Lagrangian and Hamiltonian Taylor variational integrators

Schmitt, Jérémy; Shingel, Tatiana; Leok, Melvin

doi:10.1007/s10543-017-0690-9

Cited by 9 publications

(12 citation statements)

References 19 publications

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“…In many cases, the Type I and Type II/III approaches will produce equivalent integrators. This equivalence has been established in [32] for Taylor variational integrators provided the Lagrangian is hyperregular, and in [25] for generalized Galerkin variational integrators constructed using the same choices of basis functions and numerical quadrature formula provided the Hamiltonian is hyperregular. However, Hamiltonian and Lagrangian variational integrators are not always equivalent.…”

Section: Lagrangian and Hamiltonian Mechanicsmentioning

confidence: 92%

“…Examples of variational integrators include Galerkin variational integrators [25; 27], Prolongation-Collocation variational integrators [24], and Taylor variational integrators [32]. In many cases, the Type I and Type II/III approaches will produce equivalent integrators.…”

Section: Lagrangian and Hamiltonian Mechanicsmentioning

confidence: 99%

“…We now describe the construction of Taylor variational integrators as introduced in [32] as we will use them in our numerical experiments. A discrete approximate Lagrangian or Hamiltonian is constructed by approximating the flow map and the trajectory associated with the boundary values using a Taylor method, and approximating the integral by a quadrature rule.…”

Section: Lagrangian and Hamiltonian Mechanicsmentioning

confidence: 99%

“…The following error analysis result concerning the order of accuracy of LTVIs and HTVIs were derived in [32] and [9]: Theorem 2.1. If the Lagrangian L is Lipschitz continuous in both variables, then L d (q 0 , q 1 ) approximates L E d (q 0 , q 1 ) with at least order of accuracy min (ρ + 1, s).…”

Section: Lagrangian and Hamiltonian Mechanicsmentioning

confidence: 99%

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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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Section: Lagrangian and Hamiltonian Mechanicsmentioning

confidence: 92%

Section: Lagrangian and Hamiltonian Mechanicsmentioning

confidence: 99%

Section: Lagrangian and Hamiltonian Mechanicsmentioning

confidence: 99%

Section: Lagrangian and Hamiltonian Mechanicsmentioning

confidence: 99%

See 2 more Smart Citations

Time-adaptive Lagrangian Variational Integrators for Accelerated Optimization on Manifolds

Valentin¹,

Leok²

2022

Preprint

Self Cite

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“…Remark 3.3. These discrete constrained Hamilton's equations can be thought of as the discrete Hamilton's equations coming from the augmented discrete Hamiltonians [29] for Taylor variational integrators provided the Lagrangian is hyperregular, and in [20] for generalized Galerkin variational integrators constructed using the same choices of basis functions and numerical quadrature formula provided the Hamiltonian is hyperregular.…”

Section: Discrete Constrained Variational Hamiltonian Mechanicsmentioning

confidence: 99%

Accelerated Optimization on Riemannian Manifolds via Discrete Constrained Variational Integrators

Duruisseaux,

Leok

2021

Preprint

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A variational formulation for accelerated optimization on normed spaces was introduced in Wibisono et al. [34], and later generalized to Riemannian manifolds in Duruisseaux and Leok [8]. This variational framework was exploited on normed spaces in Duruisseaux et al. [10] using time-adaptive geometric integrators to design efficient explicit algorithms for symplectic accelerated optimization, and it was observed that geometric discretizations which respect the time-rescaling invariance and symplecticity of the Lagrangian and Hamiltonian flows were substantially less prone to stability issues, and were therefore more robust, reliable, and computationally efficient. As such, it is natural to develop time-adaptive Hamiltonian variational integrators for accelerated optimization on Riemannian manifolds. In this paper, we consider the case of Riemannian manifolds embedded in a Euclidean space that can be characterized as the level set of a submersion. We will explore how holonomic constraints can be incorporated in discrete variational integrators to constrain the numerical discretization of the Riemannian Hamiltonian system to the Riemannian manifold, and we will test the performance of the resulting algorithms by solving eigenvalue and Procrustes problems formulated as optimization problems on the unit sphere and Stiefel manifold.

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Accelerated Optimization on Riemannian Manifolds via Discrete Constrained Variational Integrators

Valentin

Leok

2022

J Nonlinear Sci

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A variational formulation for accelerated optimization on normed vector spaces was recently introduced in Wibisono et al. (PNAS 113:E7351–E7358, 2016), and later generalized to the Riemannian manifold setting in Duruisseaux and Leok (SJMDS, 2022a). This variational framework was exploited on normed vector spaces in Duruisseaux et al. (SJSC 43:A2949–A2980, 2021) using time-adaptive geometric integrators to design efficient explicit algorithms for symplectic accelerated optimization, and it was observed that geometric discretizations which respect the time-rescaling invariance and symplecticity of the Lagrangian and Hamiltonian flows were substantially less prone to stability issues, and were therefore more robust, reliable, and computationally efficient. As such, it is natural to develop time-adaptive Hamiltonian variational integrators for accelerated optimization on Riemannian manifolds. In this paper, we consider the case of Riemannian manifolds embedded in a Euclidean space that can be characterized as the level set of a submersion. We will explore how holonomic constraints can be incorporated in discrete variational integrators to constrain the numerical discretization of the Riemannian Hamiltonian system to the Riemannian manifold, and we will test the performance of the resulting algorithms by solving eigenvalue and Procrustes problems formulated as optimization problems on the unit sphere and Stiefel manifold.

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Lagrangian and Hamiltonian Taylor variational integrators

Cited by 9 publications

References 19 publications

Time-adaptive Lagrangian Variational Integrators for Accelerated Optimization on Manifolds

Time-adaptive Lagrangian Variational Integrators for Accelerated Optimization on Manifolds

Accelerated Optimization on Riemannian Manifolds via Discrete Constrained Variational Integrators

Accelerated Optimization on Riemannian Manifolds via Discrete Constrained Variational Integrators

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