Given the set of discrete solution points or nodes, called the skeleton, generated by an ODE solver, we study the problem of fitting a curve passing through the nodes in the skeleton minimizing a norm of the residual vector of the ODE. We reformulate this interpolation problem as a multi-stage optimal control problem and, for the minimization of two different norms, we apply the associated maximum principle to obtain the necessary conditions of optimality. We solve the problem analytically for the Dahlquist test problem and a variant of the leaky bucket problem, in terms of the given skeleton. We also consider the Van der Pol equation, for which we obtain interpolating curves with minimal residual norms by numerically solving a direct discretization of the problem through optimization software. With the skeletons obtained by various ODE solvers of MATLAB, we make comparisons between the residuals obtained by our approach and those obtained by the MATLAB function deval.

翻译：给定由常微分方程求解器生成的离散解点集（称为骨架），我们研究如何拟合一条穿过骨架节点的曲线，以最小化常微分方程残差向量的范数。我们将该插值问题重构为多阶段最优控制问题，并针对两种不同范数的最小化，应用相应的极大值原理推导最优性必要条件。对于Dahlquist测试问题及漏桶问题的变体，我们依据给定骨架解析求解了该问题。此外，针对Van der Pol方程，我们通过优化软件数值求解问题的直接离散化形式，获得了具有最小残差范数的插值曲线。利用MATLAB多种常微分方程求解器生成的骨架，我们将所提方法获得的残差与MATLAB函数deval所得残差进行了对比分析。