Browsing by Author "McLachlan RI"

Now showing 1 - 5 of 5
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    Backward error analysis for conjugate symplectic methods
    (American Institute of Mathematical Sciences (AIMS), 2023-12-08) McLachlan RI; Offen C
    The numerical solution of an ordinary differential equation can be interpreted as the exact solution of a nearby modified equation. Investigating the behaviour of numerical solutions by analysing the modified equation is known as backward error analysis. If the original and modified equation share structural properties, then the exact and approximate solution share geometric features such as the existence of conserved quantities. Conjugate symplectic methods preserve a modified symplectic form and a modified Hamiltonian when applied to a Hamiltonian system. We show how a blended version of variational and symplectic techniques can be used to compute modified symplectic and Hamiltonian structures. In contrast to other approaches, our backward error analysis method does not rely on an ansatz but computes the structures systematically, provided that a variational formulation of the method is known. The technique is illustrated on the example of symmetric linear multistep methods with matrix coefficients.
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    Managing Aotearoa New Zealand's greenhouse gas emissions from aviation
    (Informa UK Limited, trading as Taylor & Francis Group, 2024-01-01) Callister P; McLachlan RI; Glavovic B
    Prior to COVID, the global aviation industry was growing rapidly. Growth has now resumed and is predicted to continue for at least the next three decades. Aotearoa New Zealand has particularly high aviation emissions and has been on a very rapid growth path that is incompatible with the Paris Agreement on climate change. Government, intergovernmental, nongovernmental, academic and industry sources have proposed technological innovations to address aviation emissions. These include sustainable aviation fuels, electric and hydrogen powered aircraft, and increases in efficiency. We review these and assess that none of them will lead to a significant reduction in emissions in the short to medium term. In addition, we demonstrate that even very aggressive uptake of new technology results in the New Zealand aviation sector exceeding its share of the carbon budget as determined by the Paris Agreement. Therefore, we examine the fundamental drivers of growth in aviation: the tourism and airport industries, emissions pricing and substitutes, and the distribution of air travel. Governance of this sector is challenging, but it is changing rapidly. We conclude that a national aviation action plan needs to be developed and implemented based on the ‘Avoid/Shift/Improve’ framework in use in other areas of transportation planning.
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    Structure-preserving deep learning
    (Cambridge University Press, 2021-10) Celledoni E; Ehrhardt MJ; Etmann C; McLachlan RI; Owren B; Schonlieb CB; Sherry F
    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 trade-off 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.
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    The aromatic bicomplex for the description of divergence-free aromatic forms and volume-preserving integrators
    (Cambridge University Press, 2023-08-08) Laurent A; McLachlan RI; Munthe-Kaas HZ; Verdier O
    Aromatic B-series were introduced as an extension of standard Butcher-series for the study of volume-preserving integrators. It was proven with their help that the only volume-preserving B-series method is the exact flow of the differential equation. The question was raised whether there exists a volume-preserving integrator that can be expanded as an aromatic B-series. In this work, we introduce a new algebraic tool, called the aromatic bicomplex, similar to the variational bicomplex in variational calculus. We prove the exactness of this bicomplex and use it to describe explicitly the key object in the study of volume-preserving integrators: the aromatic forms of vanishing divergence. The analysis provides us with a handful of new tools to study aromatic B-series, gives insights on the process of integration by parts of trees, and allows to describe explicitly the aromatic B-series of a volume-preserving integrator. In particular, we conclude that an aromatic Runge-Kutta method cannot preserve volume.
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    The bifurcation structure within robust chaos for two-dimensional piecewise-linear maps
    (Elsevier Ltd, 2024-07) Ghosh I; McLachlan RI; Simpson DJW
    We study two-dimensional, two-piece, piecewise-linear maps having two saddle fixed points. Such maps reduce to a four-parameter family and are well known to have a chaotic attractor throughout open regions of parameter space. The purpose of this paper is to determine where and how this attractor undergoes bifurcations. We explore the bifurcation structure numerically by using Eckstein's greatest common divisor algorithm to estimate from sample orbits the number of connected components in the attractor. Where the map is orientation-preserving the numerical results agree with formal results obtained previously through renormalisation. Where the map is orientation-reversing or non-invertible the same renormalisation scheme appears to generate the bifurcation boundaries, but here we need to account for the possibility of some stable low-period solutions. Also the attractor can be destroyed in novel heteroclinic bifurcations (boundary crises) that do not correspond to simple algebraic constraints on the parameters. Overall the results reveal a broadly similar component-doubling bifurcation structure in the orientation-reversing and non-invertible settings, but with some additional complexities.