April 2022 - NumPI - Mailing list — Dipartimento di Matematica, UniPI

[Indico] Stefano Massei, Improved parallel-in-time integration via low-rank updates and interpolation, 05/05/2022, 11:00 Europe/Rome
by leonardo.robol＠unipi.it 27 Apr '22

27 Apr '22

Title: Improved parallel-in-time integration via low-rank updates and interpolation, Speaker(s): Stefano Massei, Università di Pisa, Date and time: 5 May 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/89 Abstract -------- This work is concerned with linear matrix equations that arise from the space-time discretization of time-dependent linear partial differential equations (PDEs). Such matrix equations have been considered, for example, in the context of parallel-in-time integration leading to a class of algorithms called ParaDiag. We develop and analyze two novel approaches for the numerical solution of such equations. Our first approach is based on the observation that the modification of these equations performed by ParaDiag in order to solve them in parallel has low rank. Building upon previous work on low-rank updates of matrix equations, this allows us to make use of tensorized Krylov subspace methods to account for the modification. Our second approach is based on interpolating the solution of the matrix equation from the solutions of several modifications. Both approaches avoid the use of iterative refinement needed by ParaDiag and related space-time approaches in order to attain good accuracy. In turn, our new approaches have the potential to outperform, sometimes significantly, existing approaches. This potential is demonstrated for several different types of PDEs. -- Indico :: Email Notifier https://events.dm.unipi.it/e/89

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Ph.D. Course: An introduction to fractional calculus: fundamental ideas and numerics
by Fabio Durastante 27 Apr '22

27 Apr '22

Dear all, Monday 2 May we begin with the Ph.D. course "An introduction to fractional calculus: fundamental ideas and numerics" from 2.00 pm to 4.00 pm in the Aula Seminari. Find the latest information at: https://fdurastante.github.io/courses/introfracalculus.html#about The plan is to schedule the other lessons with those present that day. If there are people that wish to attend, but have difficulties being in person, please let me know at fabio.durastante(a)unipi.it and I'll try activating a streaming of the lecture. For any other question, feel free to write an E-mail. Best, Fabio Durastante

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[Indico] Raf Vandebril,
by leonardo.robol＠unipi.it 12 Apr '22

12 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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[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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