Symplectic Runge-Kutta schemes for adjoint equations, automatic differentiation, optimal control and more
For researchers in numerical analysis, optimal control, and automatic differentiation, this paper clarifies the hidden symplectic structure in common computational methods, but the results are largely a synthesis of existing knowledge.
This paper unifies results showing that symplectic Runge-Kutta and Partitioned Runge-Kutta methods exactly preserve quadratic first integrals, which is crucial for numerical sensitivities, optimal control, and Lagrangian mechanics. It reveals that widely used procedures like the direct method in optimal control and reverse accumulation for sensitivities implicitly employ symplectic Partitioned Runge-Kutta schemes.
It is well known that symplectic Runge-Kutta and Partitioned Runge-Kutta methods exactly preserve {\em quadratic} first integrals (invariants of motion) of the system being integrated. While this property is often seen as a mere curiosity (it does not hold for arbitrary first integrals), it plays an important role in the computation of numerical sensitivities, optimal control theory and Lagrangian mechanics, as described in this paper, which, together with some new material, presents in a unified way a number of results now scattered or implicit in the literature. Some widely used procedures, such as the direct method in optimal control theory and the computation of sensitivities via reverse accumulation imply "hidden" integrations with symplectic Partitioned Runge-Kutta schemes.