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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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Annuncio di seminario, 6/12 (Bruno Iannazzo)
by Leonardo Robol 02 Dec '21

02 Dec '21

Speaker: Bruno Iannazzo Affiliation: Università degli studi di Perugia Venue: Aula Magna, Dipartimento di Matematica Time: Lunedì, 6/12/2021, 15:00 Title: New geometries on positive-definite matrices and their relation to the power means We introduce a new family of geometries on the cone of positive definite matrices obtained from the Hessian of the power potential and provide explicit expressions for related quantities such as geodesics and distances. This generalizes in some sense the geometry obtained from the Hessian of the logarithmic potential, whose geodesic has been understood as the weighted geometric mean of two matrices. Indeed, the new geometries provide a definition of matrix power mean alternative to the existing ones. We discuss some properties of the proposed power mean and relate it with the geometric mean. Finally, we discuss on how these geometries can be used to accelerate the convergence of optimization algorithms for the matrix geometric mean of several variables. Meeting link: https://hausdorff.dm.unipi.it/b/leo-xik-xu4

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[NOMADS] Seminar - Matthew Colbrook - Dec 1 at 17:00
by Francesco Tudisco 29 Nov '21

29 Nov '21

Dear all, We are starting this week a new series of NOMADS seminars on Numerical methods, Matrix analysis and Data Science. This week's seminar will take place on *Wednesday December 1 at 17:00* (CET) in the *GSSI Main Lecture Hall* (the "red" conference hall by the entrance of the lecture rooms building). The speaker is Matthew J Colbrook from University of Cambridge and ENS Paris, with a talk on the computation of semigroups of fractional PDEs. Title and abstract are below. In person participation is encouraged, but remote attendance will also be possible via the link: https://us02web.zoom.us/j/89668910844?pwd=TmVwQUFoNklUajAzMlEyMzB6ZVZzUT09 Further info about past and future meetings are available at the webpage: https://num-gssi.github.io/seminar/ Hope to see you all on Wednesday! And, please feel free to distribute this announcement as you see fit. Francesco and Nicola ----------- Title: Computing semigroups and time-fractional PDEs with error control Abstract: We develop an algorithm that computes strongly continuous semigroups on infinite-dimensional Hilbert spaces with explicit error control. Given a generator $A$, a time $t > 0$, an arbitrary initial vector $u_0$ and an error tolerance $\epsilon > 0$, the algorithm computes $\exp(tA)u_0$ with error bounded by $\epsilon$. The (parallelisable) algorithm is based on a combination of a regularized functional calculus, suitable contour quadrature rules and the adaptive computation of resolvents in infinite dimensions. As a particular case, we deal with semigroups on $L^2(R^d)$ that are generated by partial differential operators with polynomially bounded coefficients of locally bounded total variation. For analytic semigroups, we provide a quadrature rule whose error decreases like $\exp(−cN/ log(N))$ for $N$ quadrature points, that remains stable as $N \to \infty$, and which is also suitable for infinite-dimensional operators. Finally, we extend the method to time-fractional PDEs (where it avoids singularities as $t \to 0$ and large memory consumption). Numerical examples are given, including: Schrödinger and wave equations on the aperiodic Ammann–Beenker tiling and fractional beam equations arising in the modelling of small-amplitude vibration of viscoelastic materials. The spectral analysis (which is always needed for contour methods) is considerably simplified due to an infinite-dimensional “solve-then-discretise” approach. — Francesco Tudisco Assistant Professor School of Mathematics GSSI Gran Sasso Science Institute Web: https://ftudisco.gitlab.io -- You received this message because you are subscribed to the Google Groups "nomads-list" group. To unsubscribe from this group and stop receiving emails from it, send an email to nomads-list+unsubscribe(a)gssi.it. To view this discussion on the web visit https://groups.google.com/a/gssi.it/d/msgid/nomads-list/CAO2_FWmWATOjsjz5v5…. For more options, visit https://groups.google.com/a/gssi.it/d/optout.

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Annuncio di seminario, 19/11 (Mariarosa Mazza)
by Leonardo Robol 18 Nov '21

18 Nov '21

Speaker: Mariarosa Mazza Affiliation: Università dell'Insubria Venue: Aula Magna, Dipartimento di Matematica Time: Venerdì, 19/11/2021, 10:00 Title: Caputo and Riemann-Liouville fractional derivatives: a matrix comparison Fractional derivatives are a mathematical tool that received much attention in the last decades because of their non-local behavior which has been demonstrated to be useful when modeling anomalous diffusion phenomena appearing, e.g., in imaging or electrophysiology. Two of the most famous definitions of fractional derivatives are the Riemann- Liouville and the Caputo ones. The two formulations are related by a well-known formula that expresses a Riemann-Liouville derivative as a Caputo one plus a term that depends on the function and its derivatives at the boundary. As a consequence of this relation, Riemann-Liouville and Caputo derivatives coincide only for sufficiently smooth functions that satisfy homogeneous conditions at the boundary. Aiming at uncovering how much this discrepancy reflects on their discretized counterparts, we focus on the relation between Riemann-Liouville and Caputo derivatives once approximated by a collocation method based on B-splines, i.e., high order finite elements with maximum regularity. We show that when the fractional order α ranges in (1,2) their difference in terms of matrices corresponds to a rank-1 correction whose spectral norm increases with the mesh-size n and is o(√n). On one hand, this implies that the spectral distribution for the B-spline collocation matrices corresponding to the Riemann-Liouville and Caputo derivatives coincide; on the other hand, the presence of the rank-1 correction makes the Caputo matrices worse conditioned for α tending to 1 due to a larger maximum singular value. Some linear algebra consequences of all this knowledge are discussed, and a selection of numerical experiments that validate our findings is provided together with a numerical study of the approximation behavior of B-spline collocation. Meeting link: https://hausdorff.dm.unipi.it/b/leo-xik-xu4

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