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Seminar on 20/01 (Fernando De Terán)
by Leonardo Robol 20 Jan '20

20 Jan '20

Speaker: Fernando De Terán Affiliation: Universidad Carlos III de Madrid Time: Monday, 20/01/2020, 14:00 Place: Aula Magna, Dipartimento di Matematica. Title: Flanders’ theorem for many matrices under commutativity assumptions Given two matrices $A \in C^{m \times n}$ and $B \in C^{n \times m}$ , it is known [1] that the Jordan canonical form of AB and BA can only differ in the sizes of the Jordan blocks associated with the eigenvalue zero, and the difference in the size of any two corresponding blocks is, at most, 1. Moreover, this change is exhaustive, in the sense that given two ordered lists of numbers such that the corresponding elements differ at most by 1, then there are two matrices A, B such that the fist list is the list of sizes of Jordan blocks associated with 0 in AB and the second one is the list of sizes of Jordan blocks associated with 0 in BA. The motivation of this work is the question: What happens with products of more than two matrices? First, we will see that, if we do not impose any restriction, the products ABC and CBA may have completely different eigenvalues. Therefore, some conditions must be imposed to the factors in order to be able to relate the Jordan canonical forms of any two products. By imposing some natural commutativity assumptions, we analyze the relationship between the Jordan canonical form of products, in different orders, of k square matrices $A_1, ..., A_k$. In particular, we study permuted products of $A_1, ..., A_k$ under the assumption that the graph of non-commutativity relations is a forest. Under this condition, we show that the Jordan structure of all nonzero eigenvalues is the same for all permuted products. For the eigenvalue zero, we obtain an upper bound on the difference between the sizes of Jordan blocks for any two permuted products, and we show that this bound is attainable. For k = 3 we show that, moreover, the bound is exhaustive. This is joint work with Ross A. Lippert, Yuji Nakatsukasa, and Vanni Noferini. https://www.dm.unipi.it/webnew/it/seminari/flanders%E2%80%99-theorem-many-m…

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Seminar on 11/12 (Vanni Noferini)
by Leonardo Robol 11 Dec '19

11 Dec '19

Speaker: Vanni Noferini Affiliation: Aalto University, Finland Time: Wednesday, 11/12/2019, 11:00 Place: Aula Seminari, Dipartimento di Matematica. Title: Wilkinson's bus: weak condition numbers, with applications (Joint work with Martin Lotz) Simple eigenvalues of singular matrix polynomials have infinite condition number. However, vanilla eigensolvers (blind to the singular structure) are often able to compute these eigenvalues to machine precision. We propose and analyse a more sophisticated theory of condition, with higher predicting power than the existing ones. Don't miss this talk if you would like to hear the solution of the mystery of the ill posed eigenvalue problems that are in practice solved with remarkable accuracy! https://www.dm.unipi.it/webnew/it/seminari/wilkinsons-bus-weak-condition-nu…

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Cena di Natale
by Leonardo Robol 04 Dec '19

04 Dec '19

Carissimi, anche quest'anno vorremmo provare a organizzare una cena prima di Natale; si pensava di organizzarla con un po' di anticipo in modo da poter includere anche gli ospiti che abbiamo in dipartimento in questo periodo (i.e., Vanni e Massimiliano). Ho parlato solo con alcuni di voi, e ci sarebbe abbastanza convergenza per il 9, 10 oppure l'11 di dicembre (lunedì, martedì, o mercoledì della settimana prossima). Se pensate di partecipare, vi chiederei di riempire il Doodle che trovate a questo indirizzo [1], possibilmente entro un paio di giorni così che poi possa prenotare. Per il momento c'è un certo supporto per rispolverare un classico e tornare a Montemagno (ammesso di trovare posto), ma ovviamente proposte per la location sono sembre bene accette! L'invito è esteso a numerici e simpatizzanti, chiunque abbia piacere a partecipare si può considerare incluso. A presto! -- Leonardo. [1] https://doodle.com/poll/fg22d7ezy7ziwcy2

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Seminar on 04/12 (Stefano Massei)
by Leonardo Robol 27 Nov '19

