Implicit Peer Triplets in Gradient-Based Solution Algorithms for ODE Constrained Optimal Control
Submission Status: | under review |
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- It is common practice to apply gradient-based optimization algorithms to
numerically solve large-scale ODE constrained optimal control problems. Gradients
of the objective function are most efficiently computed by approximate
adjoint variables. High accuracy with moderate computing time can be achieved
by such time integration methods that satisfy a sufficiently large number of adjoint
order conditions and supply gradients with higher orders of consistency. In
this paper, we upgrade our former implicit two-step Peer triplets constructed in
[Algorithms, 15:310, 2022] to meet those new requirements. Since Peer methods
use several stages of the same high stage order, a decisive advantage is their lack
of order reduction as for semi-discretized PDE problems with boundary control.
Additional order conditions for the control and certain positivity requirements
now intensify the demands on the Peer triplet. We discuss the construction of
4-stage methods with order pairs (4,3) and (3,3) in detail and provide three
Peer triplets of practical interest. We prove convergence for s-stage methods,
for instance, order s for the state variables even if the adjoint method and the
control satisfy the conditions for order s-1, only. Numerical tests show the
expected order of convergence for the new Peer triplets.