  We consider parameter estimation of ordinary differential equation (ODE)
models from noisy observations. For this problem, one conventional approach is
to fit numerical solutions (e.g., Euler, Runge--Kutta) of ODEs to data.
However, such a method does not account for the discretization error in
numerical solutions and has limited estimation accuracy. In this study, we
develop an estimation method that quantifies the discretization error based on
data. The key idea is to model the discretization error as random variables and
estimate their variance simultaneously with the ODE parameter. The proposed
method has the form of iteratively reweighted least squares, where the
discretization error variance is updated with the isotonic regression algorithm
and the ODE parameter is updated by solving a weighted least squares problem
using the adjoint system. Experimental results demonstrate that the proposed
method attains robust estimation with at least comparable accuracy to the
conventional method by successfully quantifying the reliability of the
numerical solutions.
