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[Indico] Construction of a sequence of orthogonal rational functions,
by leonardo.robol＠unipi.it 05 Apr '22

05 Apr '22

Title: Construction of a sequence of orthogonal rational functions, Speaker(s): Raf Vandebril, Department of Computer Science, KU Leuven, Date and time: 13 Apr 2022, 11:00 (Europe/Rome), Lecture series: Seminar on Numerical Analysis, Venue: Dipartimento di Matematica (Aula Magna). You can access the full event here: https://events.dm.unipi.it/e/86 Abstract -------- Orthogonal polynomials are an important tool to approximate functions. Orthogonal rational functions provide a powerful alternative if the function of interest is not well approximated by polynomials. Polynomials orthogonal with respect to certain discrete inner products can be constructed by applying the Lanczos or Arnoldi iteration to appropriately chosen diagonal matrix and vector. This can be viewed as a matrix version of the Stieltjes procedure. The generated nested orthonormal basis can be interpreted as a sequence of orthogonal polynomials. The corresponding Hessenberg matrix, containing the recurrence coefficients, also represents the sequence of orthogonal polynomials. Alternatively, this Hessenberg matrix can be generated by an updating procedure. The goal of this procedure is to enforce Hessenberg structure onto a matrix which shares its eigenvalues with the given diagonal matrix and the first entries of its eigenvectors must correspond to the elements of the given vector. Plane rotations are used to introduce the elements of the given vector one by one and to enforce Hessenberg structure. The updating procedure is stable thanks to the use of unitary similarity transformations. In this talk rational generalizations of the Lanczos and Arnoldi iterations are discussed. These iterations generate nested orthonormal bases which can be interpreted as a sequence of orthogonal rational functions with prescribed poles. A matrix pencil of Hessenberg structure underlies these iterations. We show that this Hessenberg pencil can also be used to represent the orthogonal rational function sequence and we propose an updating procedure for this case. The proposed procedure applies unitary similarity transformations and its numerical stability is illustrated. -- Indico :: Email Notifier https://events.dm.unipi.it/e/86

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Fwd: [Dipartimento.di] Seminar in Computational Mathematics - B.
by Federico Poloni 18 Mar '22

18 Mar '22

-------- Forwarded Message -------- Subject: [Dipartimento.di] Seminar in Computational Mathematics - B. Filipecki Date: Fri, 18 Mar 2022 13:40:32 +0100 From: Federico Poloni <federico.poloni(a)unipi.it> To: dipartimento(a)di.unipi.it Dear all, within the activities of the Computational Mathematics Lab, we are pleased to announce the following seminar. Title: Hierarchical Optimization of the Electrical Power Grid Speaker: Bartosz Filipecki, Department of Computer Science, University of Pisa Date/Time: Wednesday, 23 March 2022 17:00 Place: Aula Seminari Est, Dipartimento di Informatica Abstract: European electrical power networks operation is becoming more and more difficult due to increasing load and uncertainty associated with renewable energy sources. To address this problem, we consider a hierarchical approach to optimization of power grid operation. First, microgrid demand is aggregated to provide flexibility to the distribution system operator (DSO). At this stage, the optimal power flow problem (OPF) is solved in order to determine the flexibilities that can be made available to the transmission system operator (TSO). Afterwards, OPF is solved to determine the operation of the highest voltage grids, including additional constraints related to security and stability. The former provide protection against line failure, so that no blackout occurs. The latter make sure that the system does not deviate in the dynamic sense from the solution to the static OPF problem. The TSO level can also include Flexible Alternating-Current Transmission Systems Devices (FACTS). These provide us with advanced methods to control the power flow, but at a significant computational cost. After obtaining the solution, we propagate it back through the DSO level to the microgrids, which completes the framework. Everyone is welcome to attend, in person or on MS Teams: https://teams.microsoft.com/l/meetup-join/19%3ameeting_ZGE3ZDcyOTItOTViMS00… Mauro Passacantando Federico Poloni -- --Federico Poloni Dipartimento di Informatica, Università di Pisa https://www.di.unipi.it/~fpoloni/ tel:+39-050-2213143 ______________________________________________ Dipartimento.di mailing list Dipartimento.di(a)listgateway.unipi.it http://listgateway.unipi.it/mailman/listinfo/dipartimento.di

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[Indico] [Event reminder] Grassmann extrapolation for
by leonardo.robol＠unipi.it 17 Mar '22

