MaGIC 2002, Ustaoset: Talks and abstracts
February 4-8
- Borko Minchev:
"The order of exponential polar-type splitting for some stiff
problems"
(45 min)
Abstract:
Splitting method for the approximation of the exponential is used in
many applications. It is known that for symmetric splitting schemes
in case of stiff and nonstiff parts of ODE one may observe an order
reduction of the splitting method for sufficiently large time
steps. Similar result can be observed if we use exponential splitting
based on the generalized polar decomposition (polar-type splitting).
- Elena Celledoni:
"The double-pendulum equation: modelling and numerical methods"
(45 min)
- Bjarte Hagland:
TBA
(30-45 min)
- Stein Krogstad:
"On the computation of Lyapunov exponents for Hamiltonian and
reversible systems"
(45-60 min)
- Hans Munthe-Kaas:
"On equivariance and symmetry of numerical integrators"
(60 min)
(Report on 'work that should have been in progress')
Abstract:
In our papers we often bring forward the (obvious) point that Lie
group integrators 'stay on the correct manifold'. Other authors,
e.g. Hairer, make a point of explaining that staying on a manifold is
not very interesting unless one also have additional properties of
the integrators. He is pointing out things like time symmetry.
An other property that was implicit in early work (but not explicitly
stated!) was the property of equivariance, the numerical integrator
should respect the motions of the underlying group, e.g.
Rotate * numerical integration = numerical integration *
rotation.
This is a property that is shared by all our early integrators, such
as e.g. the CG methods, and exponential methods. For the "Kenth class of
algorithms", discussed by Kenth in his BiT article (general mappings
liealg->liegroup + Lie group action), equivariance is achieved if we
put some conditions on allowed maps from algebra to group. In general
methods based on local coordinates or retractions, the equivariance
is not necessarily preserved. We will in this talk discuss
equivariance and its implications for quality of
numerics. Furthermore, we will arrive at some resuts that are
surprising, at least to me. Here is one:
*** Retraction method
+ equivariance => method is an equivariant Kenth class algorithm
****
The opposite implication does, however, not hold. So if we
require equivariance, we have
retraction methods < Kenth
class.
Still it is useful to consider retraction methods as a class of
methods in their own right, since the inclusion above is not
practical from a computational point of view.
- Jitse Niesen:
"On the global error committed by
numerical integrators on nonautonomous oscillator"
(60 min)
Abstract:
In this talk, we show how to obtain a priori estimates for the
global error of a numerical integrator. Our approach is based on
perturbation theory and modified equations. The estimates are correct up
to a term of order h^{2p}, where h denotes the (constant) step size
and p is the order of the method.
We apply the resulting estimates to two classes of nonautonomous
oscillators. The first class consists of linear oscillators which are
amenable to the Liouville-Green approximation (this work is mainly due
to Arieh Iserles). The second class is formed by nonlinear oscillators
satisfying the Emden-Fowler equation y'' + t^v y^n
= 0. We note some
substantial differences between both classes during the computation.
Numerical results confirm the correctness of our estimates.
- Brynjulf Owren:
"Expansions of Lie-group methods and their algebraic structure"
(60 min)
- Bård Skaflestad:
TBA
(30-45 min)
- Andrzej Suslovic:
TBA
(60 min)
- Antonella Zanna:
"On global bounds for
matrix exponential splittings based on Generalized Polar
Decompositions"
(45 min)
- Krister Ålander:
"An overview of EinSum
symmetries and band tensor notation"
(60 min)
Abstract:
EinSum is a C++ package which understands index notation including the
Einstein summation convention. In this talk we will focus on its
support for tensor symmetries, and we will also discuss
generalizations to "band tensors".