Implicit Peer Triplets in Gradient-Based Solution Algorithms for ODE Constrained Optimal Control
(2024)
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.
This paper is concerned with the construction and convergence analysis
of novel implicit Peer triplets of two-step nature with four stages for nonlinear
ODE constrained optimal control problems. We combine the property of superconvergence
of some standard Peer method for inner grid points with carefully
designed starting and end methods to achieve order four for the state variables
and order three for the adjoint variables in a first-discretize-then-optimize approach
together with A-stability. The notion triplets emphasizes that these
three different Peer methods have to satisfy additional matching conditions.
Four such Peer triplets of practical interest are constructed. Also as a benchmark
method, the well-known backward differentiation formula BDF4, which is
only A(73.35)-stable, is extended to a special Peer triplet to supply an adjoint
consistent method of higher order and BDF type with equidistant nodes. Within
the class of Peer triplets, we found a diagonally implicit A(84)-stable method
with nodes symmetric in [0,1] to a common center that performs equally well.
Numerical tests with three well established optimal control problems confirm
the theoretical findings also concerning A-stability.
It is well known that in the first-discretize-then-optimize approach in the control of ordinary differential equations the adjoint method may converge under additional order conditions only.
For Peer two-step methods we derive such adjoint order conditions and pay special attention to the boundary steps.
For $s$-stage methods, we prove convergence of order s for the state variables if the adjoint method satisfies the conditions for order s-1, at least.
We remove some bottlenecks at the boundaries encountered in an earlier paper of the first author et al.
[J. Comput. Appl. Math., 262:73--86, 2014]
and discuss the construction of 3-stage methods for the order pair (3,2) in detail including some matrix background for the combined forward and adjoint order conditions. The impact of nodes having equal differences is
highlighted. It turns out that the most attractive methods are related to BDF. Three 3-stage methods are constructed which show the expected orders in numerical tests.
Method-of-lines discretizations are demanding test problems for stiff inte-
gration methods. However, for PDE problems with known analytic solution
the presence of space discretization errors or the need to use codes to compute
reference solutions may limit the validity of numerical test results. To over-
come these drawbacks we present in this short note a simple test problem with
boundary control, a situation where one-step methods may suffer from order
reduction. We derive exact formulas for the solution of an optimal boundary
control problem governed by a one-dimensional discrete heat equation and an
objective function that measures the distance of the final state from the target
and the control costs. This analytical setting is used to compare the numeri-
cally observed convergence orders for selected implicit Runge-Kutta and Peer
two-step methods of classical order four which are suitable for optimal control
problems.