Quantifying uncertainty in the numerical integration of evolution equations based on Bayesian isotonic regression

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2025年6月7日

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Yuto Miyatake,
Kaoru Irie,
Takeru Matsuda

摘要

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 $ $log$ - $\chi ^2_1$ $χ_{1}^{2}$ distribution, enabling the development of an efficient Gibbs sampling algorithm for the corresponding posterior.