SIAM/ASA Journal on Uncertainty Quantification, Volume 9, Issue 1, Page 302-331, January 2021.
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.

中文翻译：

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具有离散化误差量化的常微分方程模型的估计

SIAM / ASA不确定性量化杂志，第9卷，第1期，第302-331页，2021年1月
我们考虑从嘈杂的观察中估计常微分方程（ODE）模型的参数。对于此问题，一种常规方法是将ODE的数值解（例如Euler，Runge-Kutta）拟合到数据中。但是，这种方法不能解决数值解中的离散误差，并且估计精度有限。在这项研究中，我们开发了一种基于数据量化离散化误差的估计方法。关键思想是将离散化误差建模为随机变量，并与ODE参数同时估计其方差。所提出的方法具有迭代加权最小二乘的形式，其中，使用等渗回归算法更新离散化误差方差，并通过使用伴随系统求解加权最小二乘问题来更新ODE参数。