250 research outputs found
Pseudospectra of waveform relaxation operators
Abstract--The performance of the waveform relaxation method for solving systems of ODEs depends largely on the choices that are made for splitting, size of time window, and preconditioning. Although it is known that superlinear convergence is obtained on finite time windows, the convergence may be slow in the first few iterations. We propose the use of pseudcepectra to analyze the convergence ratio of the first few iterations when waveform relaxation is applied to linear systems of ODEs. Through pseudcepectral radii, we can examine the effect of preconditioning and overlapping on the rate of convergence. We may also use this to estimate a suitable size of the time window. Numerical experiments performed on a system of ODEs arising from the discretization of an advection-diffusion equation confirm the validity of the obtained estimates. (~) 1998 Elsevier Science Ltd. All rights reserved
Using discrete Darboux polynomials to detect and determine preserved measures and integrals of rational maps
In this Letter we propose a systematic approach for detecting and calculating
preserved measures and integrals of a rational map. The approach is based on
the use of cofactors and Discrete Darboux Polynomials and relies on the use of
symbolic algebra tools. Given sufficient computing power, all rational
preserved integrals can be found.
We show, in two examples, how to use this method to detect and determine
preserved measures and integrals of the considered rational maps.Comment: 8 pages, 1 Figur
Deep learning as optimal control problems
We briefly review recent work where deep learning neural networks have been interpreted as discretisations of an optimal control problem subject to an ordinary differential equation constraint. We report here new preliminary experiments with implicit symplectic Runge-Kutta methods. In this paper, we discuss ongoing and future research in this area
Task A and B Final Report
TASK A: TRANSMISSION OF HIGH FREQUENCY RADIO WAVES VIA
THE ARCTIC IONOSPHERE
The experimental data collected from June, 1949, through October,
1955, under "Experiment Aurora" are summarized in tables and diagrams, and the results discussed.
The monthly percentage of signal in-time is tabulated for all frequencies
and paths» and depicted in diagrams which allow a comparison
of the values for East-West and South-North propagation at each frequency.
The average monthly percentage of signal in-time for the duration
of the 6-year experiment is tabulated for each frequency and path.
The seasonal variation in signal in-tim e over short and long paths is
shown in diagrams. The relationship found between ionospheric absorption,
as measured with a vertical incidence sounder, and signal outtime
is summarized.
The average diurnal variation in the hourly median signal strength
during the different seasons of the year 1954-55 is given for all frequencies
on both short and long paths in the East-West as well as the
South-North direction. The diurnal variation in signal strength on the
4 me short paths and the 12 me long paths is compared for a year of
high solar activity (1949-50) and a year of low solar activity (1954-55).
The discussion of the data reveals that a statistically significant
difference in signal in-time for the East-West and South-North paths
exists only for the 12 me short paths. The larger percentage of signal
in-time found in the East-West direction is believed to be due to a preferential
orientation of sporadic ionization along parallels to the auroral
zone.
A study of the critical frequencies observed for the E and F -layers
shows that the difference in daytime variation of median signal strength
between the years 1949-50 and 1954-55 may be explained in terms of the
normal changes in F -layer ionization and D -layer absorption in course
of a sunspot cycle. The results indicate that in Alaska there will generally
be F2 propagation during daytime of 4 me signals over 350 km
paths throughout the solar cycle. Regular daytime F2 propagation of
12 me signals over 1100 km paths may be expected in years of reasonably
high solar activity only.
TASK B: PULSE TECHNIQUES. BACK-SCATTER AT 12 MC
A 12 me radar has been constructed and operated using A -scope
and PPI displays. Experimental results obtained during several months
of continuous operation are reviewed and discussed. Both direct backscatter
and ground back-scatter echoes, as well as possible combinations
of these modes, have been observed.
The echoes are classified in two groups according to their fading
rates, those fading rapidly being associated with aurora. Figures show
the diurnal, range and range-azimuth distribution of the observed auroral
echoes as well as some special types of echoes recorded.
The direct back-scatter echoes at 12 me associated with aurora show
characteristics consistent with those observed at YHF when allowance
is made for the frequency difference. At 12 me the fading rate is proportionally
less than at higher frequencies; and aspect sensitivity, although
weaker, still exists. The diurnal variation is similar to that
found at VHF. Several types of echoes not observed at VHF are mentioned.
