Geometric integration of non-autonomous linear Hamiltonian problems

Marthinsen, Håkon; Owren, Brynjulf

doi:10.1007/s10444-015-9425-0

Cited by 7 publications

(6 citation statements)

References 28 publications

(54 reference statements)

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“…Ideally, we want to design optimisation methods that preserve these rates, i.e., are "rate-matching", and are also numerically stable. As we will see, such geometric integrators can be constructed by leveraging the shadow Hamiltonian property of symplectic methods on higher-dimensional conservative Hamiltonian systems [9] (see also [98,99]). This holds not only on R 2d but on general settings, namely on arbitrary smooth manifolds [9,10].…”

Section: Rate-matching Integrators For Smooth Optimisationmentioning

confidence: 99%

“…The reason for doing this procedure, called symplectification, is purely theoretical: since the theory of symplectic integrators only accounts for conservative systems, we can now extend this theory to dissipative settings by applying a symplectic integrator to (13) and then fixing the relevant coordinates (17) in the resulting method. Geometrically, this corresponds to integrating the time flow exactly [9,98]. In [9] such a procedure was defined under the name of presymplectic integrators, and these connections hold not only for the specific example above but also for general non-conservative Hamiltonian systems.…”

Section: Rate-matching Integrators For Smooth Optimisationmentioning

confidence: 99%

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Geometric Methods for Sampling, Optimisation, Inference and Adaptive Agents

Barp,

Da Costa,

França

et al. 2022

Preprint

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In this chapter, we identify fundamental geometric structures that underlie the problems of sampling, optimisation, inference and adaptive decision-making. Based on this identification, we derive algorithms that exploit these geometric structures to solve these problems efficiently. We show that a wide range of geometric theories emerge naturally in these fields, ranging from measure-preserving processes, information divergences, Poisson geometry, and geometric integration. Specifically, we explain how (i) leveraging the symplectic geometry of Hamiltonian systems enable us to construct (accelerated) sampling and optimisation methods, (ii) the theory of Hilbertian subspaces and Stein operators provides a general methodology to obtain robust estimators, (iii) preserving the information geometry of decision-making yields adaptive agents that perform active inference. Throughout, we emphasise the rich connections between these fields; e.g., inference draws on sampling and optimisation, and adaptive decision-making assesses decisions by inferring their counterfactual consequences. Our exposition provides a conceptual overview of underlying ideas, rather than a technical discussion, which can be found in the references herein.

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Section: Rate-matching Integrators For Smooth Optimisationmentioning

confidence: 99%

Section: Rate-matching Integrators For Smooth Optimisationmentioning

confidence: 99%

Geometric Methods for Sampling, Optimisation, Inference and Adaptive Agents

Barp,

Da Costa,

França

et al. 2022

Preprint

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“…From the outset, it might not be so clear which geometric features such a non-autonomous Hamiltonian system has. There seem to be at least two ways to understand this problem [97]. Let us assume that u = (z, p) ∈ T * R M ≡ R M ⊕ R M with 'positions' z and 'momenta' p forming the phase space.…”

Section: Hamiltonian Vector Fieldsmentioning

confidence: 99%

Structure-preserving deep learning

Celledoni¹,

Ehrhardt

Etmann³

et al. 2021

Eur. J. Appl. Math

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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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“…Note that these equations are still satisfied along the flow of X H . A presymplectic integrator is, according to Definition 4.1, any symplectic integrator Ψ h for which q 0 , p 0 are integrated exactly (see also [33,34]). Let us denote y ≡ (q µ , p µ ), z ≡ (q i , p i ) and y • ≡ y(s = 0).…”

Section: "Rate-matching" Geometric Integratorsmentioning

confidence: 99%

Optimization on manifolds: A symplectic approach

França¹,

Barp²,

Jordan³

2021

Preprint

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There has been great interest in using tools from dynamical systems and numerical analysis of differential equations to understand and construct new optimization methods. In particular, recently a new paradigm has emerged that applies ideas from mechanics and geometric integration to obtain accelerated optimization methods on Euclidean spaces. This has important consequences given that accelerated methods are the workhorses behind many machine learning applications. In this paper we build upon these advances and propose a framework for dissipative and constrained Hamiltonian systems that is suitable for solving optimization problems on arbitrary smooth manifolds. Importantly, this allows us to leverage the well-established theory of symplectic integration to derive "rate-matching" dissipative integrators. This brings a new perspective to optimization on manifolds whereby convergence guarantees follow by construction from classical arguments in symplectic geometry and backward error analysis. Moreover, we construct two dissipative generalizations of leapfrog that are straightforward to implement: one for Lie groups and homogeneous spaces, that relies on the tractable geodesic flow or a retraction thereof, and the other for constrained submanifolds that is based on a dissipative generalization of the famous RATTLE integrator.

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Geometric integration of non-autonomous linear Hamiltonian problems

Cited by 7 publications

References 28 publications

Geometric Methods for Sampling, Optimisation, Inference and Adaptive Agents

Geometric Methods for Sampling, Optimisation, Inference and Adaptive Agents

Structure-preserving deep learning

Optimization on manifolds: A symplectic approach

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