27 Nov '19

Speaker: Stefano Massei Affiliation: EPF Lausanne, Switzerland Time: Wednesday, 04/12/2019, 11:00 Place: Sala Seminari Est, Dipartimento di Informatica. Title: Rational Krylov for Stieltjes matrix functions: convergence and pole selection Evaluating the action of a matrix function on a vector, that is x = f(M)v, is an ubiquitous task in applications. When the matrix M is large, subspace projection method, such as the rational Krylov method, are usually employed. In this work, we provide a quasi-optimal pole choice for rational Krylov methods applied to this task when f(z) is either Cauchy-Stieltjes or Laplace-Stieltjes (or, which is equivalent, completely monotonic) for a positive definite matrix M. Then, we consider the case when the argument M has the Kronecker structure M = I⊗A - B^T⊗I, and is applied to a vector obtained vectorizing a low-rank matrix. This finds application, for instance, in solving fractional diffusion equation on rectangular domains. We introduce an error analysis for the numerical approximation of x. Pole choices and explicit convergence bounds are given also in this case. https://www.dm.unipi.it/webnew/it/seminari/rational-krylov-stieltjes-matrix…

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Seminar on 27/11 (Gerald Williams)
by Leonardo Robol 27 Nov '19

27 Nov '19

Speaker: Gerald Williams Affiliation: University of Essex, UK Time: Wednesday, 27/11/2019, 11:00 Place: Aula Magna, Dipartimento di Matematica Title: The role of circulant matrices in the study of cyclically presented groups A circulant matrix is a square matrix in which each row is obtained from its predecessor by a cyclic shift by one column. A cyclically presented group is one that is defined by a group presentation (ie a description of a group in terms of generators and defining relations) that admits a similar cyclic symmetry. Prominent examples are the Fibonacci groups, where the presentations' defining relations mimic the Fibonacci recurrence relation. Questions that arise when studying cyclically presented groups ask when such a group is finite or perfect, when it is 3-manifold group or knot group; if so, what more can we say about the group. Knowledge of the determinant, rank, and the Smith normal form of a related circulant matrix can help to answer these questions. In this talk I will discuss how well-known properties of circulant matrices have been used to gain insight into the structure of cyclically presented groups, and discuss recent and ongoing work with Vanni Noferini that is advancing the theory to provide new tools for studying such groups. https://www.dm.unipi.it/webnew/it/seminari/role-circulant-matrices-study-cy…

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Seminar on 20/11 (Enrico Facca)
by Leonardo Robol 19 Nov '19

19 Nov '19

Speaker: Enrico Facca Affiliation: Scuola Normale Superiore, Pisa Time: Wednesday, 20/11/2019, 11:00 Place: Aula Tonelli, SNS Title: A bio-inspired optimization tool, the Dynamic Monge-Kantorovich model. Numerical solutions and applications In this talk I will present the Dynamical Monge-Kantorovich model, a PDE system coupling an elliptic equation with a diffusion coefficient that change in time according to a non-linear dynamics. The steady state of this system has been related to the different Optimal Transport problems, offering an efficient numerical scheme for their solution. We will explore some examples of application, like identification of the Cut-Locus of a point on manifold and the study of complex natural network. The simplicity of the model allows simple but effective transformations of its core equations, that have been shown to be connected with different optimization problems, such as the Basis Pursuit Problem and the Shape Optimization Problem. https://www.dm.unipi.it/webnew/it/seminari/bio-inspired-optimization-tool-d…

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Lectio Luca Trevisan, "A Theory of Spectral Clustering"
by Federico Poloni 17 Nov '19