TASK B: VISUAL OBSERVATIONS OF THE AURORA
Analysis is made of the visual auroral data obtained at five stations
in Alaska during the observing period of 1954-55. Graphs giving the
percentage occurrence of aurora at each station as a function of latitude
and time of day are presented. Graphs showing the variation of auroral
occurrence with geomagnetic latitude as a function of magnetic K index
are also given. The conclusions drawn from the 1954-55 data are substantially
the same as those based on the 1953-54 data discussed in an
earlier report.List of Figures – List of Tables – Section I Purposes – Section II Abstract – Section III Publications, Reports and Conferences – Section IV Factual Data : 1 Task A. Transmission of High Frequency of Radio Waves Via the Arctic Ionosphere ; 2 Task B. Pulse Techniques Back-Scatter at 12 Mc. ; 3 Task B Visual Observations of the Aurora – Section V Conclusions – Section VI Recommendations – Section VII PersonnelYe
Deep learning as optimal control problems: Models and numerical methods
We consider recent work of Haber and Ruthotto 2017 and Chang et al. 2018,
where deep learning neural networks have been interpreted as discretisations of
an optimal control problem subject to an ordinary differential equation
constraint. We review the first order conditions for optimality, and the
conditions ensuring optimality after discretisation. This leads to a class of
algorithms for solving the discrete optimal control problem which guarantee
that the corresponding discrete necessary conditions for optimality are
fulfilled. The differential equation setting lends itself to learning
additional parameters such as the time discretisation. We explore this
extension alongside natural constraints (e.g. time steps lie in a simplex). We
compare these deep learning algorithms numerically in terms of induced flow and
generalisation ability
Structure-preserving deep learning
Over the past few years, deep learning has risen to the foreground as a topic
of massive interest, mainly as a result of successes obtained in solving
large-scale image processing tasks. There are multiple challenging mathematical
problems involved in applying deep learning: most deep learning methods require
the solution of hard optimisation problems, and a good understanding of the
tradeoff between computational effort, amount of data and model complexity is
required to successfully design a deep learning approach for a given problem. A
large amount of progress made in deep learning has been based on heuristic
explorations, but there is a growing effort to mathematically understand the
structure in existing deep learning methods and to systematically design new
deep learning methods to preserve certain types of structure in deep learning.
In this article, we review a number of these directions: some deep neural
networks can be understood as discretisations of dynamical systems, neural
networks can be designed to have desirable properties such as invertibility or
group equivariance, and new algorithmic frameworks based on conformal
Hamiltonian systems and Riemannian manifolds to solve the optimisation problems
have been proposed. We conclude our review of each of these topics by
discussing some open problems that we consider to be interesting directions for
future research
Backward error analysis and the substitution law for Lie group integrators
Butcher series are combinatorial devices used in the study of numerical
methods for differential equations evolving on vector spaces. More precisely,
they are formal series developments of differential operators indexed over
rooted trees, and can be used to represent a large class of numerical methods.
The theory of backward error analysis for differential equations has a
particularly nice description when applied to methods represented by Butcher
series. For the study of differential equations evolving on more general
manifolds, a generalization of Butcher series has been introduced, called
Lie--Butcher series. This paper presents the theory of backward error analysis
for methods based on Lie--Butcher series.Comment: Minor corrections and additions. Final versio
On post-Lie algebras, Lie--Butcher series and moving frames
Pre-Lie (or Vinberg) algebras arise from flat and torsion-free connections on
differential manifolds. They have been studied extensively in recent years,
both from algebraic operadic points of view and through numerous applications
in numerical analysis, control theory, stochastic differential equations and
renormalization. Butcher series are formal power series founded on pre-Lie
algebras, used in numerical analysis to study geometric properties of flows on
euclidean spaces. Motivated by the analysis of flows on manifolds and
homogeneous spaces, we investigate algebras arising from flat connections with
constant torsion, leading to the definition of post-Lie algebras, a
generalization of pre-Lie algebras. Whereas pre-Lie algebras are intimately
associated with euclidean geometry, post-Lie algebras occur naturally in the
differential geometry of homogeneous spaces, and are also closely related to
Cartan's method of moving frames. Lie--Butcher series combine Butcher series
with Lie series and are used to analyze flows on manifolds. In this paper we
show that Lie--Butcher series are founded on post-Lie algebras. The functorial
relations between post-Lie algebras and their enveloping algebras, called
D-algebras, are explored. Furthermore, we develop new formulas for computations
in free post-Lie algebras and D-algebras, based on recursions in a magma, and
we show that Lie--Butcher series are related to invariants of curves described
by moving frames.Comment: added discussion of post-Lie algebroid
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