard in 1984.

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Numerical software is being reinvented to provide opportunities to tune dynamically the accuracy of computation to the requirements of the application, resulting in savings of memory, time, and energy. Floating point computation in science and engineering has a history of “oversolving” relative to expectations for many models. So often are real datatypes defaulted to double precision that GPUs did not gain wide acceptance until they provided in hardware operations not required in their original domain of graphics. However, computational science is now reverting to employ lower precision arithmetic where possible. Many matrix operations considered at a blockwise level allow for lower precision and many blocks can be approximated with low rank near equivalents. This leads to smaller memory footprint, which implies higher residency on memory hierarchies, leading in turn to less time and energy spent on data copying, which may even dwarf the savings from fewer and cheaper flops. We provide examples from several application domains, including a look at entries that earned Gordon Bell Prize finalist status in 2022 and 2023.

Contatti:
alfio.quarteroni@polimi.it
paola.antonietti@polimi.it

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Il tema dell'argomentazione, nella sua declinazione nei diversi ambiti disciplinari, è chiaramente espresso nelle indicazioni Nazionali, ma spesso disatteso nella pratica scolastica.
Educare all'argomentazione presenta problemi didattici specifici che, nell'ambito matematico, si articolano con i problemi didattici relativi sia alla costruzione dei concetti che all'introduzione ad una prospettiva teorica, ed in particolare alla pratica della dimostrazione.
Da qualche decennio la ricerca in didattica della matematica affronta i problemi che emergono in ambito scolastico in relazione all'argomentare, al dimostrare e al loro mutuo rapporto.
Il seminario ha l'obiettivo di discutere alcuni di questi problemi didattici proponendo, alla luce dalla ricerca in questo ambito, qualche esempio di buone pratiche da sviluppare in classe.

Sauro Succi serves as Senior Research Executive and Principal Investigator at the Center for Life Nano-Neuro Sciences at la Sapienza of the Italian Institute of Technology and Research Affiliate of the Physics Department of Harvard University. He holds a degree in nuclear engineering from the University of Bologna and a PhD in physics from EPFL. His research activity covers a broad range of topics related to the mathematical modeling and simulation of complex states of flowing matter, such as thermonuclear plasmas, fluid turbulence, soft matter as well as quantum and subnuclear fluids such as quark gluon plasmas. He is best known for his contributions to the early inception, development and application of the Lattice Boltzmann method, for which he has received a number of international awards, including the Alexander von Humboldt Award in Physics (2002), the Raman Chair of the Indian Academy of Sciences (2012), the APS Aneesur Rahman Prize in Computational Physics (2017), and the CECAM Berni Alder Prize (2019) for exceptional contributions to the microscopic simulation of matter. He is also a ERC Advanced Grant and PoC awardee (2017,2022), an elected member of Academia Europaea (2015) and a Honorary Professor at University College London (2022).

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We present a pedagogical introduction to the current state of quantum computing algorithms for the simulation of classical fluids. Different strategies, such as quantum nonlinear ODE solvers, Carleman-Koopman linearization, functional Liouville formulations and variational quantum eigensolvers, will be briefly discussed and commented on. We shall then focus on two distinct quantum computing implementations of the Carleman-Koopman procedure, one based on quantum linear algebra solvers and the other on a second-quantized Dirac-Majorana formulation of the Lattice Boltzmann equation. Finally, we shall offer a brief mid-term perspective of quantum versus exascale computing for fluid flows.

Contatto: alfio.quarteroni@polimi.it

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Quale forma assume una catena omogenea appesa ai suoi estremi quando è sottoposta alla sola forza peso? Nel XVII secolo questa domanda ha impegnato le menti di grandi scienziati, che hanno individuato nella cosiddetta catenaria (una curva esprimibile in termini di funzioni iperboliche) la risposta. Per via di questa sua particolarità, essa è stata utilizzata in architettura, soprattutto da Antoni Gaudí. In questo seminario vedremo qualche esempio di questo utilizzo e mostreremo che la catenaria svolge un importante ruolo anche nella descrizione delle forme assunte dalle pellicole di acqua saponata.

