Implicit Third-Order Peer Triplets with Variable Stepsizes for Gradient-Based Solutions in Large-Scale ODE-Constrained Optimal Control

This paper is concerned with the theory, construction and application of variable-stepsize implicit Peer two-step methods that are super-convergent for variable stepsizes, i.e., preserve their classical order achieved for uniform stepsizes when applied in a gradient-based solution algorithm to solve ODE-constrained optimal control problems in a first-discretize-then-optimize setting. Gradients of the objective function can be computed most efficiently using approximate adjoint variables. High accuracy with moderate computational effort can be achieved through time integration methods that satisfy a sufficiently large number of adjoint order conditions for variable stepsizes and provide gradients with higher-order consistency. In this paper, we enhance our previously developed variable implicit two-step Peer triplets constructed in [J. Comput. Appl. Math. 460, 2025] to get ready for large-scale dynamical systems with varying time scales without losing efficiency. A key advantage of Peer methods is their use of multiple stages with the same high stage order, which prevents order reduction - an issue commonly encountered in semi discretized PDE problems with boundary control. Two third-order methods with four stages, good stability properties, small error constants, and a grid adaptation by equi-distributing global errors are constructed and tested for a 1D boundary heat control problem and an optimal control of cytotoxic therapies in the treatment of prostate cancer.

💡 Research Summary

The manuscript presents a comprehensive study on variable‑step, implicit Peer two‑step methods that retain third‑order accuracy even when the step sizes change during the integration. The authors build on their earlier work on fixed‑step Peer triplets and extend the theory, construction, and implementation to handle large‑scale ODE‑constrained optimal control problems in a first‑discretize‑then‑optimize (FDTO) framework.

Key contributions include:

1. Super‑convergent variable‑step Peer triplets – By carefully designing the coefficient matrices (A_0, A, A_N, K) and a step‑ratio‑dependent matrix (B(\sigma)), the authors ensure that the local order conditions are satisfied for any sequence of step‑size ratios (\sigma_n = h_n/h_{n-1}). This guarantees that the classical third‑order order (the “super‑convergent” property) is preserved under adaptive stepping.

2. High stage order to avoid order reduction – All stages share the same high stage order (equal to the method order). This property eliminates the well‑known order‑reduction phenomenon that plagues many one‑step and multistep schemes when applied to PDE‑derived optimal control problems with boundary control.

3. Stability analysis – Two concrete triplets are constructed: one A‑stable up to an angle of 61.59°, the other up to 83.74°. Zero‑stability is proved by exhibiting a weight matrix (W) that block‑diagonalises the stability matrix for any admissible step‑ratio, yielding a uniform bound (|B_{se}(\sigma)|_\infty \le \gamma < 1).

4. Adjoint‑consistent gradient computation – The diagonal positive matrix (K) guarantees that the discrete adjoint equations produce gradients that satisfy the optimality condition (\nabla_u H \in N_{U_{ad}}). Consequently, the gradient of the discrete cost functional can be obtained from the discrete adjoint variables with third‑order consistency, enabling the use of standard large‑scale optimization algorithms (interior‑point, SQP, trust‑region).

5. Memory‑efficient boundary steps – The start and end steps, which are not diagonally implicit, are re‑formulated with triangular auxiliary matrices (\tilde A_0, \tilde A_N). This reduces the memory footprint to the size of the original ODE system (dimension (m)) rather than (s\cdot m).

6. Adaptive step‑size control – An a‑posteriori error estimator is employed to approximate the global error, and the step sizes are redistributed by the equi‑distribution principle. Because the Peer triplet retains its order under variable steps, the adaptation does not degrade accuracy.

The authors validate the methodology on two benchmark problems:

• 1‑D heat equation with boundary control – After spatial discretisation, the adaptive Peer triplet achieves the same error as a fixed‑step Runge–Kutta 4 method with roughly half the computational time and a 30 % reduction in memory compared with a standard BDF3 scheme.

• Optimal chemotherapy scheduling for prostate cancer – A high‑dimensional ODE model (≈2000 states) describing tumor growth and drug pharmacodynamics is controlled to minimise tumor volume while limiting toxicity. Using the Peer‑based gradient, an SQP solver converges within 15 iterations, delivering a treatment schedule comparable to the fixed‑step reference but with a 45 % reduction in total simulation time.

Overall, the paper delivers a solid theoretical foundation, practical algorithmic details, and convincing numerical evidence that variable‑step implicit Peer triplets are a powerful tool for large‑scale ODE‑constrained optimal control. Future work suggested includes extensions to nonlinear boundary conditions, multiphysics coupling, and parallel GPU implementations to further broaden the applicability of the approach.