This paper presents a new Bayesian framework for quantifying discretization errors in numerical solutions of ordinary differential equations. By modelling the errors as random variables, we impose a monotonicity constraint on the variances, referred to as discretization error variances. The key to our approach is the use of a shrinkage prior for the variances coupled with variable transformations. This methodology extends existing Bayesian isotonic regression techniques to tackle the challenge of estimating the variances of a normal distribution. An additional key feature is the use of a Gaussian mixture model for the $\log$-$\chi^2_1$ distribution, enabling the development of an efficient Gibbs sampling algorithm for the corresponding posterior.

翻译：本文提出了一种新的贝叶斯框架，用于量化常微分方程数值解的离散化误差。通过将误差建模为随机变量，我们对离散化误差方差施加单调性约束。该方法的核心在于结合变量变换对方差使用收缩先验。此方法将现有贝叶斯保序回归技术扩展至估计正态分布方差的挑战。另一个关键特征是采用高斯混合模型逼近$\log$-$\chi^2_1$分布，从而能够为相应后验开发高效的吉布斯采样算法。