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L’introduzione del calcolo letterale nell’ultimo anno della scuola secondaria di primo grado spesso si rivela come il momento in cui si manifesta la distanza tra l’essere in grado di svolgere una procedura e l’averla compresa, e che in genere segna una svolta irreversibile verso l’abbandono di interesse e impegno da parte degli studenti. Una delle cause dell’insorgere di questa distanza è la relativa facilità d’uso del linguaggio algebrico e delle sue regole sintattiche, che rischia di oscurarne la profondità semantica. Nel seminario si intende proporre alcuni spunti di riflessione affinché l’introduzione della “lettera” sia un passaggio di maturazione nell’acquisizione del linguaggio della Matematica nella pienezza della sua potenzialità.

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The two function theories of monogenic and of slice monogenic functions with values in a Clifford algebra have been extensively studied in the literature. Although these functions are very different in nature they can be related via the celebrated Fueter-Sce mapping theorem or via Radon transform. In this talk we introduce a new class of functions, that of generalized partial-slice monogenic functions, which includes both of them as special cases. For this class of functions we can prove main properties such as the identity theorem, a Representation Formula, two different types of extension theorems, the Cauchy integral formula. The talk is based on a joint work with Zhenghua Xu.

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Geomstats is an open-source Python package for computations, and statistics on nonlinear manifolds. We provide object-oriented and extensively unit-tested implementations. Manifolds can be equipped with Riemannian metrics with associated exponential and logarithmic maps, geodesics, and parallel transport. Some manifolds can also be endowed with additional mathematical structures, such as Lie group, or fiber bundle structures. Statistics and learning algorithms provide methods for estimation, clustering, and dimensionality reduction on manifolds. All associated operations are supported in different backends, namely NumPy, Autograd, and PyTorch.
In this talk, we briefly introduce the main concepts in Riemannian geometry and discuss the package design. We show that Geomstats provides reliable building blocks to both foster research in differential geometry and statistics and democratize the use of Riemannian geometry in statistics and machine learning. The source code is freely available under the MIT license at github.com/geomstats/geomstats.

Contatti:
paola.antonietti@polimi.it
edie.miglio@polimi.it
marco.verani@polimi.it

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La teoria dei giochi, nata verso la metà del secolo scorso, è una disciplina matematica che ambisce a studiare il comportamento degli esseri umani (e non solo) nelle situazioni di interazione, in cui sia necessario compiere delle scelte strategiche. Negli ultimi decenni diversi matematici hanno vinto il premio Nobel per l'economia grazie alle loro ricerche nell'ambito della teoria dei giochi, che si è dimostrata estremamente proficua ed efficace in svariati ambiti di applicazione: politica, medicina, informatica. Nonostante ciò, le ipotesi della teoria e alcuni dei concetti fondamentali, come quello di equilibrio di Nash, sono semplici e possono essere accessibili a tutti.
In questo seminario illustreremo cosa sia la teoria dei giochi, fornendo oltre agli elementi teorici della disciplina e alcuni esempi di applicazione anche degli spunti didattici per affrontare l'argomento con gli studenti della scuola secondaria.

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We consider the 2D Euler equations on $\mathbb{R}^2$ in vorticity form, with unbounded initial vorticity, perturbed by a suitable non-smooth Kraichnan transport noise, with regularity index $\alpha\in (0,1)$.
We show weak existence for every $\dot{H}^{-1}$ initial vorticity. Thanks to the noise, the solutions that we construct are limits in law of a regularized stochastic Euler equation and enjoy an additional $L^2([0,T];H^{-\alpha})$ regularity.
For every $p>3/2$ and for certain regularity indices $\alpha \in (0,1/2)$ of the Kraichnan noise, we show also pathwise uniqueness for every $L^p$ initial vorticity. This result is not known without noise.
Joint work with Michele Coghi.

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A Product-Quotient surface is the minimal resolution of singularities of a quotient of a product of curves by the action of a finite group of automorphisms.
Product-Quotient surfaces are introduced by Catanese in a famous paper of 2000 and then extensively studied by several authors. Indeed, they are revealed to be a very useful tool for building new examples of algebraic surfaces and exploring their geometry in an accessible way. Hence a classification of them (by fixing some invariants, e.g. K^2 and \chi), is not only interesting by itself, but also also highly practical in various contexts.