17 Nov '19

Segnalo la presenza di questo colloquio, che non fa parte della nostra serie di seminari ma ha un tema che potrebbe essere di interesse per gli iscritti alla lista. È una "lectio magistralis" che fa parte delle celebrazioni per i 50 anni del corso di laurea in informatica. Si terrà nell'aula magna del polo Pontecorvo (edificio E) lunedì 18 novembre alle 15:30. Cordialmente, -federico -------- Forwarded Message -------- Subject: [Dipartimento.di] Lectio Luca Trevisan Date: Fri, 15 Nov 2019 09:55:55 +0100 From: Gian Luigi Ferrari <gian-luigi.ferrari(a)unipi.it> To: CDD <cdd(a)di.unipi.it>, dipartimento(a)di.unipi.it Lo avevo comunicato in consiglio ma mi sono scordato di mandarlo via mail. Rimedio Title: A Theory of Spectral Clustering Speaker: Luca Trevisan Abstract: Spectral clustering algorithms find clusters in a given network by exploiting properties of the eigenvectors of matrices associated with the network. As a first step, one computes a spectral embedding, that is a mapping of nodes to points in a low-dimensional real space; then one uses geometric clustering algorithms such as k-means to cluster the points corresponding to the nodes. Such algorithms work so well that, in certain applications unrelated to network analysis, such as image segmentation, it is useful to associate a network to the data, and then apply spectral clustering to the network. In addition to its application to clustering, spectral embeddings are a valuable tool for dimension-reduction and data visualization. The performance of spectral clustering algorithms has been justified rigorously when applied to networks coming from certain probabilistic generative models. A more recent development, which is the focus of this lecture, is a worst-case analysis of spectral clustering, showing that, for every graph that exhibits a certain cluster structure, such structure can be found by geometric algorithms applied to a spectral embedding. Such results generalize the graph Cheeger’s inequality (a classical result in spectral graph theory), and they have additional applications in computational complexity theory and in pure mathematics. ——— Bio: Luca Trevisan is a professor of Computer Science at Bocconi University. Luca studied at the Sapienza University of Rome, he was a post-doc at MIT and at DIMACS, and he was on the faculty of Columbia University, U.C. Berkeley, and Stanford, before returning to Berkeley in 2014 and, at long last, moving back to Italy in 2019. Luca's research is in theoretical computer science, and it is focused on computational complexity and graph algorithms. Luca received the STOC'97 Danny Lewin (best student paper) award, the 2000 Oberwolfach Prize, and the 2000 Sloan Fellowship. He was an invited speaker at the 2006 International Congress of Mathematicians. He is a recipient of a 2019 ERC Advanced Grant. ______________________________________________ Dipartimento.di mailing list Dipartimento.di(a)listgateway.unipi.it http://listgateway.unipi.it/mailman/listinfo/dipartimento.di

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Seminar on 13/11 (Massimiliano Fasi)
by Leonardo Robol 12 Nov '19

12 Nov '19

Speaker: Massimiliano Fasi Affiliation: Department of Mathematics, The University of Manchester Time: Wednesday, 13/11/2019, 11:00 Place: Aula Seminari, Dipartimento di Matematica Title: Handling the conditioning of extreme-scale matrices In order to assess experimentally the stability of algorithms for the solution of systems of linear equations, it is typically desirable to have a certain degree of control over the condition number of the test matrices being used. If the tests are being performed at scale (e.g., in an HPL benchmark run for a TOP500 submission), it is necessary to ensure that generating the test data will take up only a negligible portion of the overall execution time required to solve the linear system. We develop two new techniques that satisfy these requirements and can be used to efficiently construct extremely large matrices with preassigned 2-norm condition number. Focusing on distributed memory environments, we discuss how these can be implemented in a communication-avoiding fashion. https://www.dm.unipi.it/webnew/it/seminari/handling-conditioning-extreme-sc…

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Colloquio della Classe di Scienze - Alfio Quarteroni
by Leonardo Robol 23 Oct '19

23 Oct '19

Dear all, Alfio Quarteroni will give a presentation at SNS next Wednesday at 15:00, which might be of interest to some of you. The topic is "Taking mathematics to heart"; you can find more information, including venue and abstract, on the SNS website [1]. [1] https://www.sns.it/it/evento/taking-mathematics-to-heart Best wishes, Leonardo Robol.

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Seminari autunnali NumPI
by Leonardo Robol 09 Oct '19

09 Oct '19

Carissimi, prossimamente si terranno alcuni seminari numerici, al Dipartimento di Matematica / Informatica e in Normale, come l'anno scorso. Invito chi di voi fosse potenzialmente interessato a partecipare a compilare il Doodle [1], in modo da poter scegliere un giorno ed un'ora che non coincida con i vostri impegni / lezioni (o almeno, che lo faccia per il minor numero possibile di persone). Per comodità, il Doodle è per date della prossima settimana, ma _non ci sarà_ davvero un seminario la settimana prossima! Probabilmente i primi saranno a fine ottobre / inizio di novembre. Grazie, a presto. [1] https://doodle.com/poll/5p2fm3brdrnu6h4f -- Leonardo Robol <leonardo.robol(a)unipi.it>

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