17 Mar '22

Please note that the event "Grassmann extrapolation for Born-Oppenheimer Molecular Dynamics" will start on 18 Mar 2022, 11:00 (Europe/Rome). It will take place at Dipartimento di Matematica (Aula Magna). You can access the full event here: https://indico.cs.dm.unipi.it/e/76 Description ----------- Born-Oppenheimer Molecular Dynamics (BOMD) is a powerful, yet very demanding technique in computational quantum chemistry. Performing a BOMD simulation require, for each time step, to compute the energy and forces for a quantum mechanical system, which can also be embedded in a classical environment. When Density Functional Theory is used as a quantum mechanical model, the most time-consuming step for each energy/forces evaluation is the solution to the non-linear eigenvalue problem that stems from the discretization of the Kohn-Sham equations. Limiting the number of iterations required to achieve a satisfactory convergence of the procedure is therefore paramount to reduce the cost - and therefore extend the applicability - of such simulations. The most important factor in reducing the number of iteration is starting from a good guess, usually in the form of a one-body reduced density matrix. In a BOMD simulation it is possible to extrapolate information from previous steps into a new guess, however, this is not straightforward due to the nonlinear constraints that the density matrix must satisfy. In this contribution, we address this issue by performing the extrapolation on the tangent plane to the Grassmann manifold where the density is defined, by mapping the manifold to its tangent plane with a locally bijective map, which allows us to perform a linear extrapolation and then map the result back to the manifold, ensuring thus that all the physical properties of the density matrix are correctly enforced. Preliminary benchmark calculations show that the strategy is general and robust, and that even when using a naive linear extrapolation we obtain a strategy that outperforms the current state-of-the-art methods. Streaming link: https://hausdorff.dm.unipi.it/b/leo-xik-xu4 -- Indico :: Email Notifier https://indico.cs.dm.unipi.it/e/76

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[Indico] [Event reminder] Grassmann extrapolation for
by leonardo.robol＠unipi.it 08 Mar '22

08 Mar '22

Please note that the event "Grassmann extrapolation for Born-Oppenheimer Molecular Dynamics" will start on 18 Mar 2022, 11:00 (Europe/Rome). It will take place at Dipartimento di Matematica (Aula Magna). You can access the full event here: https://indico.cs.dm.unipi.it/e/76 Description ----------- Born-Oppenheimer Molecular Dynamics (BOMD) is a powerful, yet very demanding technique in computational quantum chemistry. Performing a BOMD simulation require, for each time step, to compute the energy and forces for a quantum mechanical system, which can also be embedded in a classical environment. When Density Functional Theory is used as a quantum mechanical model, the most time-consuming step for each energy/forces evaluation is the solution to the non-linear eigenvalue problem that stems from the discretization of the Kohn-Sham equations. Limiting the number of iterations required to achieve a satisfactory convergence of the procedure is therefore paramount to reduce the cost - and therefore extend the applicability - of such simulations. The most important factor in reducing the number of iteration is starting from a good guess, usually in the form of a one-body reduced density matrix. In a BOMD simulation it is possible to extrapolate information from previous steps into a new guess, however, this is not straightforward due to the nonlinear constraints that the density matrix must satisfy. In this contribution, we address this issue by performing the extrapolation on the tangent plane to the Grassmann manifold where the density is defined, by mapping the manifold to its tangent plane with a locally bijective map, which allows us to perform a linear extrapolation and then map the result back to the manifold, ensuring thus that all the physical properties of the density matrix are correctly enforced. Preliminary benchmark calculations show that the strategy is general and robust, and that even when using a naive linear extrapolation we obtain a strategy that outperforms the current state-of-the-art methods. Streaming link: https://hausdorff.dm.unipi.it/b/leo-xik-xu4 -- Indico :: Email Notifier https://indico.cs.dm.unipi.it/e/76

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Annunci di seminari NumPI
by Leonardo Robol 08 Mar '22

08 Mar '22

Cari tutti, d'ora in poi -ammesso che la cosa funzioni bene- useremo la piattaforma Indico per inviare gli annunci dei seminari sulla mailing list, dato che lo fa in automatico (reminder inclusi, cosa che io tendo a dimenticare piuttosto facilmente). A breve dovrebbe arrivarne uno. Il formato dell'annuncio sarà leggermente diverso rispetto al solito, ma si tratta sempre dei seminari NumPi. A presto, -- Leonardo.

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Seminar announcement
by Fabio Durastante 07 Mar '22

07 Mar '22

Speaker: Nicola Guglielmi Affiliation: Gran Sasso Science Institute Venue: Aula Volterra, Scuola Normale Superiore *Remark. *Il seminario si terrà in presenza. Chi fosse interessato a partecipare deve scrivere a classi(a)sns.it _entro le 9:00 di Lunedì __ __7 marzo 2022_ *Abstract. *Stability and robustness analysis of linear continuous and discrete dynamical systems is a vast and active interdisciplinary research area. Although the word stability is used in many different contexts, in this talk it is meant to indicate the broad spectrum of issues that arise in the analytical and numerical study of dynamical and control systems, especially in relation to qualitative information of the system under study. These issues result from the need to develop stable and reliable systems robustly preserving essential qualitative properties. The analysis of these features is often based on eigenvalue optimization or on pseudospectral measures of a certain matrix A, where the structure of A (e.g., sparse, Hamiltonian, nonnegative, Toeplitz, etc.) plays an important role. The main goal of this talk is to show how these structured distances can be approximated. The proposed method is a two-level iterative algorithm, where in an inner iteration a gradient flow drives perturbations to the original matrix of a fixed size into a minimum of a functional that depends on eigenvalues (and possibly eigenvectors), and in an outer iteration the perturbation size is optimized such that the functional reaches some target value (for example O). An interesting point - related to certain low-rank properties of extremizers - lies in the possibility of considering gradient flows on low-rank manifolds. The talk is mainly inspired by joint works with Christian Lubich (Tuebingen). Meeting link: https://hausdorff.dm.unipi.it/b/leo-xik-xu4

