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