Minimizing Residuals in ODE Integration Using Optimal Control

TL;DR

This paper introduces a novel method for fitting curves through ODE solution points by formulating the problem as an optimal control task, minimizing residuals to improve interpolation accuracy.

Contribution

It reformulates residual minimization in ODE interpolation as an optimal control problem and provides analytical and numerical solutions for various test cases.

Findings

Analytical solutions for Dahlquist and leaky bucket problems.

Numerical residual minimization for Van der Pol equation.

Comparison shows reduced residuals with the proposed method.

Abstract

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.