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Seminar announcement
by Fabio Durastante 23 Feb '22

23 Feb '22

Speaker: Leonardo Robol Affiliation: Università di Pisa Venue: Aula Magna, Dipartimento di Matematica Time: Thursday, 24/02/2022, 11:00 Title: Sampling the eigenvalues of random matrices Random matrices (that is, matrices with random variables as entries) appear in several areas of mathematics and physics. Often, one is interested in sampling their eigenvalues (which are again random variables); we briefly recap the main tools that can be used to sample the joint eigenvalue distribution for Hermitian matrices from the Gaussian unitary Ensemble (GUE) with O(n^2) flops, and discuss how similar ideas can be used to efficiently perform the analogous operation for Haar-distributed orthogonal or unitary matrices. The latter operation requires to find a suitable compact canonical form for upper Hessenberg unitary matrices obtained by reducing random orthogonal/unitary matrices, and then efficiently operate on this compact representation to compute the eigenvalues. Meeting link: https://hausdorff.dm.unipi.it/b/leo-xik-xu4

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Annuncio di seminario 10/02
by Fabio Durastante 10 Feb '22

10 Feb '22

Speaker: Milo Viviani Affiliation: Scuola Normale Superiore Venue: Aula Magna, Dipartimento di Matematica Time: Thursday, 10/02/2022, 11:00 Title:Solving cubic matrix equations arising in conservative dynamics Spatial semi-discretization of conservative PDEs can be described as flows in suitable matrix spaces, which in turn leads to the need to solve polynomial matrix equations, a classical and important topic both in theoretical and in applied mathematics. Solving these equations numerically is challenging due to the presence of several conservation laws which our finite models incorporate and which must be retained while integrating the equations of motion. In the last thirty years, the theory of geometric integration has provided a variety of techniques to tackle this problem. These numerical methods require to solve both direct and inverse problems in matrix spaces. We present two algorithms to solve a cubic matrix equation arising in the geometric integration of isospectral flows. This type of ODEs includes finite models of ideal hydrodynamics, plasma dynamics, and spin particles, which we use as test problems for our algorithms. Meeting link:https://hausdorff.dm.unipi.it/b/leo-xik-xu4

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Corso di dottorato: Low-rank approximation techniques for matrices
by Leonardo Robol 14 Jan '22

14 Jan '22

Buongiorno, segnalo il corso di dottorato "Low rank approximation techniques for matrices and tensors", che sarà tenuto a partire del prossimo 7 febbraio da Stefano Massei presso il dipartimento di Informatica. È un corso concentrato in 3 settimane; in quel periodo Stefano sarà ospite del Dipartimento di Matematica. Riporto qui sotto ulteriori informazioni, che in ogni caso trovate sulla pagina web del corso di dottorato in Computer Science [1]. ===================================================================== Low rank approximation techniques for matrices and tensors Lecturers: Stefano Massei (TU/e, The Netherlands) Period: From February 7th to February 26th, 2022 Schedule: 6 hours the first and second week, 4 hours the third week To attend the course send an email to s.massei(a)tue.nl Matrices and tensors are the heart of many scientific and industrial computational problems. In this course we will survey numerical methods that leverage low-rank structures in matrices and tensors in order to reduce the consumption of computational resources while maintaining a satisfactory accuracy. The course will touch the following topics: * Low-rank tensor formats: Tucker and tensor train * Deterministic and randomized algorithms for low-rank approximation of matrices and tensors * Trace and diagonal estimation of black box matrices [1] https://dottorato.di.unipi.it/phd-programme/teaching/phd-courses-a-y-2021-2…

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Spostamento mailing list
by Leonardo Robol 09 Dec '21

09 Dec '21

Cari tutti, stiamo provando a spostare la mailing list NumPi su un nuovo server, per permettere anche ai non-unipi di iscriversi in autonomia. Per ora è un tentativo sperimentale, ma se leggete questo messaggio è già un buon segno; l'indirizzo a cui scrivere cambia, e diventa numpi(a)lists.dm.unipi.it. Se vediamo che tutto va bene, faremo puntare il vecchio alias (numpi(a)di.unipi.it) a questo nuovo indirizzo, in modo che il passaggio sia il più indolore possibile. Le vecchie iscrizioni e preferenze sono state importate; segnalatemi pure ogni problema che doveste notare! A presto, -- Leonardo Robol.

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