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Symplectic Runge-Kutta Schemes for Adjoint Equations, Automatic Differentiation, Optimal Control, and More

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dc.affiliation.dptoUC3M. Departamento de Matemáticases
dc.contributor.authorSanz Serna, Jesús María
dc.contributor.funderMinisterio de Ciencia e Innovación (España)es
dc.date.accessioned2021-04-30T10:18:23Z
dc.date.available2021-04-30T10:18:23Z
dc.date.issued2016
dc.description.abstractThe study of the sensitivity of the solution of a system of differential equations with respect to changes in the initial conditions leads to the introduction of an adjoint system, whose discretization is related to reverse accumulation in automatic differentiation. Similar adjoint systems arise in optimal control and other areas, including classical mechanics. Ad-joint systems are introduced in such a way that they exactly preserve a relevant quadratic invariant (more precisely, an inner product). Symplectic Runge-Kutta and partitioned Runge-Kutta methods are defined through the exact conservation of a differential geometric structure, but may be characterized by the fact that they preserve exactly quadratic invariants of the system being integrated. Therefore, the symplecticness (or lack of symplecticness) of a Runge-Kutta or partitioned Runge-Kutta integrator should be relevant to understanding its performance when applied to the computation of sensitivities, to optimal control problems, and in other applications requiring the use of adjoint systems. This paper examines the links between symplectic integration and those applications and presents in a new, unified way a number of results currently scattered among or implicit in the literature. In particular, we show how some common procedures, such as the direct method in optimal control theory and the computation of sensitivities via reverse accumulation, imply, probably unbeknownst to the user, "hidden" integrations with symplectic partitioned Runge-Kutta schemes.en
dc.format.extent31
dc.identifier.bibliographicCitationSanz-Serna, J. M. (2016). Symplectic Runge--Kutta Schemes for Adjoint Equations, Automatic Differentiation, Optimal Control, and More. SIAM Review, 58(1), 3–33.en
dc.identifier.doihttps://doi.org/10.1137/151002769
dc.identifier.issn0036-1445
dc.identifier.publicationfirstpage3
dc.identifier.publicationissue1
dc.identifier.publicationlastpage33
dc.identifier.publicationtitleSIAM Reviewen
dc.identifier.publicationvolume58
dc.identifier.urihttps://hdl.handle.net/10016/32518
dc.identifier.uxxiAR/0000017707
dc.language.isoeng
dc.publisherElsevieren
dc.relation.projectIDGobierno de España. MTM2010-18246-C03-01es
dc.rights© 2016, Society for Industrial and Applied Mathematics.en
dc.rights.accessRightsopen access
dc.subject.ecienciaMatemáticases
dc.subject.otherRunge-Kutta methodsen
dc.subject.otherPartitioned Runge-Kutta methodsen
dc.subject.otherSymplectic integrationen
dc.subject.otherHamiltonian systemsen
dc.subject.otherVariational equationsen
dc.subject.otherAdjoint equationsen
dc.subject.otherComputation of sensitivitiesen
dc.subject.otherLagrange multipliersen
dc.subject.otherAutomatic differentiationen
dc.subject.otherOptimal controlen
dc.subject.otherLagrangian mechanicsen
dc.subject.otherReflected and transposed Runge-Kutta schemesen
dc.subject.otherDifferential-algebraic problemsen
dc.subject.otherConstrained controlsen
dc.subject.otherOrder conditionsen
dc.subject.otherDiscrete mechanicsen
dc.subject.otherIntegratorsen
dc.subject.otherSystemsen
dc.titleSymplectic Runge-Kutta Schemes for Adjoint Equations, Automatic Differentiation, Optimal Control, and Moreen
dc.typeresearch article*
dc.type.hasVersionAM*
dspace.entity.typePublication
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