Speaker: Cecilia Pagliantini
Affiliation: TU Eindhoven
Venue: Sala Seminari, Dipartimento di Matematica
Time: Monday, 25/10/2021, 15:00
Title: Structure-preserving dynamical model order reduction of
parametric Hamiltonian systems
In real-time and many-query simulations of parametric differential
equations, computational methods need to face high computational costs
to provide sufficiently accurate and stable numerical solutions. To
address this issue, model order reduction techniques aim at
constructing low-complexity high-fidelity surrogate models that allow
rapid and accurate solutions under parameter variation. In this talk,
we will consider reduced basis methods (RBM) for the model order
reduction of parametric Hamiltonian dynamical systems describing
nondissipative phenomena. The development of RBM for Hamiltonian
systems is challenged by two main factors: (i) failing to preserve the
geometric structure encoding the physical properties of the dynamics,
such as invariants of motion or symmetries, might lead to instabilities
and unphysical behaviors of the resulting approximate solutions; (ii)
the \emph{local} low-rank nature of transport-dominated and
nondissipative phenomena demands large reduced spaces to achieve
sufficiently accurate approximations. We will discuss how to address
these aspects via a structure-preserving nonlinear reduced basis
approach based on dynamical low-rank approximation. The gist of the
proposed method is to evolve low-dimensional surrogate models on a
phase space that adapts in time while being endowed with the geometric
structure of the full model.
Meeting link: https://hausdorff.dm.unipi.it/b/leo-xik-xu4
Cari tutti,
a breve ricominceremo con i tradizionali seminari NumPi, questa volta
in forma "ibrida".
Per tutti quelli che vorranno ci sarà la possibilità di seguire in
presenza in Aula Magna del Dipartimento di Matematica (compilando
questa form: https://forms.gle/AaMgajhKhroXdHqY6)
Per gli altri, organizzeremo lo streaming tramite il solito link (
https://hausdorff.dm.unipi.it/b/leo-xik-xu4)
Qui sotto trovate l'annuncio del primo seminario; a breve vi farò avere
un Doodle per provare a scegliere un giorno che torni il più comodo
possibile a tutti, da utilizzare per quelli successivi.
Trovate tutte le informazioni sulla pagina dei seminari del sito NumPi
(https://numpi.dm.unipi.it/seminars)
A presto, -- Leonardo.
Speaker: Igor Simunec
Affiliation: Scuola Normale Superiore, Pisa
Time: Friday, 15/10/2021, 16:30
Title: Computation of generalized matrix functions with rational Krylov
methods
Generalized matrix functions [3] are an extension of the notion of
standard matrix functions to rectangular matrices, defined using the
singular value decomposition instead of an eigenvalue decomposition. In
this talk, we consider the computation of the action of a generalized
matrix function on a vector and we present a class of algorithms based
on rational Krylov methods [2]. These algorithms incorporate as a
special case previous methods based on the Golub-Kahan
bidiagonalization [1]. By exploiting the quasiseparable structure of
the projected matrices, we show that the basis vectors can be updated
using a short recurrence, which can be seen as a generalization to the
rational case of the Golub-Kahan bidiagonalization. We also prove error
bounds that relate the error of these methods to uniform rational
approximation on an interval containing the singular values of the
matrix. The effectiveness of the algorithms and the accuracy of the
bounds is illustrated with numerical experiments.
This is joint work with Angelo A. Casulli (Scuola Normale Superiore).
[1] F. Arrigo, M. Benzi, and C. Fenu, Computation of generalized matrix
functions, SIAM J. Matrix Anal. Appl. 37 (2016), no. 3, 836–860.
[2] A. A. Casulli, I. Simunec, Computation of generalized matrix
functions with rational Krylov methods, arXiv:2107.12074 (2021).
[3] J. B. Hawkins and A. Ben-Israel, On generalized matrix functions,
Linear and Multilinear Algebra 1 (1973), no. 2, 163–171.
Meeting link: https://hausdorff.dm.unipi.it/b/leo-xik-xu4
Cari tutti,
a partire dal prossimo seminario (dopo quello di Igor venerdì)
cercheremo di scegliere un orario compatibile con gli impegni di tutti,
e mantere una cadenza indicativamente bisettimanale.
Ho creato un Doodle [1] per la settimana 25/10 - 29/10, dove chiederei
(a chi è interessato a seguire i seminari in presenza e/o online) di
indicare le preferenze di orario.
Sebbene il Doodle sia per quella settimana specifica, è da intendersi
come le preferenze in base ai vostri impegni settimanali.
Vi chiederei di farci avere una risposta entro venerdì, così che
possiamo organizzare il seminario seguente basandoci sulle risposte.
Grazie!
A presto, -- Leonardo (e Fabio).
[1] https://doodle.com/poll/dtvyu6uakv3exdp8
