隐式三阶同伴三元组与可变步长用于大规模ODE约束最优控制的梯度基解法
Implicit Third-Order Peer Triplets with Variable Stepsizes for Gradient-Based Solutions in Large-Scale ODE-Constrained Optimal Control
AI总结：
本文提出了一种隐式三阶同伴三元组方法，用于高效求解大规模ODE约束最优控制问题，通过可变步长和高阶一致性提升计算精度。
摘要：
本文关注可变步长隐式Peer两步方法的理论、构造和应用，这些方法在可变步长下具有超收敛性，即在梯度基解算法中求解ODE约束最优控制问题时，能保持其在均匀步长下获得的经典阶数。通过近似伴随变量最有效地计算目标函数的梯度。通过满足足够多的伴随阶条件的时间积分方法，可以在中等计算努力下实现高精度。在本文中，我们增强了之前在[J. Comput. Appl. Math. 460, 2025]中开发的可变隐式两步Peer三元组，以适应具有变化时间尺度的大规模动态系统，而不牺牲效率。Peer方法的一个关键优势是使用多个相同高阶阶的阶段，这可以防止阶数降低——这是半离散化PDE问题中边界控制时常见的问题。构造并测试了两种三阶方法，具有良好的稳定性特性、小误差常数以及通过等分布置全局误差的网格适应性，用于一维边界热控制问题和前列腺癌治疗中的细胞毒性疗法最优控制。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
