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

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 discretisation is related to reverse accumulation in automatic differentiation. Similar adjoint systems arise in optimal control and other areas, including classical Mechanics. Adjoint 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 understand 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. The article presents in a new, unified way a number of results now scattered 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.

Learning Adaptive Regularization for Image Labeling Using Geometric Assignment

“…Symplectic Partitioned Runge-Kutta Methods. In this section, we recall basic concepts of numerical integration from [HLW06,SS16] in order to prepare Section 2.3. Symplectic schemes are typically applied to Hamiltonian systems in order to conserve certain quantites, often with a physical background.…”

“…Proof of Theorem 2.7. For the proof of this theorem we follow the suggested outline of [SS16]: State the Lagrangian of the nonlinear problem (2.21) and apply the following lemma, which is a slightly different version of Lemma 3.5 in [SS16]. Then, the Euclidean gradient of Φ with respect to p at p 0 is given by…”

“…Conditions and ways to resolve this dilemma have been studied in the optimal control literature [Hag00,Ros06]. See also the recent survey [SS16] and references therein. We provide the corresponding background in Sections 2.2 and 2.3 including detailed proof of Theorem 2.7 that is merely outlined in [SS16].…”

“…See also the recent survey [SS16] and references therein. We provide the corresponding background in Sections 2.2 and 2.3 including detailed proof of Theorem 2.7 that is merely outlined in [SS16]. The application to the linear assignment flow (Section 3) requires considerable work, taking into account that the state equation (1.1b) derives from the full nonlinear geometric assignment flow (Section 4).…”

Line Integral Solution of Differential Problems

“…Moreover, since the Butcher matrix in (49) satisfies (10), then all quadratic Casimirs turn out to be conserved as well. The conservation of all quadratic invariants, in turn, is an important property as it has been observed in [58]. Let us then sketch the choice of the parameter α to gain energy conservation (we refer to [53] for full details).…”