During the talk, I will introduce one of the still open problems on the degree of the canonical map of surfaces of general type and I will show which is the role that Product-Quotient surfaces have been played on the results that I have given.
Hence I will provide a brief overview on Product-Quotient surfaces and I will describe the most important tools that are developed by some authors to produce a classification of them via a computational algebra system (e.g. MAGMA).

I will introduce the results I have obtained to provide a more performant algorithm. The main result is a theorem that allows us to move from a database of G-coverings of the projective line (in pairs), that already exists and it has been produced in a recent work by Conti, Ghigi and Pignatelli, to a database of families of Product-Quotient surfaces.

I have used this to produce a huge list of families of Product-Quotient surfaces with pg=3,q=0, and K^2 high. The classification is almost complete for K^2 equal to 32, with very few exceptions.
Finally, if time permits, I will give more details of what I did with this list to study the already mentioned open question on the degree of the canonical map of surfaces of general type.

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Algebraic Multigrid (AMG) is the preconditioner of choice in several applications due to its effectiveness in solving linear systems arising from the discretization of PDEs. The key feature of AMG is its optimality that is its ability to guarantee, at least in some problems, a convergence rate that is independent of the mesh size. This feature is obtained through a good interplay between the smoother and the interpolation. Unfortunately, for difficult problems such as those arising from linear elasticity or diffusion problems with strong contrasts in the coefficients, standard smoothers and interpolation techniques are not enough to ensure fast convergence. In these cases, improving the prolongation operator by minimizing its energy norm may greatly enhance AMG convergence. In this talk, we show how energy minimization can be seen as a constrained minimization problem, where the constraint is twofold: the prolongation must be sparse, and its range must contain the operator near-kernel. To solve this problem, we propose an iterative algorithm based on the null-space method and preconditioning to speed-up the set-up time. We will show, with the aid of a few numerical experiments arising from real-world applications, that this approach is perfectly parallel and can lead to convergence rates significantly higher than those from standard interpolation techniques at a reasonable set-up cost.


Contatto: luca.formaggia@polimi.it

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Solving differential problems using full order models (FOMs), like the finite element method, incurs prohibitively computational costs in real-time simulations and multi-query routines. Reduced order modeling aims at replacing FOMs with reduced order models (ROMs), that exhibit significantly reduced complexity while retaining the ability to capture the essential physical characteristics of the system.
Traditional ROMs, rooted in the assumption of linear modal superimposition, such as proper orthogonal decomposition (POD), can prove inadequate when addressing nonlinear time-dependent parametrized partial differential equations (PDEs), especially for scenarios involving coherent structures evolving over time. In response, an alternative approach is proposed, based on deep learning (DL) algorithms, leveraging tools like convolutional neural networks (CNNs), to construct an efficient nonlinear surrogate. In the resulting DL-ROM both the nonlinear trial manifold and the nonlinear reduced dynamics are learned non-intrusively through DL models trained on a dataset of FOM snapshots corresponding to different parameter values. Accuracy and efficiency of the DL-ROM technique are evaluated across various applications, demonstrating the capability to perform real-time computations for new queries.
After presenting some recent achievements in this field, a quick overview of the ongoing research, sketching a list of open issues of potential interest, will be provided.

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Neurodegenerative disorders are an increasing health and societal issue worldwide. Many of these diseases (such as Alzheimer’s and Parkinson’s) are related to an impairment of the waste clearance system in the brain. Indeed, in physiological conditions, brain activity produces waste proteins that are collected and removed by the Cerebrospinal Fluid (CSF), the main component of the glymphatic system. In this talk, I present a multiphysics model coupling Stokes equations for the CSF flow in the brain ventricles and subarachnoid space with dynamic Multiple-Network Poroelastic (MPE) equations for the perfusion of the brain tissue by CSF and blood [Corti et al. 2023; Causemann et al. 2022]. The model is discretized by a discontinuous Galerkin method based on polytopal grids, which can naturally comply with the geometrical complexity of the domain. An a priori analysis of the method is provided, together with numerical tests in realistic geometries. Then, perspectives are discussed on further developments of the model, its numerical discretization, and its employment to address clinical questions.