Adaptive Hamiltonian Variational Integrators and Applications to Symplectic Accelerated Optimization

Valentin, Duruisseaux,; Schmitt, Jérémy; Leok, Melvin

doi:10.1137/20m1383835

Cited by 20 publications

(43 citation statements)

References 35 publications

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“…For future work, it would be interesting to explore the generalization of the proposed method to symmetric spaces, by applying the generalized polar decomposition [15]. This may be of particular interest in the construction of accelerated optimization algorithms on symmetric spaces, following the use of time-adaptive variational integrators for symplectic optimization [6] based on the discretization of Bregman Lagrangian and Hamiltonian flows [16,5]. Examples of symmetric spaces include the space of Lorentzian metrics, the space of symmetric positive-definite matrices, and the Grassmannian.…”

Section: Discussionmentioning

confidence: 99%

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High-order symplectic Lie group methods on <inline-formula><tex-math id="M1">$ SO(n) $</tex-math></inline-formula> using the polar decomposition

Shen¹,

Tran²,

Leok³

2022

JCD

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<p style='text-indent:20px;'>A variational integrator of arbitrarily high-order on the special orthogonal group <inline-formula><tex-math id="M2">\begin{document}$ SO(n) $\end{document}</tex-math></inline-formula> is constructed using the polar decomposition and the constrained Galerkin method. It has the advantage of avoiding the second order derivative of the exponential map that arises in traditional Lie group variational methods. In addition, a reduced Lie–Poisson integrator is constructed and the resulting algorithms can naturally be implemented by fixed-point iteration. The proposed methods are validated by numerical simulations on <inline-formula><tex-math id="M3">\begin{document}$ SO(3) $\end{document}</tex-math></inline-formula> which demonstrate that they are comparable to variational Runge–Kutta–Munthe-Kaas methods in terms of computational efficiency. However, the methods we have proposed preserve the Lie group structure much more accurately and and exhibit better near energy preservation.</p>

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Section: Discussionmentioning

confidence: 99%

“…Recall that dP A (B) involves solving the Lyapunov equation (6). We aim to compute dP * A , so we define the following two maps,…”

Section: 22mentioning

confidence: 99%

High-order symplectic Lie group methods on <inline-formula><tex-math id="M1">$ SO(n) $</tex-math></inline-formula> using the polar decomposition

Shen¹,

Tran²,

Leok³

2022

JCD

Self Cite

View full text Add to dashboard Cite

show abstract

“…For future work, it would be interesting to explore the generalization of the proposed method to symmetric spaces, by applying the generalized polar decomposition [15]. This may be of particular interest in the construction of accelerated optimization algorithms on symmetric spaces, following the use of time-adaptive variational integrators for symplectic optimization [6] based on the discretization of Bregman Lagrangian and Hamiltonian flows [5,16]. Examples of symmetric spaces include the space of Lorentzian metrics, the space of symmetric positive-definite matrices, and the Grassmannian.…”

Section: Discussionmentioning

confidence: 99%

High-order symplectic Lie group methods on $SO(n)$ using the polar decomposition

Shen¹,

Tran²,

Leok³

2022

Preprint

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A variational integrator of arbitrarily high-order on the special orthogonal group SO(n) is constructed using the polar decomposition and the constrained Galerkin method. It has the advantage of avoiding the second-order derivative of the exponential map that arises in traditional Lie group variational methods. In addition, a reduced Lie-Poisson integrator is constructed and the resulting algorithms can naturally be implemented by fixed-point iteration. The proposed methods are validated by numerical simulations on SO(3) which demonstrate that they are comparable to variational Runge-Kutta-Munthe-Kaas methods in terms of computational efficiency. However, the methods we have proposed preserve the Lie group structure much more accurately and and exhibit better near energy preservation.

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“…This variational framework was later extended to the Riemannian manifolds setting in [8]. This variational framework and the time-rescaling property of the Bregman family were exploited on vector spaces [9] and Riemannian manifolds [6; 7], by using time-adaptive geometric integrators to design efficient, explicit algorithms for symplectic accelerated optimization. It was observed that a careful use of adaptivity and symplecticity could result in a significant gain in computational efficiency, by simulating higher-order Bregman dynamics using the computationally efficient lower-order Bregman integrators applied to the time-rescaled dynamics.…”

Section: Introductionmentioning

confidence: 99%

“…There has been a great effort to circumvent this problem, and there have been many successes, including methods based on the Poincaré transformation [12; 41]: a Poincaré transformed Hamiltonian in extended phase space is constructed which allows the use of variable time-steps in symplectic integrators without losing the nice conservation properties associated to these integrators. In [9], the Poincaré transformation was incorporated in the Hamiltonian variational integrator framework which provides a systematic method for constructing symplectic integrators of arbitrarily high-order based on the discretization of Hamilton's principle [14; 27], or equivalently, by the approximation of the generating function of the symplectic flow map. The Poincaré transformation was at the heart of the construction of time-adaptive geometric integrators for Bregman Hamiltonian systems which resulted in efficient, explicit algorithms for accelerated optimization in [9].…”

Section: Introductionmentioning

confidence: 99%

Time-adaptive Lagrangian Variational Integrators for Accelerated Optimization on Manifolds

Valentin¹,

Leok²

2022

Preprint

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A variational framework for accelerated optimization was recently introduced on normed vector spaces and Riemannian manifolds in Wibisono et al. [38] and Duruisseaux and Leok [8]. It was observed that a careful combination of time-adaptivity and symplecticity in the numerical integration can result in a significant gain in computational efficiency. It is however well known that symplectic integrators lose their near energy preservation properties when variable time-steps are used. The most common approach to circumvent this problem involves the Poincaré transformation on the Hamiltonian side, and was used in Duruisseaux et al. [9] to construct efficient explicit algorithms for symplectic accelerated optimization. However, the current formulations of Hamiltonian variational integrators do not make intrinsic sense on more general spaces such as Riemannian manifolds and Lie groups. In contrast, Lagrangian variational integrators are well-defined on manifolds, so we develop here a framework for time-adaptivity in Lagrangian variational integrators and use the resulting geometric integrators to solve optimization problems on normed vector spaces and Lie groups.

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Adaptive Hamiltonian Variational Integrators and Applications to Symplectic Accelerated Optimization

Cited by 20 publications

References 35 publications

High-order symplectic Lie group methods on <inline-formula><tex-math id="M1">$ SO(n) $</tex-math></inline-formula> using the polar decomposition

High-order symplectic Lie group methods on <inline-formula><tex-math id="M1">$ SO(n) $</tex-math></inline-formula> using the polar decomposition

High-order symplectic Lie group methods on $SO(n)$ using the polar decomposition

Time-adaptive Lagrangian Variational Integrators for Accelerated Optimization on Manifolds

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