NumPI October 2020

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1 participants
4 discussions

Seminar on 03/11 (Angelo Casulli)
by Leonardo Robol 28 Oct '20

28 Oct '20

Speaker: Angelo Casulli Affiliation: SNS Time: Tuesday, 03/11/2020, 11:00 Meeting link: https://hausdorff.dm.unipi.it/b/leo-xik-xu4 Title: Rank-structured QR for Chebyshev rootfinding The computation of the roots of polynomials expressed in the Chebyshev basis has a lot of applications, for instance, it is useful in the computation of real roots of smooth functions. We present an algorithm for the rootfinding of Chebyshev polynomials based on an improvement of the QR iteration presented in [Eidelman, Y., Gemignani, L., and Gohberg, I., Numer. Algorithms , 47.3 (2008): pp. 253-273]. We introduce an aggressive early deflation strategy, and we show that the rank-structure allows to parallelize the algorithm avoiding data dependencies which would be present in the unstructured QR. The method exploits the particular structure of the colleague linearization to achieve quadratic complexity and linear storage requirements. The (unbalanced) QR iteration used for Chebyshev rootfinding does not guarantee backward stability on the polynomial coefficients, unless the vector of coefficients satisfy ||p|| ~ 1, an hypothesis which is almost never verified for polynomials approximating smooth functions. Even though the presented method is mathematically equivalent to the latter algorithm, we show that exploiting the rank structure allows to guarantee a small backward error on the polynomial, up to an explicitly computable amplification factor ɣ(p), which depends on the polynomial under consideration. We show that this parameter is almost always of moderate size, making the method accurate on several numerical tests, in contrast with what happens in the unstructured unbalanced QR. We also discuss the connection between the size of this amplification factor and the existence of a good balancing. This provides some insight on why the accuracy of our method is often very close to the balanced QR iteration. https://www.dm.unipi.it/webnew/it/seminari/rank-structured-qr-chebyshev-roo…

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Reminder: Seminar on 20/10 (Simon Telen)
by Leonardo Robol 19 Oct '20

19 Oct '20

Dear all, this is just a reminder of tomorrow's seminar by Simon Telen. Best, -- Leonardo. Speaker: Simon Telen Affiliation: MPI Leipzig Time: Tuesday, 20/10/2020, 11:00 Meeting link: https://hausdorff.dm.unipi.it/b/leo-xik-xu4 Title: Solving Polynomial Systems via Truncated Normal Forms Systems of polynomial equations arise from many problems in applied mathematics. Solving such systems is considered a challenging computational problem. An important class of numerical solving methods converts the problem into a system of coupled eigenvalue problems. In order to perform this conversion one has to choose a representation for an algebra naturally associated to the equations. Standard choices (coming from Gröbner bases, border bases, resultants ...) may show some very bad, numerical behaviour, even for generic systems. Truncated normal form methods generalise all the aforementioned approaches and are designed to avoid these numerical issues. I will introduce the theory and give many examples. This is joint work with Bernard Mourrain and Marc Van Barel. https://www.dm.unipi.it/webnew/it/seminari/solving-polynomial-systems-trunc…

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Seminar on 20/10 (Simon Telen)
by Leonardo Robol 13 Oct '20

13 Oct '20

Speaker: Simon Telen Affiliation: MPI Leipzig Time: Tuesday, 20/10/2020, 11:00 Meeting link: https://hausdorff.dm.unipi.it/b/leo-xik-xu4 Title: Solving Polynomial Systems via Truncated Normal Forms Systems of polynomial equations arise from many problems in applied mathematics. Solving such systems is considered a challenging computational problem. An important class of numerical solving methods converts the problem into a system of coupled eigenvalue problems. In order to perform this conversion one has to choose a representation for an algebra naturally associated to the equations. Standard choices (coming from Gröbner bases, border bases, resultants ...) may show some very bad, numerical behaviour, even for generic systems. Truncated normal form methods generalise all the aforementioned approaches and are designed to avoid these numerical issues. I will introduce the theory and give many examples. This is joint work with Bernard Mourrain and Marc Van Barel. https://www.dm.unipi.it/webnew/it/seminari/solving-polynomial-systems-trunc…

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Seminar on 06/10 (Milo Viviani)
by Leonardo Robol 06 Oct '20

06 Oct '20

Dear all, below you find the first announcement for the NumPI seminars. The seminar will be next week on Tuesday, and will be online using the conference platform available at the math department. You find the link below (which will stay the same for all conferences, assuming it works well). When entering the conference you will be asked if you want to only listen or to join with microphone; I suggest you choose the latter: you will be muted anyway after joining, but will be able to ask questions at the end if you wish. Best wishes, -- Leonardo. Speaker: Milo Viviani Affiliation: Scuola Normale Superiore, Pisa Time: Tuesday, 06/10/2020, 11:00 Meeting link: https://hausdorff.dm.unipi.it/b/leo-xik-xu4 Title: Symplectic methods for isospectral flows and 2D ideal hydrodynamics The numerical solution of non-canonical Hamiltonian systems is an active and still growing field of research. At the present time, the biggest challenges concern the realization of structure preserving algorithms for differential equations on infinite dimensional manifolds. Several classical PDEs can indeed be set in this framework, and in particular the 2D hydrodynamical Euler equations. In this talk, I will present some results I have obtained during my PhD studies. In particular, I will show how to derive a new class of numerical schemes for Hamiltonian and non-Hamiltonian isospectral flows, in order to solve the 2D hydrodynamical Euler equations. The use of a conservative scheme has revealed new insights in the 2D ideal hydrodynamics, showing clear connections between geometric mechanics, statistical mechanics and integrability theory. https://www.dm.unipi.it/webnew/it/seminari/symplectic-methods-isospectral-f…

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NumPI October 2020