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…
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…
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…
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…
