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

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Show simple item record Sanz Serna, Jesús María 2021-04-30T10:18:23Z 2021-04-30T10:18:23Z 2016
dc.identifier.bibliographicCitation Sanz-Serna, J. M. (2016). Symplectic Runge--Kutta Schemes for Adjoint Equations, Automatic Differentiation, Optimal Control, and More. SIAM Review, 58(1), 3–33.
dc.identifier.issn 0036-1445
dc.description.abstract The 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.
dc.format.extent 31
dc.language.iso eng
dc.publisher Elsevier
dc.rights © 2016, Society for Industrial and Applied Mathematics.
dc.subject.other Runge-Kutta methods
dc.subject.other Partitioned Runge-Kutta methods
dc.subject.other Symplectic integration
dc.subject.other Hamiltonian systems
dc.subject.other Variational equations
dc.subject.other Adjoint equations
dc.subject.other Computation of sensitivities
dc.subject.other Lagrange multipliers
dc.subject.other Automatic differentiation
dc.subject.other Optimal control
dc.subject.other Lagrangian mechanics
dc.subject.other Reflected and transposed Runge-Kutta schemes
dc.subject.other Differential-algebraic problems
dc.subject.other Constrained controls
dc.subject.other Order conditions
dc.subject.other Discrete mechanics
dc.subject.other Integrators
dc.subject.other Systems
dc.title Symplectic Runge-Kutta Schemes for Adjoint Equations, Automatic Differentiation, Optimal Control, and More
dc.type article
dc.subject.eciencia Matemáticas
dc.rights.accessRights openAccess
dc.relation.projectID Gobierno de España. MTM2010-18246-C03-01
dc.type.version acceptedVersion
dc.identifier.publicationfirstpage 3
dc.identifier.publicationissue 1
dc.identifier.publicationlastpage 33
dc.identifier.publicationtitle SIAM Review
dc.identifier.publicationvolume 58
dc.identifier.uxxi AR/0000017707
dc.contributor.funder Ministerio de Ciencia e Innovación (España)
dc.affiliation.dpto UC3M. Departamento de Matemáticas
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